If the universe is infinite, does that mean that everything exists somewhere?

[Chalnoth]

Im going to look that up that sounds really interesting.

Here is his webpage on the subject, if you're interested:
http://space.mit.edu/home/tegmark/toe_frames.html

Includes links to the more in-depth treatments of this idea.

 

[Freeman Dyson]

Can I jump in and say that it is my understanding that infinity is a mathematical concept which doesn't exist in the real world of science, but which may possibly exist (according to mathematical theories).

Right. Infinities exist in math. This was debated throughout history for a while but now math is considered to have infinities. I recently read a good book on infinity.

The Infinite Book: A Short Guide to the Boundless, Timeless and Endless

https://www.amazon.com/dp/0375422277/?tag=pfamazon01-20

It covers nearly everything discussed in this thread. From my understanding, when infinities pop up in the physical world, scientists tend to think of them as a flaw in the theory/measurement. Like how the Big Bang shows infinite properties, it is thought that maybe when a proper theory of quantum gravity is applied to the Big Bang, the infinities will be smoothed down to the finite. Scientists generally don't like infinities in the physical world from what this book says. Infinities don't really exist, they are markers of error in our methods.

 

[ViewsofMars]

It looks like I need to return to this topic. This is common knowledge known by reputable scientists. The Big Bang Theory (the new standard model as mentioned in my previous post) often noted as "Cosmic Evolution" (Professor Chaisson (scientist), Wright Center for Science Education - Cosmic Evolution - http://www.tufts.edu/as/wright_center/cosmic_evolution/docs/splash.html ) is based on Science! Thank goodness for that! :)

I'm going to add onto my previous posting to this topic, since it now appears to me that more is needed in understanding what is *science*. I don't wish to get into a debate with people here. Hopefully, people will be able to read, understand what they are reading, and learn. :)

I'm providing three quotes from TalkOrigins that apply to our discussion, though I recommend a thorough reading of the article for possible future discussions on various topics in physicsforums.com.

[snip]
What is meant by scientific evidence and scientific proof? In truth, science can never establish 'truth' or 'fact' in the sense that a scientific statement can be made that is formally beyond question. All scientific statements and concepts are open to re-evaluation as new data is acquired and novel technologies emerge. Proof, then, is solely the realm of logic and mathematics (and whiskey). That said, we often hear 'proof' mentioned in a scientific context, and there is a sense in which it denotes "strongly supported by scientific means". Even though one may hear 'proof' used like this, it is a careless and inaccurate handling of the term. Consequently, except in reference to mathematics, this is the last time you will read the terms 'proof' or 'prove' in this article.

[snip]

Now, to answer the question "What is the scientific method?" - very simply (and somewhat naively), the scientific method is a program for research which comprises four main steps. In practice these steps follow more of a logical order than a chronological one:

1.Make observations.
2.Form a testable, unifying hypothesis to explain these observations.
3.Deduce predictions from the hypothesis.
4.Search for confirmations of the predictions;
if the predictions are contradicted by empirical observation, go back to step (2).
Because scientists are constantly making new observations and testing via those observations, the four "steps" are actually practiced concurrently. New observations, even if they were not predicted, should be explicable retrospectively by the hypothesis. New information, especially details of some process previously not understood, can impose new limits on the original hypothesis. Therefore, new information, in combination with an old hypothesis, frequently leads to novel predictions that can be tested further.

Examination of the scientific method reveals that science involves much more than naive empiricism. Research that only involves simple observation, repetition, and measurement is not sufficient to count as science. These three techniques are merely part of the process of making observations (#1 in the steps outlined above). Astrologers, wiccans, alchemists, and shamans all observe, repeat, and measure — but they do not practice science. Clearly, what distinguishes science is the way in which observations are interpreted, tested, and used.

[snip]

In contrast, Newton's scientific theory of universal gravitation makes specific predictions about what should be observed. Newton's theory predicts that the force between two masses should be inversely proportional to the square of the distance between them (otherwise known as the "inverse square law"). In principle, we could take measurements which indicated that the force is actually inversely proportional to the cube of the distance. Such an observation would be inconsistent with the predictions of Newton's universal theory of gravitation, and thus this theory is testable. Many anti-evolutionists, such as the "scientific" creationists, are especially fond of Karl Popper and his falsifiability criterion. These cynics are well known for claiming that evolutionary theory is unscientific because it cannot be falsified. In this article, these accusations are met head on. Each of the evidences given for common descent contains a section providing examples of potential falsifications, i.e. examples of observations that would be highly unlikely if the theory is correct.
[snip]

http://www.talkorigins.org/faqs/comdesc/sciproof.html

The following should be helpful. It is from the United States National Academy of Sciences (Advisors to the Nation on Science, Engineering, and Medicine).

Is Evolution a Theory or a Fact?

It is both. But that answer requires looking more deeply at the meanings of the words "theory" and "fact."

In everyday usage, "theory" often refers to a hunch or a speculation. When people say, "I have a theory about why that happened," they are often drawing a conclusion based on fragmentary or inconclusive evidence.

The formal scientific definition of theory is quite different from the everyday meaning of the word. It refers to a comprehensive explanation of some aspect of nature that is supported by a vast body of evidence.

Many scientific theories are so well-established that no new evidence is likely to alter them substantially. For example, no new evidence will demonstrate that the Earth does not orbit around the sun (heliocentric theory), or that living things are not made of cells (cell theory), that matter is not composed of atoms, or that the surface of the Earth is not divided into solid plates that have moved over geological timescales (the theory of plate tectonics). Like these other foundational scientific theories, the theory of evolution is supported by so many observations and confirming experiments that scientists are confident that the basic components of the theory will not be overturned by new evidence. However, like all scientific theories, the theory of evolution is subject to continuing refinement as new areas of science emerge or as new technologies enable observations and experiments that were not possible previously.

One of the most useful properties of scientific theories is that they can be used to make predictions about natural events or phenomena that have not yet been observed. For example, the theory of gravitation predicted the behavior of objects on the moon and other planets long before the activities of spacecraft and astronauts confirmed them. The evolutionary biologists who discovered Tiktaalik predicted that they would find fossils intermediate between fish and limbed terrestrial animals in sediments that were about 375 million years old. Their discovery confirmed the prediction made on the basis of evolutionary theory. In turn, confirmation of a prediction increases confidence in that theory.

In science, a "fact" typically refers to an observation, measurement, or other form of evidence that can be expected to occur the same way under similar circumstances. However, scientists also use the term "fact" to refer to a scientific explanation that has been tested and confirmed so many times that there is no longer a compelling reason to keep testing it or looking for additional examples. In that respect, the past and continuing occurrence of evolution is a scientific fact. Because the evidence supporting it is so strong, scientists no longer question whether biological evolution has occurred and is continuing to occur. Instead, they investigate the mechanisms of evolution, how rapidly evolution can take place, and related questions.
http://www.nationalacademies.org/evolution/TheoryOrFact.html

We should also be mindful of this from NASA.

Tests of Big Bang: Expansion
NASA Official: Dr. Gary F. Hinshaw (scientists)
Page Updated: Tuesday, 10-14-2008

The Big Bang model was a natural outcome of Einstein's General Relativity as applied to a homogeneous universe. However, in 1917, the idea that the universe was expanding was thought to be absurd. So Einstein invented the cosmological constant as a term in his General Relativity theory that allowed for a static universe. In 1929, Edwin Hubble announced that his observations of galaxies outside our own Milky Way showed that they were systematically moving away from us with a speed that was proportional to their distance from us. The more distant the galaxy, the faster it was receding from us. The universe was expanding after all, just as General Relativity originally predicted! Hubble observed that the light from a given galaxy was shifted further toward the red end of the light spectrum the further that galaxy was from our galaxy.

The Hubble Constant

The specific form of Hubble's expansion law is important: the speed of recession is proportional to distance. The expanding raisin bread model at left illustrates why this is important. [Please view the "expanding raisin bread model" by clinking on the link below.] If every portion of the bread expands by the same amount in a given interval of time, then the raisins would recede from each other with exactly a Hubble type expansion law. In a given time interval, a nearby raisin would move relatively little, but a distant raisin would move relatively farther - and the same behavior would be seen from any raisin in the loaf. In other words, the Hubble law is just what one would expect for a homogeneous expanding universe, as predicted by the Big Bang theory. Moreover no raisin, or galaxy, occupies a special place in this universe - unless you get too close to the edge of the loaf where the analogy breaks down.

The current WMAP results show the Hubble Constant to be 73.5 +/-3.2 (km/sec)/Mpc. If the WMAP data is combined with other cosmological data, the best estimate is 70.8 +/- 1.6 (km/sec)/Mpc.
http://map.gsfc.nasa.gov/universe/bb_tests_exp.html

A review of my mgs. 39 might be helpful. A segment from that post was from a "scientist (physicist) -" from NASA, Is the Universe Infinite? Here is a quote from him, but please review the entire website.

"However, the results of the WMAP mission and observations of distant supernova have suggested that the expansion of the universe is actually accelerating which implies the existence of a form of matter with a strong negative pressure, such as the cosmological constant. This strange form of matter is also sometimes referred to as the "dark energy". If dark energy in fact plays a significant role in the evolution of the universe, then in all likelihood the universe will continue to expand forever." http://map.gsfc.nasa.gov/universe/uni_shape.html

Also, "Mathematicians" are not scientists. "Physicists" are scientists that know mathamatics. :)

Have a good day,
Mars

 

[Entropee]

Thanks Chalnoth I actually couldn't find it on my own lol

 

[Axuality]

Can I jump in and say that it is my understanding that infinity is a mathematical concept which doesn't exist in the real world of science, but which may possibly exist (according to mathematical theories).

Of all words, 'infinity' is tied as the most striking example of a word that can have only one definition. For example, I'm sure we all know that the number of integers is not infinite, it is unlimited.

The only definition of infinity which is rational, is 'the summation of all things'. There can be only one infinity,- in much the same way that there can be only one reality, incidentally.

That is important, because upon reflection, it reveals startling things about the nature of reality, which reveals additional startling things about the natures of science and math.

 

[Chalnoth]

Of all words, 'infinity' is tied as the most striking example of a word that can have only one definition. For example, I'm sure we all know that the number of integers is not infinite, it is unlimited.

Uh, that's not strictly true. Mathematically speaking, different infinities can and often do have rather different characters.

The number of integers, for instance, is called a "countably infinite" number. Any set of numbers which can be mapped one-to-one onto integers is also countably infinite. Sets which cannot be mapped onto the integers (such as the reals) are uncountably infinite, which means that there are, for instance, more real numbers than there are integers (by contrast, there are no fewer positive integers than total integers).

The only definition of infinity which is rational, is 'the summation of all things'. There can be only one infinity,- in much the same way that there can be only one reality, incidentally.

That is important, because upon reflection, it reveals startling things about the nature of reality, which reveals additional startling things about the natures of science and math.

Sorry, but definitions are arbitrary. There is never only one rational definition.

 

[Axuality]

Uh, that's not strictly true. Mathematically speaking, different infinities can and often do have rather different characters.

The number of integers, for instance, is called a "countably infinite" number. Any set of numbers which can be mapped one-to-one onto integers is also countably infinite. Sets which cannot be mapped onto the integers (such as the reals) are uncountably infinite, which means that there are, for instance, more real numbers than there are integers (by contrast, there are no fewer positive integers than total integers).


Sorry, but definitions are arbitrary. There is never only one rational definition.

Ha ha, you obviously are intelligent, so not for one moment would I forget that.

Perhaps I should have said that there "should" be only one definition of the word 'infinity'.
I know and understand what you told me about 'infinity'. And I recognize that everything you said was correct. What I am saying to you though, is that the understanding of the concept of TOTAL infinity makes impossible the logical use of the word infinity in the phrase "countable infinities". I mean if we want to call a horse a horse, and also call a cow a horse, we can do it. But it makes things less clear, not more clear.

And when you tell me that "definitions are arbitrary" I know what you mean of coures, but I respectfully chuckle to realize that the word 'definitions' is somehow based on the word 'definite' which would make the statement kind of like saying 'definite is arbitrary', which in some sense is rather contradictory. :)

More seriously though, I disagree that there is "never only one rational definition". While on the surface that seems, and IS correct, I'm not on the surface with this definition thing.

In fact, I construe and extrapolate to conclude that IF that statement is true, then by it's own truth, it is not ALWAYS true. Hence it is not true at all.

Forgive me, I don't think I'm 'smarter' than you. I think in fact, that I'm not smart ENOUGH to convey to you that I'm am talking about a slightly different aspect of 'definition' than you are.

I would beg you to simply consider (for just a moment) the definition of 'infinity' as being the entire collection of all things which compose reality (matter, energy, thoughts, et al). --that's what infinity is; what it means. If we want to call a cow a horse, then we can use the word 'infinity' to mean something else also. :)

 

[Chalnoth]

The problem, Axuality, is that you're abusing what it means to define a word.

First, as I said, there is never anyone rational definition: all definitions are arbitrary. And furthermore, words in the English language tend to be extremely context-sensitive. The important point isn't that words have rational definitions, but rather that words are understood. That is to say, words are defined by how they are used by people. This means that if you are to use a word, it is a darned good idea to understand how people will interpret that word.

So when you go and use a definition of infinite as "the summation of all things", that strikes me as rather ridiculous as nobody uses that definition. Infinite is, by large, an intrinsically mathematical term (except when it colloquially used to mean "really really big"). In mathematics, there are a few different classes of infinites. And the fact is, we do not yet know for sure whether or not various parts of our universe match one of these different classes of infinities.

Thus if you want to talk about the "summation of all things", if you wish to be understood, you should use the word "universe" instead of "infinity".

 

[Axuality]

The problem, Axuality, is that you're abusing what it means to define a word.

First, as I said, there is never anyone rational definition: all definitions are arbitrary.


if you want to talk about the "summation of all things", if you wish to be understood, you should use the word "universe" instead of "infinity".

Hi. If we're are going to discuss any further, I need you to understand that I respect your intelligence. Therefore, I will speak as if I know that you will not be offended.

I started a response to you which became too long for you to read and for me to write, so I'm shortening it. :)

I do not agree with what you say.

#1 In the ultimate, the word 'universe' and the word 'infinity' are identical in meaning. If you doubt this, you are not looking large enough.

The universe is larger in scope(not physical scale, but 'scope') than is imagined by physics. Quantum theory is approaching a conclusion on the subject which will substantiate this.

Maybe I should have said in the first place "There is only one infinity". There are many definitions of the word 'infinity', but there is only one infinity. To understand this, you must be able to separate the concept behind a word from the definition of that word. You may well doubt that that is possible or makes sense, but that is okay if you doubt it. ;)

#2 The statement that 'all definitions are arbitrary' is self-contradictory.- much as the statement 'Truth does not exist' is self-contradictory. I'm going to abstain from any attempt at long proof of that, and if you don't choose to believe it, that is okay. :) I had to put it out there. ( let me make a quick offering of "proof" --'if all definitions are arbitrary, or relative, then there ARE no definitions, there are only 'word assignments'. Maybe we need a new definition of the word 'definition'.

Again, you're obviously a smart guy or girl, and I hope I've been able to speak directly and unoffendingly, if not very diplomatically. :)

 

[Deuterium2H]

I was reading about infinity - and aren't there differing orders of infinity, and even sets of infinite numbers that still exclude other numbers?

Like the set of all even numbers is infinite. But it does not include the number 1, 3, 5, 7, etc. So even if the universe was infinite, there could still be an "everthing" that doesn't exist?

Perhaps this is more of a philosophical question.

Of all the replies, only JnWaco has correctly answered the original poster's question. And Chalnoth also looks to be on the same track.

This problem is invariably answered incorrectly by most physicists (even the best), simply because they are not specialists in Set Theory, or to be more succinct, transfinite Set Theory. The fact is this. If the Universe is infinite, it may only be "countably" infinite, or equal in cardinality to Aleph Nought (countably infinite = a denumerably infinite set). However, a countably infinite set (= Aleph Nought) is the "smallest" infinity, and is not necessarily exhaustive. As JWaco mentioned, the set of Even Numbers is infinite, yet it is missing an infinite amount of numbers (specifically, all the Odd numbers). A denumerable infinite Set could contain every countable (ordinal) number...with the exception of the number three "3". It is still infinite, but it does not contain all the numbers (in this case, "3"). In fact, just like the Odds, you could instead remove all the Prime Numbers (which are infinite) from the set of Natural Numbers (N), yet you still are left with an infinite set...all the numbers that aren't Prime.

So again, if the Universe is infinite, with cardinality equal to Aleph Nought, then while it may be infinite, it is NOT NECESSARILY exhaustive. That is to say, it is NOT true that every possibility necessarily exists. While it is NECESSARY that the Universe be infinite in order for there to exist every possibility, it is NOT SUFFICIENT.

However, if the Universe has a cardinality equal to the Continuum (= 2^Aleph Nought), then it is possible that it is exhaustive and that it is possible that everything exists somewhere...as the Original Post questions.

In conclusion, it all comes down to the question: If the Universe is infinite, is it countably infinite (i.e. denumerable, equal in cardinality to the Natural Numbers = Aleph Nought), or is it an Aleph greater then Aleph Nought? Only if the infinite Universe is greater in cardinality then Aleph Nought can there exist the sufficient condition/possibility that everything exists somewhere.

 

[xantox]

A countably infinite set (= Aleph Nought) is the "smallest" infinity, and is not necessarily exhaustive [..] A denumerable infinite Set could contain every countable (ordinal) number...with the exception of the number three "3". It is still infinite, but it does not contain all the numbers (in this case, "3"). [..] Only if the infinite Universe is greater in cardinality then Aleph Nought can there exist the sufficient condition/possibility that everything exists somewhere. [..] It all comes down to the question: If the Universe is infinite, is it countably infinite (i.e. denumerable, equal in cardinality to the Natural Numbers = Aleph Nought), or is it an Aleph greater then Aleph Nought?

An uncountably infinite set is also not necessarily "exhaustive", eg it could also not contain "3".

This problem is invariably answered incorrectly by most physicists (even the best), simply because they are not specialists in Set Theory

Instead of assuming that physicists don't know about set theory, consider that they may take into account the additional constraints of the full physical theories – this must be the case when attempting to answer a physical question, set theory alone won't be sufficient to answer it. Eg if cosmological inflation is assumed, then a condition of ergodicity and randomness could apply on the initial conditions of an infinite universe. In that case, and adding to this that the number of states in a finite volume at finite temperature is also finite, all that can exist physically [1] and within certain temperature limits would exist somewhere.
___
[1] the question is implicitly about physical existence – it would be probably meaningless to require that physical "exhaustivity" should include unphysical states.

 

[Deuterium2H]

Xantox,

I never stated that an uncountably infinite set was necessarily exhaustive. I stated that it was possibly exhaustive. I specifically used the example of an uncountably infinite set
(e.g. the cardinality of the Continuum) to contrast it with a countably infinite set. And I explained that if the Universe was only countably infinite, that it was a necessary but NOT sufficient condition that "everything exists somewhere". I then provided an example.

Now I agree with you that an uncountably infinite Set may also not be exhaustive. For example, the Set of Real Numbers in the interval [0,1] is uncountable, but not exhaustive. This set also does not contain the number "3". Nevertheless, it is also the case that an uncountable Set of the same Cardinality (2^Aleph Nought) may be exhaustive.
For example the Power Set of |N| = P(N) is the Set of ALL subsets of the Natural numbers, and thus definitely does contain the number "3". Things get very tricky when dealing with transfinite Sets.

Finally, I respectfully make the comment that your citation of "unphysical" states has no meaning in Cosmology. By definition, the Universe contains everything that is physical, and nothing that is non physical. I presume that by your term "unphysical" you technicall mean non-physical. While I agree that non-physical states arise as mathematical constructs in Quantum Field Theory and String Theory, these non-physical states are eliminated by employing gauge symmetry methods. In any event, your example of a finite phase space ("the number of states in a finite volume at finite temperature is also finite") is irrelevant for two reasons. Firstly, because a phase space can also be infinite. Secondly, and more importantly, the very subject of this topic/original post posits that the Universe is Infinite.

 

[Godswitch]

Surely there are exotic elements we know nothing about that would allow for seemingly improbable situations

 

[Deuterium2H]

Surely there are exotic elements we know nothing about that would allow for seemingly improbable situations

Even so, this still would not make it a necessary and sufficient condition for "everything to exist somewhere".

 

[xantox]

Xantox, I never stated that an uncountably infinite set was necessarily exhaustive. I stated that it was possibly exhaustive.

Yes, but how "exhaustivity" is defined here? The set of all real numbers does contain all real numbers. But it does not contain complex numbers. Is it "exhaustive" then? To define exhaustivity we should also define the space of states. If it is the integers, then the set of all integers is countably infinite and exhaustive. A dice has only 6 states. We can say in probability theory that the 6 outcomes of a rolling dice are collectively exhaustive.

Finally, I respectfully make the comment that your citation of "unphysical" states has no meaning in Cosmology. By definition, the Universe contains everything that is physical, and nothing that is non physical. I presume that by your term "unphysical" you technicall mean non-physical.

The term "unphysical" is the one most commonly used in the literature – see http://arxiv.org/find/all/1/all:+unphysical/0/1/0/all/0/1 for some usage. Indeed it is just a synonym for "non physical". No big deal anyway on which spelling we use. An unphysical state is something we can come up mathematically but that is against the laws of physics. Like traveling faster than the speed of light. So that it has probably no meaning to require that for "everything to exist" we need to include things that would travel faster than the speed of light. Once we exclude all unphysical states, what remains can be well only countably infinite.

In any event, your example of a finite phase space ("the number of states in a finite volume at finite temperature is also finite") is irrelevant for two reasons. Firstly, because a phase space can also be infinite. Secondly, and more importantly, the very subject of this topic/original post posits that the Universe is Infinite.

The meaning of saying that the number of states of finite volume at finite temperature is finite, is that as a consequence, the number of states of an universe behaving that way, when we assume it to be infinite, is countably infinite.

 

[Chalnoth]

Yes, but how "exhaustivity" is defined here? The set of all real numbers does contain all real numbers. But it does not contain complex numbers. Is it "exhaustive" then? To define exhaustivity we should also define the space of states. If it is the integers, then the set of all integers is countably infinite and exhaustive. A dice has only 6 states. We can say in probability theory that the 6 outcomes of a rolling dice are collectively exhaustive.

One interesting thing is that an infinite subset of all integers is also exhaustive, such as, for instance, the set of all even integers (because the set of all even integers has a one-to-one relationship with the set of all integers, it is equivalent to the set of all integers).

The term "unphysical" is the one most commonly used in the literature – see http://arxiv.org/find/all/1/all:+unphysical/0/1/0/all/0/1 for some usage. Indeed it is just a synonym for "non physical". No big deal anyway on which spelling we use. An unphysical state is something we can come up mathematically but that is against the laws of physics. Like traveling faster than the speed of light. So that it has probably no meaning to require that for "everything to exist" we need to include things that would travel faster than the speed of light. Once we exclude all unphysical states, what remains can be well only countably infinite.

I strongly suspect that an actual TOE would include no unphysical states.

 

[Deuterium2H]

One interesting thing is that an infinite subset of all integers is also exhaustive, such as, for instance, the set of all even integers (because the set of all even integers has a one-to-one relationship with the set of all integers, it is equivalent to the set of all integers).
.

Woops, a bit of clarification is required, here, Chalnoth. You are correct that the Set of all Even, natural numbers has the same size (i.e. cardinality) as the Set of all natural numbers (N)...however, the two sets are not "equal", in the sense that they do not contain identical members. The Set of even natural numbers does not exhaust all the natural numbers. However, taking the Power Set of N would ensure that you exhaust all the Natural numbers.

 

[Deuterium2H]

The meaning of saying that the number of states of finite volume at finite temperature is finite, is that as a consequence, the number of states of an universe behaving that way, when we assume it to be infinite, is countably infinite.

Not necessarily. What if the number of finite volumes in the Universe is itself uncountable. Then, the Universe would be uncountably infinite.

 

[Chalnoth]

Woops, a bit of clarification is required, here, Chalnoth. You are correct that the Set of all Even, natural numbers has the same size (i.e. cardinality) as the Set of all natural numbers (N)...however, the two sets are not "equal", in the sense that they do not contain identical members. The Set of even natural numbers does not exhaust all the natural numbers. However, taking the Power Set of N would ensure that you exhaust all the Natural numbers.

If the two sets have a one-to-one correspondence, however, the two sets are identical in every way. That is, in any sort of mathematical structure where I use the set of all natural numbers, I can also use the set of all even numbers and everything will always work out the same, as long as I carry through the effects of that correspondence.

 

[Deuterium2H]

If the two sets have a one-to-one correspondence, however, the two sets are identical in every way. That is, in any sort of mathematical structure where I use the set of all natural numbers, I can also use the set of all even numbers and everything will always work out the same, as long as I carry through the effects of that correspondence.

Hi Chalnoth,

I must disagree. I believe you are confusing equivalence in Set Cardinality with Set equality. While two sets may have the same Cardinality, they are not necessarily equal. For example, take the finite sets X = {1,a,3,4,5}
and the set Y = {1,2,3,4,5}.

The two sets are equal in cardinality. That is |X| = |Y|...where |X| stands for the cardinality of set X. Both sets are equipotent.

However, the sets are not equal...that is, X does not equal Y, because set X has the member "a" whereas set Y has a member "2".

By the definition of Sets, two Sets are equal if and only if they have the same elements.

The Set of Rational numbers has the same cardinality as the set of Natural numbers. Both sets have a Cardinality = Aleph Nought. However, try as one might, you will never find the element "1/3" in the Set of Natural Numbers. The two sets are not equal in membership, although the are "equal" in size. Technically, one can only use the equality sign when comparing the cardinality of these sets, i.e.:
|Q| = |N| is a true statement. However, {Q} = {N} is NOT a true statement.

Another example would be the Set of Algebraic Numbers. They can be put in a one-to-one correspondence, and thus have the same Cardinality as the Natural Numbers. In fact, the Set of Natural numbers is a proper subset of the Set of Algebraic Numbers, even though they are equal in size/cardinality. However, if one were tasked to pick out squareroot(2) from the Set of Natural numbers, one would be at a loss.

 

[Chalnoth]

Hi Chalnoth,

I must disagree. I believe you are confusing equivalence in Set Cardinality with Set equality. While two sets may have the same Cardinality, they are not necessarily equal. For example, take the finite sets X = {1,a,3,4,5}
and the set Y = {1,2,3,4,5}.

The two sets are equal in cardinality. That is |X| = |Y|...where |X| stands for the cardinality of set X. Both sets are equipotent.

However, the sets are not equal...that is, X does not equal Y, because set X has the member "a" whereas set Y has a member "2".

By the definition of Sets, two Sets are equal if and only if they have the same elements.

The Set of Rational numbers has the same cardinality as the set of Natural numbers. Both sets have a Cardinality = Aleph Nought. However, try as one might, you will never find the element "1/3" in the Set of Natural Numbers. The two sets are not equal in membership, although the are "equal" in size. Technically, one can only use the equality sign when comparing the cardinality of these sets, i.e.:
|Q| = |N| is a true statement. However, {Q} = {N} is NOT a true statement.

Another example would be the Set of Algebraic Numbers. They can be put in a one-to-one correspondence, and thus have the same Cardinality as the Natural Numbers. In fact, the Set of Natural numbers is a proper subset of the Set of Algebraic Numbers, even though they are equal in size/cardinality. However, if one were tasked to pick out squareroot(2) from the Set of Natural numbers, one would be at a loss.

Well, yes, this is strictly true. But since we're talking about this in the context of a physical law (assuming, for a moment, that we're trying to keep this on the topic of the original post), then set equality is not the proper metric.

Consider in the context of physical law, a set would be one component of the full mathematical structure. Let's imagine, for the sake of argument, that the full mathematical structure we are talking about is an algebra. I can define an algebra with the set of natural numbers combined with addition. If you give me any set that has a one-to-one correspondence with the natural numbers, I can define an algebra in such a way that the behavior of this other set is identical to the behavior of the algebra with natural numbers (though the operator may, depending upon the set, look nothing like addition).

In the end, this doesn't matter for the physics. What we call a specific number is irrelevant. It is only the interrelationships that matter for defining the behavior of the mathematical structure.

 

[Deuterium2H]

Well, yes, this is strictly true. But since we're talking about this in the context of a physical law (assuming, for a moment, that we're trying to keep this on the topic of the original post), then set equality is not the proper metric.

Consider in the context of physical law, a set would be one component of the full mathematical structure. Let's imagine, for the sake of argument, that the full mathematical structure we are talking about is an algebra. I can define an algebra with the set of natural numbers combined with addition. If you give me any set that has a one-to-one correspondence with the natural numbers, I can define an algebra in such a way that the behavior of this other set is identical to the behavior of the algebra with natural numbers (though the operator may, depending upon the set, look nothing like addition).

In the end, this doesn't matter for the physics. What we call a specific number is irrelevant. It is only the interrelationships that matter for defining the behavior of the mathematical structure.

Fair enough. So getting back on topic, do we both agree, then, that it is not a necessary and sufficient condition that the Universe be infinite in order that "everything exists somewhere". The crux of my argument was to rebut the commonly held belief (even amongst some physicists) that an infinite universe implies that all possibile states exist and that somewhere out there is an exact duplicate of myself typing this very post.

 

[Chalnoth]

Fair enough. So getting back on topic, do we both agree, then, that it is not a necessary and sufficient condition that the Universe be infinite in order that "everything exists somewhere".

Yes. You also need a physical mechanism to explore all possibilities.

The crux of my argument was to rebut the commonly held belief (even amongst some physicists) that an infinite universe implies that all possibile states exist and that somewhere out there is an exact duplicate of myself typing this very post.

Well, that particular possibility is a necessary consequence of an infinite universe combined with inflation.

 

[Deuterium2H]

Well, that particular possibility is a necessary consequence of an infinite universe combined with inflation.

Chalnoth...you lost me. Perhaps I misunderstand your statement...however, it seems you are now stating that "everything exists somewhere" as being a necessary consequence of an infinite Universe combined with inflation.

As already discussed, an infinite Universe is not a sufficient condition that "everthing exists somewhere", and inflation does not change this in the least. Inflation is just an exponential expansion. If the Universe was created infinite, then inflation doesn't make it a higher power of infinity...nor does it in any way change what may be a countably infinite Universe into an uncountably infinite Universe.

 

[xantox]

I strongly suspect that an actual TOE would include no unphysical states.

I believe the same.

Not necessarily. What if the number of finite volumes in the Universe is itself uncountable. Then, the Universe would be uncountably infinite.

Note that you said in your first message "If the Universe is infinite, it may only be "countably" infinite". Anyway, it is possible to map to naturals all permutations of state at finite temperature of any causally connected patch of the universe (observable universe). That is all which may exist, within known law of physics.

So getting back on topic, do we both agree, then, that it is not a necessary and sufficient condition that the Universe be infinite in order that "everything exists somewhere". The crux of my argument was to rebut the commonly held belief (even amongst some physicists) that an infinite universe implies that all possibile states exist and that somewhere out there is an exact duplicate of myself typing this very post.

This part is fine. The problem was on the other part, where you said that a necessary condition for all possible states to exist is that "the universe must be infinitely uncountable". I don't see any reason for that to be true.

 

[Chalnoth]

Chalnoth...you lost me. Perhaps I misunderstand your statement...however, it seems you are now stating that "everything exists somewhere" as being a necessary consequence of an infinite Universe combined with inflation.

Not quite. Now, for the sake of argument, I think it would be better to think of eternal inflation instead of "infinite universe plus inflation". They have the same implications here anyway, except that eternal inflation is at least somewhat more reasonable as a physical theory.

With that out of the way, the argument here isn't that everything happens somewhere, but that eternal inflation is a mechanism for exploring some subset of the possible parameter space. Now, you can actually calculate that the possible configurations for a universe like our own is quite finite. And since eternal inflation produces an infinite number of Hubble volumes, and those Hubble volumes form a finite set of possible configurations, any configuration that eternal inflation explores once will be explored an infinite number of times.

 

[Deuterium2H]

I believe the same.


This part is fine. The problem was on the other part, where you said that a necessary condition for all possible states to exist is that "the universe must be infinitely uncountable". I don't see any reason for that to be true.

?? Didn't I say necessary but NOT sufficient. Certainly, the Universe has to be infinite (that is a necessary condition) for there to exist the possibility that "everything exists somewhere." But that in and of itself does not make it a sufficient condition. That is what I have been arguing all along. I don't think I had stated anywhere that an uncountably infinite Universe was a necessary AND sufficient condition for this to occur...if I did, that was an unintentional mistake.

 

[xantox]

?? Didn't I say necessary but NOT sufficient. Certainly, the Universe has to be infinite (that is a necessary condition) for there to exist the possibility that "everything exists somewhere." But that in and of itself does not make it a sufficient condition. That is what I have been arguing all along. I don't think I had stated anywhere that an uncountably infinite Universe was a necessary AND sufficient condition for this to occur...if I did, that was an unintentional mistake.

Yes, but the problem is that it is not even a necessary condition. A countably infinite universe can be exhaustive, too.

 

[Sage Lee]

Ok - first of all, I apologize for what is likey to be a very long-winded example of my rambulitis.

It will soon be clear that I know absolutely nothing about any of this; I can hardly follow half of the jargon that you guys throw around so casually. I only came across this thread (and forum) by googling the question that is in the thread title, because I'm just crazy like that and found myself thinking about infinity (again), and I wanted to hear some smart-people thoughts on the matter.

But I quickly found myself over my head. I don't know what Hubble volume is; I don't know what TOE stands for, I don't really know what the Copenhagen interpretation is (although I'm sure I've read all about these concepts on Wikipedia at some point or another, because that's just what I do.) I suppose I could go and refresh my Wiki knowledge (and I probably will, sigh), but I know that if I try I will inevitably find something I don't understand within the explantion of what I'm trying to understand, which will lead me to delve into an explantion of that, which will of course contain another term or concept I don't understand, and so on, until I have 50 pages of advanced physics concepts opened on my web browser and a throbbing mental headache. The problem lies in the fact that there probably aren't too many laymen that are interested in discussing the finer points of such complicated topics, but there's at least one (hai dere!) So basically, what I'm trying to say is: be gentle.

So, all these different interpretations of infinity, countable and uncountable, etc etc... these just seems like different ways of putting a limit on infinity, which by (my) definition should have no limits. For instance, the example of how a set containing only even numbers could be infinite and yet not exhuastive... that was a great explanation, but it still seems to me that a finite limit has been put on the (my) basic concept of infinity. It's like saying an "infinite line"... to me that seems like a misnomer, simply because the phrase itself puts a finite parameter (a line) on infinity. Put another way, it's like saying infinity, but in only one direction. Which (to me) means it's not actually "infinite" at all, it just happens to go on forever in that one direction.

In my mind, imagining infinity (ha!) is more like picturing a sphere that expands outwards in all directions and never stops. In fact, time itself is kind of like this infinite line I mentioned, and by existing in the first place it already tells my feeble brain that a true infinity isn't possible in our observable universe. If infinity truly existed, physically, it seems to me that it would be everything, everywhere, EVER... happening all at once (and everywhere at once.) Over and over and over again, until my head assploded.

I'm realizing now that my defintion of infinity (everything) is the exact opposite of the definition of zero (nothing). I don't know if this is intuitive or if there's some mathematical basis for that, or if it's simply just incorrect, but that's how I've always defined infinity: on a number line, it's the polar opposite of zero, and to extend that concept in a philisophical sense is to make it the polar opposite of nothing.

But let's assume that we're only talking about infinite physical space. Time, whether I like it or not, seems to exist, even if only to keep everything from happening at once. So with this one boundary in place (time), let's assume that physical space goes on forever. I've always taken to heart the concept of "the closer you get to infinity, the probability of x happening approaches 1." And by extension, if you actually could get to infinity, then the probability of x happening, somewhere, sometime, must equal 1.

And I still just can't get past this. How is this not true? What exactly am I missing about this concept of infinity? Using that one example along the lines of "different blurry versions of myself that all slightly vary outwards from point A (the "real" me, from now) and some get hit by the car, or meet the girl, and some don't, blah blah blah" but then you assume this has been going on since the beginning of the universe (or dare I say, since even before that? Maybe it's been going on forever? Maybe the universe itself has infinite variations, an infinite amount of which evolved life similar to ours, or nothing like ours, and likewise, an infinite amount of universes that never were, so to speak.)

Here though, I must clarify once again that when I say infinity, I'm talking about something that all variations are encompassed within. I'm aware of the many-worlds theory, but in my definition of infinity, every world (or dimension, or variation, or whatever) is included within that term. I guess I'm saying that if infinity exists in any real sense, everything that exists, wherever it may be, is contained within that infinity. It's impossible for anything to exist outside it... well, because there is no "outside," it goes on forever, durrr.

So I can't help but stand by the concept that if our reality were infinite, everything would be happening within it. Everything meaning anything that any of us can think of, along with an infinite amount of things we could never possibly think of. And I just don't understand why it's assumed that all reality, even if infinite, would have to conform to our known laws. I don't understand why it's assumed that everything was once connected, as someone put it, to our reality (or something to that effect) and therefore must follow the laws we (think we) know. I mean, from what I understand, there's already contradictions in the "rules" when we try to relate them to very very small or very very large objects (the so far fruitless search for a unified theory), so it follows, for me, that our rules might conceivably not apply once we go even bigger (or smaller.) And when we're talking infinitley bigger (or smaller), well, it seems like everything we (think we) know could be up in the air.

I remember first thinking about this when I was about 14 (I'm 32). I read some sci-fi book that touched on the concept of "everything must exist within infinity", and I thought about it for a long while. The concept just made sense, and it still does, which is why I can't get past this. At the time though, I "proved" to myself that infinty can't exist. I did this by thinking of something that *should* exist, but obviously didn't. I thought of a planet full of alternate "me's" (an infinite amount of them). I then thought of a planet full of "me's" that had found a way to bend time and space and traverse dimensions with but a thought. I then thought of one of these "me's" that could observe (the real) me, and had the power to appear before me, and make himself known to me, and then I thought of a "me" who chose to do just that. And since I never appeared before myself, I thought I had proved that infinity didn't exist. (I then realized that there would be an infinite amount of these me's who would appear before me, as well as an infinite amount of anyone else, and everything else, appearing in front of anyone and everything else, and so on, and that's when I decided that true infinity would mean an unimaginable blur of everything happening all at once, everywhere at once.)

I am now old enough to undertand that the only thing I "proved" is that I didn't understand what the hell I was talking about. But the problem is that I still don't understand, because everything I just said still makes perfect sense to me. Even if we're in finite space, even if time is the only thing that's infinite, it seems to me that sometime, everything I can think of (and everything I can't) must exist, eventually. But again, I'm thinking of a "me" from the future who has figured out how to travel back here to my time, and of course there are an infinite amount of them at some point in the timeline, all of whom can travel back to this exact moment and have the power to make themselves observable to me, and... boom, everything at once.

Anyway, so I guess it comes back to these "rules" or "laws" that we have observed, and whether or not they can ever be broken, given infinity. Can infinty be separated into sections that can't ever co-exist? I contend that it can't; eventually they must (or already have.) Eventually, given infinity, all of our rules must be broken. So a *true* inifinty cannot exist in any physical sense.

Anyway, sorry for all that. I can never be concise in things like this, for 2 reasons: 1) I don't understand enough of the technical jargon to properly sum up complex thoughts with one or two terms, and 2) I have no idea what I'm talking about.

I guess I'll sum it up my questions here at the bottom for those not inclined to read this whole thing:

Can someone explain to me, as you would to a child, why an infinite universe "isn't sufficient" for *everything* existing? By the same token, why would an infinite timeline be insufficient for everything existing, eventually? Why can't laws (traveling back through time, or across dimensions, and all the rest) be broken, given infinite time or space? Why can't a four-sided triangle exist just because I can't conceptualize it? In infinity, even that should be there somewhere, even if our feeble, logical minds would snap if they ever actually tried to understand it. (To be clear, my whole argument is that these things don't exist, but only because infinity doesn't either, at least beyond a theoretical concept.)

But on that same note, is it possible that logic itself is only a limitation of the human perspective, rather than some universal, infallible ideal? (This is a question I asked someone in another forum recently, where a bunch of people got to arguing about whether or not God exists (and on a poker forum, believe it or not.) One guy (basically) said "No, because [too many things about that] are not logical." Which got me thinking about illogicalities, and the possible limits of human thinking/perception, and about how if God does exist, he could pretty much violate any rule we can think of, because let's face it, he's God. A bit off topic here, but I'm just reiterating the concept of "just because it doesn't make sense to us doesn't mean it's not true.")

In closing, can I just say I ****ing hate v-bulletin? To my great consternation, I swear that everybody uses it now. Someone hurry up and write something better.

 

[Chalnoth]

Can someone explain to me, as you would to a child, why an infinite universe "isn't sufficient" for *everything* existing? By the same token, why would an infinite timeline be insufficient for everything existing, eventually?

Well, consider a simple case: a list of numbers. If the list of numbers is infinite in length, does this mean every number is represented? Nope. Consider this list:
{1,2,3,4,5,1,2,3,4,5,1,2,3,4,5,...}

The list, in this example, repeats the numbers 1-5 an infinite number of times, but it still only includes the numbers 1-5. The same sort of thing could potentially be the case with reality where, for whatever reason, it is unable to access certain possible configurations, even if it is infinite in size.

Why can't laws (traveling back through time, or across dimensions, and all the rest) be broken, given infinite time or space?

If we define a law of nature as an accurate description of nature, then by definition it cannot possibly be broken: whatever nature does, an accurate law describes it. Now, the laws which we have discovered are all wrong in some regard, meaning that they don't always describe what nature does. But this is because we don't yet know the full laws of nature.

Why can't a four-sided triangle exist just because I can't conceptualize it?

A triangle is defined as having three sides. So saying a four-sided triangle is the same as saying, "a four-sided, three-sided polygon." It is an improper use of language.

But on that same note, is it possible that logic itself is only a limitation of the human perspective, rather than some universal, infallible ideal? (This is a question I asked someone in another forum recently, where a bunch of people got to arguing about whether or not God exists (and on a poker forum, believe it or not.) One guy (basically) said "No, because [too many things about that] are not logical." Which got me thinking about illogicalities, and the possible limits of human thinking/perception, and about how if God does exist, he could pretty much violate any rule we can think of, because let's face it, he's God. A bit off topic here, but I'm just reiterating the concept of "just because it doesn't make sense to us doesn't mean it's not true.")

Basic logic just assumes one thing: logic is consistent. That is to say, whenever you have a definitive statement, that statement is always either true or false. We may not always know which, but it is always one or the other. By only allowing statements in the logic that are either true or false, the laws of logic that can be derived are absolute and inviolable.

One can potentially consider logics that allow for ambiguous or meaningless statements, but often it is easier to just not allow those statements.

 

[Deuterium2H]

Ok - first of all, I apologize for what is likey to be a very long-winded example of my rambulitis.

It will soon be clear that I know absolutely nothing about any of this; I can hardly follow half of the jargon that you guys throw around so casually. I only came across this thread (and forum) by googling the question that is in the thread title, because I'm just crazy like that and found myself thinking about infinity (again), and I wanted to hear some smart-people thoughts on the matter.

But I quickly found myself over my head. I don't know what Hubble volume is; I don't know what TOE stands for, I don't really know what the Copenhagen interpretation is (although I'm sure I've read all about these concepts on Wikipedia at some point or another, because that's just what I do.) I suppose I could go and refresh my Wiki knowledge (and I probably will, sigh), but I know that if I try I will inevitably find something I don't understand within the explantion of what I'm trying to understand, which will lead me to delve into an explantion of that, which will of course contain another term or concept I don't understand, and so on, until I have 50 pages of advanced physics concepts opened on my web browser and a throbbing mental headache. The problem lies in the fact that there probably aren't too many laymen that are interested in discussing the finer points of such complicated topics, but there's at least one (hai dere!) So basically, what I'm trying to say is: be gentle.

Sage,

There is absolutely nothing here to be embarrassed or uncomfortable about. In fact, you are in good Company. From at least the time of the ancient Greeks (and most likely much earlier) up until the late 19th Century, mankind has struggled with the the metaphysical and mathematical concept of infinity. In fact, it wasn't until well into the beginning of the 20th century that Georg Cantor's revolutionary work on Set Theory and Transfinite numbers was put on firm axiomatic foundations, and accepted by the mainstream mathematical community. If you can just imagine the breadth of time that has passed since antiquity (3,000 plus years), in which many of the GREATEST mathematical minds in history struggled with the seemingly paradoxical characteristics of the infinite, then this fact should humble us all.

Just to add a bit more context to the problem of infinity represents what is now called one of the Great "crisis" in Mathematics. And in a way, the concept of infinity was directly or indirectly involved in each great crisis.

The first great "crisis" was the discovery, by the Greeks, of the Irrational Numbers. How this came to be, and how they dealt with them (or perhaps more aptly put, ignored them), entire books have been written. The theory of Irrational numbers is intimately tied up with the Theory of Real Numbers, which in itself is intimately tied up with Set Theory, and the concept of completed, infinite Sets.

The second great "crisis" involved the fact that the development of the Calculus had no rigourous foundations, even though Newton and Liebniz's methods worked, and solved previously intractable physical problems. Key to both Newton's and Liebniz's Calculus was the concept of infinitesmals, as well as the approach to a Limit. Both are inexorably wrapped up with the concept of infinity. It wasn't until Cauchy, Bolzano and Weierstrass (in the early 1800's) that Calculus was put more or less on a firm foundation...despite the fact that there as yet existed no rigorous foundation for the Real Numbers (and, by consequence, Irrationals, Rationals, and even the Natural Numbers).

The third "crisis" involved the "discovery" and development of Non-Euclidean Geometry, by Gauss, Riemann, and others. Again, the Infinite reared it's head, as non-Euclidean geometries were predicated upon assuming the falsification of Euclid's fifth postulate (parallel line postulate).

The last great "crisis" involved the very foundations of Mathematics, and at it's very heart was the development of Set Theory and Transfinite numbers. Again, entire books have been written on this topic. Suffice it to say that Cantor's Set Theory and transfinite numbers shook the very pillars of mathematics, and eventually led to Godel's Incompleteness Theorem(s), which set limits on what was trully "knowable" in mathematics. In short, within a given mathematical system, certain logical statements can neither be proved nor disproved.

So, all these different interpretations of infinity, countable and uncountable, etc etc... these just seems like different ways of putting a limit on infinity, which by (my) definition should have no limits. For instance, the example of how a set containing only even numbers could be infinite and yet not exhuastive... that was a great explanation, but it still seems to me that a finite limit has been put on the (my) basic concept of infinity. It's like saying an "infinite line"... to me that seems like a misnomer, simply because the phrase itself puts a finite parameter (a line) on infinity. Put another way, it's like saying infinity, but in only one direction. Which (to me) means it's not actually "infinite" at all, it just happens to go on forever in that one direction.

Sage, you may be mixing up two concepts...that of a Line, and that of a Line Segment. By it's very nature, a Line (in the strict geometric sense) is infinite in length. A Line Segment is bounded, and of finite length. A line that starts at a point, and goes on forever in one direction is just as infinite as one that goes in both directions. When dealing with Infinity, our natural intuition is of no help...and in fact only get's us in trouble. As an example, I just previously claimed that a line segment is finite. And in one sense, it is, in that it is both bounded and has a definite, finite extent. However, that same "finite" line segment is composed of an infinite number of points. For those unfamiliar with Set Theory, it comes as a real shock to learn that there are EXACTLY the same number of points on the line interval from [0,1] as there are on an interval twice as long [0,2]. No more, no less. In fact, there are the same number of points. In math-speak, we say that there is a one-to-one correspondence between the set of Real numbers in the interval [0,1] and the interval [0,2]. How can we prove this? We establish a Function that maps each and every Real number in the smaller interval with those in the larger interval. That function would be:
y = f(x) = 2x

That is to say, take any Real number "x" in [0,1], and double it, via the the function f(x) = 2x. The result is that you will have paired of each Real number in the smaller interval with exactly one Real number in the larger interval. Technically, this is called a bijection, which is "one-to-one" and "onto". When dealing with infinite sets, the phrase "the whole is always greater then one of it's parts" is no longer valid. In fact, the very definition of a infinity (i.e. an Infinite Set) is any Set that can be put in a one-to-one correspondence with at least one of it's proper Subsets. Another example would be the Set of all Natural Numbers and a proper Subset of just the Even Numbers. Both of these Sets contain exactly the same number of members, and are the same "size" (otherwise known as Cardinality). We know this because we can "count" by making a one-to-one correspondence between each Natural number and each Even number, like so:

1 -> 2
2 -> 4
3 -> 6
4 -> 8
5 -> 10

Each Natural Number is matched with exactly one Even number, and vice versa.

In my mind, imagining infinity (ha!) is more like picturing a sphere that expands outwards in all directions and never stops. In fact, time itself is kind of like this infinite line I mentioned, and by existing in the first place it already tells my feeble brain that a true infinity isn't possible in our observable universe. If infinity truly existed, physically, it seems to me that it would be everything, everywhere, EVER... happening all at once (and everywhere at once.) Over and over and over again, until my head assploded.

What you just described happens to be one of the great stumbling blocks in the mathematical history of Infinity. Just as you described a sphere that expands outwards in all direction, and never stops, is exactly how pre-Cantorian mathematicians conceived infinity. They only accepted a "potential" infinity. A potential infinity was any process that could be continued indefinitely, and never ends or completes, such as the sequence of numbers: 1, 2, 3, 4, 5...
An actual or "completed" infinity is thinking of those same numbers, but taken as a complete, single Set, i.e.: {1,2,3,4,5...}
"A set is a many that allows itself to be thought of as a one."
The difference between a "potential" and an "actual" infinity may seem subtle, but it lies at the core of modern mathematics. Once infinite sets are taken as completed wholes, they can be manipulated and worked with.

Perhaps the single biggest surprise, when first learning transfinite Set theory, is that not all infinite Sets are equal. That is to say, there exists larger sizes of infinity. The smallest infinite Set is the Set of Natural Numbers, which is equal in size to the Set of Integers, which is equal in size to the Set of Rational Numbers. They all are equal in size, and all of the aforementioned numbers comprise the smallest Infinity, also called a "countable" or "denumerable" infinity, and all are designated by the Cardinal number Aleph-Nought. It is quite counter-intuitive to think that the Set of Rational Numbers is no greater in size then the counting numbers...especially when you consider that between any two Natural numbers (e.g. number "2" and number "3") there are an infinite number of Rational numbers. Futhermore, between any two Rational numbers there are an infinite amount of more Rational numbers. Yet, the number of Rationals is exactly the same as the number of Naturals. The Set of Natural numbers is bijective with, and can be put in a one-to-one correspondence with the Set of Integers, the Set of Rationals, and even the Set of Algebraic Irrationals.

As mentioned previously, there exists greater Infinite sets (in fact, an infinite number). The Set of Real numbers is one example. It's size is greater then the Natural/Integer/Rational numbers. Mathematically, the Set of Real numbers = ( 2 ^ |N| )...where |N| is the Cardinality (size) of the Natural numbers. Another way of stating this is that the Set of Reals is equal to the Set of ALL Subsets of Natural Numbers.

Can someone explain to me, as you would to a child, why an infinite universe "isn't sufficient" for *everything* existing? By the same token, why would an infinite timeline be insufficient for everything existing, eventually? Why can't laws (traveling back through time, or across dimensions, and all the rest) be broken, given infinite time or space? Why can't a four-sided triangle exist just because I can't conceptualize it? In infinity, even that should be there somewhere, even if our feeble, logical minds would snap if they ever actually tried to understand it. (To be clear, my whole argument is that these things don't exist, but only because infinity doesn't either, at least beyond a theoretical concept.)

In order to understand this, you need to understand the formal, logical distinction between what is a "necessary" condition, and what is a "necessary AND sufficient" condition. They are not the same. I guess the best way to explain is through an analogy and example.
To say that X is a necessary condition for Y is to say that it is impossible to have Y without X. In other words, the absence of X guarantees the absence of Y.
Example: Having four sides is a Necessary condition for being a Square.
Notice, however, it is not a Sufficient condition. For example, a Rectangle has four sides, as does a Rhombus, but they are not necessarily Squares. A Rectangle has four equal angles, but may not have four equal sides. Conversely, a Rhombus has four equal sides, but may not have four equal angles.
Compare/contrast the above example to the following:
A quadrilateral with four equal sides and four equal angles is a both Necessary and Sufficient condition for being a Square.
-Or- another way of phrasing this: A quadrilateral that is BOTH a Rectangle AND a Rhobus is a Necessary and Sufficient Condition for being a Square.

Now, getting back to your question as to how an Infinite Universe isn't a "Sufficient" condition for "Everything existing somewhere"...
It is a Necessary condition that the Universe be Infinite in order for there to exist the possibility that "everything exists somewhere". This is obviously trivially true, because if it were not infinite, then it would be finite, and a finite Universe cannot be a Necessary condition for everything existing somewhere. So, as a minimum, it is a Necessary condition that the Universe be Infinite in order for this possibility to exist. However, that is not a Sufficient condition. As discussed in earlier posts in this thread, the Universe may be "countably" infinite...that is to say, having the same size (Cardinality) as the countably Infinite Set of Natural Numbers ( |N| ). However, the Set of all Even numbers is just as big (i.e. the same size) as the Set of all Natural Numbers, yet the former Set is missing an infinite amount of numbers...that is, the Odd numbers. So, these two sets have exactly the same NUMBER of elements (members), but these two sets are not "identical", and only one of these sets "exhaust" all the Natural numbers, whereas the other set does not.

With that said, I am not exactly certain what would be both a Necessary and Sufficient condition for an infinite Universe to ensure that "everything exists somewhere". From a purely mathematical perspective, I might argue that the Universe would need to have the Cardinality of the Continuum (= the Set of Real numbers). However, one could equally argue that that, in and of itself, may not even be a Sufficient condition. The tiny interval [0,1] on the Real number line is everywhere Dense and Continuous, and this segment contains an equal number of points as in the entire Real Number line. In fact, it contains in equal number of points as on a plane. Moreover, it contains just as many points as on any finite n-dimensional space. Nevertheless, despite the equipollence of the interval [0,1] with the entire Real Number line, it is not "exhaustive". It doesn't contain the number "2", or "pi", or "e", or for that matter any Real number greater then one or less then zero.

All this gobbledygook ultimately comes down to the conclusion that, even though the Universe may be infinite, it does not necessarily follow that "everything exists somewhere".

 

[Sage Lee]

Well, consider a simple case: a list of numbers. If the list of numbers is infinite in length, does this mean every number is represented? Nope. Consider this list:
{1,2,3,4,5,1,2,3,4,5,1,2,3,4,5,...}

The list, in this example, repeats the numbers 1-5 an infinite number of times, but it still only includes the numbers 1-5. The same sort of thing could potentially be the case with reality where, for whatever reason, it is unable to access certain possible configurations, even if it is infinite in size.

Okay... honestly, I kind of rolled my eyes (at first) when I saw that you just rehashed the same example as had been stated previously, using a finite list of numbers (except this time you said "1 through 5" instead of "only even numbers". Because this is where I suffer a disconnect: you're saying, it's infinite!... eeeeexcept it stops at 5. To my way of thinking, saying it stops at 5 is already cheating, because if it stops at 1 on one end and 5 on the other end, it's not really infinite, is it? It's limited and therefore in a sense *finite* in that it can only use five numbers.

If we were to use numbers to represent an infinite reality, I would've thought that we must by definition have no limits on the numbers we choose to use, if we're talking about infinite. Like, an infinite representation using only numbers would by definition have to include all numbers, positive and negative, odd and even, real and imaginary, all integers and complex numbers and everything in between; it would go on forever in all directions, with no finite "bookends," so to speak (1 and 5). And there would be an infinite amount of 1's and 2's and 6's and 10's and an infinite amount of each negative number and imaginary number and so on... or else it wouldn't be an accurate representation of "infinite."

However, funnily enough, your italicized "could" made me realize what you're trying to say more than the reiteration of the example itself: "just because the universe is infinite in one sense doesn't mean it's infinite in all senses." ("Sense" of course isn't the right word here, but you probably get what I'm trying to say. Perhaps it would be better to say that "just because the universe is infinite along one dimension (space) doesn't mean it's infinite along infinite dimensions...?") I think this is what you're saying, although you use the word "exhaustive" instead of "infinite along infinite dimensions," when all along I've used "infinity" synonymously with "exhaustive" or "without any limits at all."

So I can see that if we're only talking about being infinite along one dimension or whatever, then infinite space, for instance, is indeed not enough to imply an all-inclusive set. So what would be sufficient for all things (and non-things) to exist? What kind of infinity am I thinking of that implies supreme inclusivity? Is there a term to address this concept (because "infinity" obviously doesn't quite cut it), or am I just wading in too-murky waters here? Are we talking about infinite space and infinite time and infinite [insert vague dimensions that I know nothing about here]? Infinite infinitivity? Okay, that sounds dumb, I'm just trying to be clear because I'm not so sure that I'm being clear at all.

A triangle is defined as having three sides. So saying a four-sided triangle is the same as saying, "a four-sided, three-sided polygon." It is an improper use of language.

Yes, yes, of course... believe me, I do understand this. But there is still a part of me that wants to say "just because my puny mind can't comprehend the existence of something that has only three sides while still *somehow* having four sides doesn't mean that it's not possible." I mean, I don't honestly think it is possible, but as I said, a part of me romanticizes that this could simply be a function of the human mind's inability to think outside the proverbial box rather than a testament to the supreme infallibility of logic. Which is why I joked that we would go mad if we ever actually comprehended a "three sided and yet four sided" thing, to get across the idea that it's possible (although extremely unlikely) that such a thing could actually exist whether or not we understand it.

But I do of course understand what you are saying, which is basically summed up here:

Basic logic just assumes one thing: logic is consistent. That is to say, whenever you have a definitive statement, that statement is always either true or false. We may not always know which, but it is always one or the other. By only allowing statements in the logic that are either true or false, the laws of logic that can be derived are absolute and inviolable.

One can potentially consider logics that allow for ambiguous or meaningless statements, but often it is easier to just not allow those statements.

And yet the bolded has always bothered me, because yes it is easier but not necessarily correct. The fact remains that a contradiction can't really exist... except, of course, by some kind of magic or supreme omnipotence beyond my ability to understand. Which is kind of what I was getting at. I always think of these things in the context of a supreme omnipotence - if there was a supremely omnipotent God, could he draw a square circle? Could he make a burrito so hot that even he couldn't eat it? Of course not, that doesn't make any sense... except that maybe - just maybe - he could. Because, duh, he's frickin God isn't he? He could, theoretically, create a reality that we perceive, that seems to behave in a certain way but that isn't at all indicative of how things might actually be outside our sphere of observation.

But whatever, I do understand that it's kind of pointless to talk about things in such a way, we can only use what we have (or what we can observe, or what we can comprehend.) It's just easy for me to talk like this considering that there is just so much that has been shown to be incorrect as our observational capabilities have grown that it's hard for me to accept that anything at all is set in stone. I mean, I even recall recently reading an article about a paper by somebody or another postulating that gravity doesn't really exist. It was full of concepts and equations that I don't know enough about to properly ponder, so I didn't really try, and I guess the whole idea has gotten resistance from some other smart people, but I can only assume the original writer of the paper is pretty smart too and is convinced of the work, so I guess only time will tell if he/she/they can prove their thoughts or not. But in this sense, who knows what might be proven as a falsehood, given enough time?

Anyway, it seems to me this whole thing does indeed prove that the universe isn't infinite in totality; it isn't infinite along all dimensions or whatever (but again, I don't really know how to properly say what I'm trying to say here) or else I'd have an infinite amount of past and future and present "me's" (and an infinite amount of everything else) occupying the entirety of an infinite amount of space. To use the example of a line again, a line running east to west that goes on forever will never, ever, go north or south. Nor will it ever go up or down. It can never escape its own boundaries of being just a flat, unbending line, and I have trouble with infiinity being used in the context of something that has such obviously finite boundaries.

So reality might be spatially infinite, but that doesn't mean it's not finite in the sense that it's still limited to certain configurations (only 1 through 5; only east to west.) I would still ask though, if there is a term to properly describe this concept of an all-inclusive infinity, because I have a feeling that "infinite along all dimensions" isn't really saying what I mean to say; I can only hope you understand what I'm trying to get at. Is "exhaustive" all we have for that? Maybe I could say then that while the universe may be infinite, it's not possible for the universe to be exhaustive, or else everything would exist all at once in some unimaginable blur of... well, everything at once. Would this be correct, and if so, is there a better way to say it? If incorrect, what assumptions am I making here that I shouldn't be?

I actually came up with my own term for an "all-inclusive infinity" a long time ago when I was trying to *prove* a theory (again using this term loosely since I'm a half-wit in these matters) that reality doesn't need an observer to exist on it's own. (In other words, that reality can exist whether or not God is watching, because I heard that some very smart physicists were beginning to think that he or someone must be observing or else we wouldn't exist, and that got me thinking.) I don't remember what the term was, but it sounded cool. Dimensional Infinitum, or something like that. Forgive me, I tend to pull these things out of my ***.

Anyway, the whole idea that something needs to be observed in order to exist has never sat well with me, so I came up with the aforementioned and half-baked theory one time when I was quite ill and admittedley feeling a bit loopy; I lovingly refer to this theory as "Masturbational Existentiality."

***Much of what follows will likely be nonsense, so read on at your own risk; however, I feel compelled to share this simply because I can and because no one I know would ever humour (much less understand) me. But again, forgive me for being such an amateur and for my illusions of grandeur. (Plus my computer crashed a while back and I lost all of the nonsense I'd written on the subject, so this is all from memory and as such, probably a bit more wishy-washy than I would hope.) I can only hope that what I'm about to say is at least entertaining, in some fashion or another.***

My theory - M.E. for short - postulated and attempted to prove, among other things, that:

1) Reality *is* whether or not anyone *else* is watching (measuring/observing, whatever)
2) In order to exist in observable reality, something must be capable of observing itself
3) Particles have free will, and so does a tomato
4) Reality is finite in some sense, and because of this, everything in reality could be defined in terms of a tomato

The tomato thing is intentionally ludicrous, but this is all, of course, tongue-in-cheek, or I wouldn't call it Masturbational Existentiality. (The name is taken from the fact that *if* some kind of observance is indeed necessary in order to exist, and *if* something can exist all on it's own, without any outside observation, then something that exists must be capable of observing (interacting with) itself... interacting with itself, get it? Masturbational Existentiality. Also, I'm essentially stroking my own ego by even pretending to think competently about things like this, so there's a double meaning there: I'm stroking myself. I'm sorry, but I still find all of this funny and yet deadly serious at the same time. ZOMG an existing contradiction!)

I used the following and quite logical statement to *prove* that no outside observation is necessary for reality to exist, all on it's own:

If there is a rock, than there is a rock.

I still laugh at myself every time I write this, because it still seems quite inarguable while still of course being nonsense. I mean, if there's a rock, then there is indeed a rock, right? Conversely, if there is not a rock, then there is not a rock; it's still true both ways, which, if I recall, is important when dealing with logical statements. I've no doubt that anyone who's fluent in logic will gladly inform me that there is some name for this type of ridiculously stupid obviousness, and it's probably not one said with fondness, and yet I can't help but detect a whiff of profundity there. Though perhaps it would be clearer for me to say, "If there is only a rock, then there is only a rock." To be more precise, it doesn't matter if there is anything else with which to use as a frame of reference; if a rock exists, then dammit, it exists. (I ultimately changed the rock to a tomato, because I found that funnier, which is how the poor tomato became involved.)

And to say that "if reality is finite you could describe it in terms of a tomato" is simply to say that if one could *somehow* observe all of a finite reality at once, and furthermore, had an intelligence far greater than and could calculate infinitely faster than the greatest theoretical supercomputer, then it should theoretically be possible for this intelligence to assign a value to every property of everything in existence as it relates to everything else in existence. So while a tomato has the obvious properties of being "red" and "soft" and "vegetable", so must I have some kind of less obvious value for these properties, even if my value is zero or even negative (or even imaginary? I've never really understood the concept of imaginary numbers, although I've never really put much effort into understanding them.)

So I figured that if you must pick a "ground zero," so to speak, with which to find common denominators for the entirety of reality, you might as well start with a rock, or better yet, a tomato-why-not.

Of course, above I only mentioned physical properties. I first starting thinking of all of this by assuming that if there was a supremely omnipotent God, one who could observe all of a finite reality at the same time, he could potentially see everything as one huge, unimaginably complicated and constantly changing mathematical equation. My "redness," my "softness," but also, since we're talking about a supremely omnipotent God who observes *all* of reality, we have to include "my love for my cat," "my anger at being splashed by that puddle," "this thought I'm thinking right now;" emotions, thoughts, and all sorts of intangible things that I can't observe but that are a part of reality as we know it nonetheless (and as such is a part of what a supremely omnipotent God should be able to observe and therefore assign a value to.) Because it seems to me that even thoughts, emotions, etc. exist in some sense, even if I can't prove that, and even if they're neither observable nor measurable. I mean, I'm fairly certain I'm thinking right now...? (Or maybe I just think that I'm thinking... errrr... *head assplode*)

Of course, if I understand what I'm saying correctly here (admittedley doubtful), than this approach would necessitate finding what would probably be close to an infinite amount of common denominators (properties?) between observable (what we ourselves can observe) and unobservable reality. (To be clear, I'm saying that "observable reality"+"unobservable reality" = "reality," the totality of which a supremely omnipotent God could observe). Which isn't really possible, but theoretically, as I stated before, if there was such a thing as infinite wisdom combined with supreme perception, it seems it could be possible if you realize that most of the values assigned to the properties of intangibles would have a negative or zero (or possibly even imaginary?) value when applied to tangibles, and vice versa... but those values would still, in some sense, exist. Err, maybe.

It got really out of hand when I considered that consciousness, and by extension, free will, as part of the totality of reality, would have to have a place in this huge mythical equation describing all of reality. I then decided that it would be ridiculously impossible for me (or possibly anyone who's not completely insane) to write "free will" as an equation. But then I got into reading about "choice functions" (or whatever they may be called, something about an infinite number of bins and deciding which bin to place a package in or something to that effect) and that's where I gave up because I was in danger of losing my mind (and didn't really understand what I was reading anyway, since the more complex "formulas" in math are basically just sentences and truths written in a language I don't know how to read.)

It is also interesting (to me, anyway) to note that *if* some kind of observation is in fact necessary to exist, and *if* in fact a rock that exists does so with or without an outside observer and by implication must be observing itself in order to exist, then I have assumed a certain amount of consciousness on the part of the rock. Err, excuse me... tomato. (I called it "awareness," rather than consciousness, to make it sound less stupid, but really, it might as well be the same thing.) But this is when I started thinking that if free will exists, than everything within reality must have some kind of value for free will, including the tomato and all of it's smallest particles - and also, in order to exist, particles must have some kind of fundamental awareness of themselves even if it's so miniscule that we could never hope to comprehend or measure it. (I also became fond of thinking that the reason the smallest observable particles sometimes seem to flicker in and out of existence (I read this somewhere, I believe) is simply because they aren't aware enough of themselves or their environment to understand the difference between existing and not existing. Sometimes, they cease to exist because they choose to, but more importantly, because they don't really know any better.)

I just realized that I'm basically expounding on the classic "I think therefore I am," although really I'm saying that "A particle thinks, and therefore it is... except that sometimes it doesn't, and therefore, at those moments, it isn't." This is probably all nonsense, but dammit, it's poetic nonsense. But I made the leap that if true, then choice, or free will, these intangible things, may *be* the Higgs Boson (that's the thing that gives matter its form, right? That thing we can't seem to find? If I recall correctly and if I'm not being too simplistic.) What I mean to say is that maybe we can't find it because we're looking for an intangible, a choice: matter is able to take a certain form simply because its smallest particles, in some rudimentary sense, develop enough of an awareness to continue existing, and then, in some abstract sense, choose to take a particular form.

Anyway, sorry, I'm rambling and off-topic here, and I better stop with this because I'm beginning to confuse myself, which is probably a sign that I'm delusional. Maybe this all belongs in a different thread... perhaps "humour?" I really didn't intend to go into all of that, but once I started I couldn't help but try to explain myself. I mean, it's not often I get the chance to show to physics (or logic) buffs just how little I actually understand about their respective fields. I'm quite sure that most of the above is nonsensical and makes conclusive leaps that it shouldn't, and I probably contradict myself without realizing it, but I'm not convinced that this is because I'm wrong, it may just be that I'm incapable of proving or disproving anything because I don't know enough to detail logical steps from premise to conclusion. But I do realize that both could be true; I may be in danger of being insane, and completely unschooled, but I'm fairly certain I'm not stupid.

Against my better judgement, and at the risk of embarassing myself, I'm about to hit "submit reply." Just do me the favor of laughing with me, and not at me. (And sorry for the novel; that's just what I do. I've always been under the impression that the more ways I can repeat myself, the clearer I will be. It's a condition.)

 

[Sage Lee]

Deuterium, I saw that you posted while writing the above, and have not yet had time to do more than skim. I'm going to have to read it a few times before responding to any of it, because although I think I get the gist of most of what you are trying to explain, I still got a little lost amongst the gobbledygook. (You only thought you knew the meaning of that word till I showed up...)

But if you'll excuse me, the poker staking forum I belong to is having an interesting discussion about why or why not God exists, what he would be like, and the overall nature of good and evil. I'm currently trying to explain why I think it might not be possible for Heaven to be "better" than our current condition. (My attentions tend to wander, so, like the butterfly, I must float...)

 

[Sage Lee]

Also, while waiting for a response in that other forum (sometimes I wonder if any of those guys actually ever play poker) I couldn't help but click a "related thread" link that I found below, and I found this guy talking about his theory of infinite-infinite:

https://www.physicsforums.com/showthread.php?t=65278

This is basically what I feel is "disproven" by the simple fact that all of reality isn't happening everywhere and at the same time, over and over again. There HAS to be some kind of limit to reality, or it would be an inconceivable and chaotic blur of all possibilities happening at once, running into each other (occupying the same space and time. And everything else.)

In my humble opinion, anyway. I can't prove that. Sigh.

 

[Chalnoth]

Okay... honestly, I kind of rolled my eyes (at first) when I saw that you just rehashed the same example as had been stated previously, using a finite list of numbers (except this time you said "1 through 5" instead of "only even numbers". Because this is where I suffer a disconnect: you're saying, it's infinite!... eeeeexcept it stops at 5. To my way of thinking, saying it stops at 5 is already cheating, because if it stops at 1 on one end and 5 on the other end, it's not really infinite, is it? It's limited and therefore in a sense *finite* in that it can only use five numbers.

It doesn't stop, though. It keeps repeating over and over again, endlessly.

In a real-world scenario, this would be like there being an observable universe somewhere far away that is absolutely identical to our own. If the universe is infinite, in fact, we know this must be the case, because due to quantum mechanics there are only a finite number of possible configurations. So if it is infinite in space, then the real universe would actually behave very much like the repeating number line, except that the repetition would be more chaotic than orderly.

If we were to use numbers to represent an infinite reality, I would've thought that we must by definition have no limits on the numbers we choose to use, if we're talking about infinite. Like, an infinite representation using only numbers would by definition have to include all numbers, positive and negative, odd and even, real and imaginary, all integers and complex numbers and everything in between; it would go on forever in all directions, with no finite "bookends," so to speak (1 and 5). And there would be an infinite amount of 1's and 2's and 6's and 10's and an infinite amount of each negative number and imaginary number and so on... or else it wouldn't be an accurate representation of "infinite."

From quantum mechanics we find that the total number of possible configurations of a given region of the universe is finite. It's a very large number, but a finite one nonetheless.

Yes, yes, of course... believe me, I do understand this. But there is still a part of me that wants to say "just because my puny mind can't comprehend the existence of something that has only three sides while still *somehow* having four sides doesn't mean that it's not possible."

Well, no, because in this case a triangle is an abstract mathematical construct. It isn't a real object. Because it is an abstract mathematical construct, with a very specific definition, we can say absolutely that it doesn't have four sides.

And yet the bolded has always bothered me, because yes it is easier but not necessarily correct.

No, that's not the right way to look at things. Our choice of logic is more or less arbitrary. One choice of logic is no more or less correct than another. But one choice may be more useful than another under certain situations.