Is Infinity Real? Exploring the Concept of Endlessness

[bayan]

i know that Infinite can exist in mathematical world (As it is imposible to count the very last number at any given instance. i.e in a frozen time having an end.)

But when it come to reality is there anything such as infinite? I mean people used to believe that universe was infinite but now based on theory we say it has a limit. because (again based on theory, we say universe is expanding).

So as I said before is there anything such as infinite in real world? (not beig able to reach the end of it even if no time exists.)

 

[russ_watters]

If the universe is expanding and is going to expand forever, that's a natually occurring infinity.

 

[Tournesol]

The cosmos could be infinite now and expanding.

 

[sneez]

"The cosmos could be infinite now and expanding"

Infinite and expanding ? infinity + 1 = infinity or am i missing something ?

 

[Tournesol]

Things are getting futher apart. (On the cosmological scale. Rulers don't expand).

 

[<<<GUILLE>>>]

think in the longest number you can.

right now, that number is infinite for you.

but here comes your clever friend and tell you,

"Add one to that number"

Now, you have a bigger number than the old one, so this number is the biggest you know, and so, it is infinite for you.

I'm trying to say that infinite is a concept as well as zero, not a number.

 

[bayan]

what is the defenition of infinite? Isn't it defined as the gratest? (meaning that nothing bigger than that)

If so at a given instant you can mesuare the length of universe thus giving it a length which no longer is infintite.

However if infinite is defined by someother meaning please tell me so I can clarify this to myself

 

[Tournesol]

Infinite basically means without end. An very loose illustration of the expanding universe
would be

1,2,3,4...

1,3,5,7...

1,4,7,10...

note the dots!

 

[The Rev]

I have always thought that in a world of finites, infinity would be incompatible. For example, if the universe is infinite, and the planet Earth is finite, how does one express the proportion in size?

The Earth is $$\frac{1}{\infty}$$ of the universe?
So is any other object in the universe, if the universe itself is infinite.

It's a little like that proof that says because $$4*0=0$$ and $$3*0=0$$, then $$4=3$$

The addition of infinity into the equation makes it nonsense.

$$\phi$$

The Rev

 

[dextercioby]

Sorry to "crash the party",but let me add that,if physicists would be fully convinced that the mathematical infinite has (under certain circumstances,cosmology is one of them) physical meaning,then why would they search for theories whose equations would be infinite-free??

Daniel.

 

[Picklehead]

Would the infinite set of all even positive whole numbers, and the infinite set of all natural numbers be two different 'orders' of infinities? It seems silly, but wouldn't one be in some sense 'more' infinite then the other?

 

[Crosson]

Would the infinite set of all even positive whole numbers, and the infinite set of all natural numbers be two different 'orders' of infinities? It seems silly, but wouldn't one be in some sense 'more' infinite then the other?

Yes, there are different "orders" of infinity. The word for this is cardinality.

In order to talk meaningfully about infinite sets of different cardinalities, we need a definition.

"Two infinite sets have the same cardinality if they can be put into one-to-one correspondence".

In the example you gave, the natural numbers and the even natural numbers have the same cardinality. This is because their is a one to one correspondence, i.e. for each n in the natural numbers there exists 2*n in the even numbers.

So the answer is yes, different orders of infinity exist; and no, you haven't found one.

The set of natural numbers, integers and fractions all have the same cardinality. The real numbers, however, belong to a "higher order" of infinity.

 

[Picklehead]

Where do imaginary numbers fit into there?

How would we recognize an occasion when an infinity coming out of theoretical physics is in fact correct?

 

[Gokul43201]

Where do imaginary numbers fit into there?

Imaginary numbers are nothing much more special than the reals (multiplying a real by i gives an imaginary number) and clearly have the same cardinality. Complex numbers are the sum of a real and imaginary number (a special ordered pair of reals) and so they too have the same cardinality as the reals (in just the same way that rationals have the same cardinality as the integers).

How would we recognize an occasion when an infinity coming out of theoretical physics is in fact correct?

One way is from experiment.

 

[Gokul43201]

I'm trying to say that infinite is a concept as well as zero, not a number.

Both ARE numbers. "Infinity" is not a real number. Zero, most certainly is.

 

[Tournesol]

Sorry to "crash the party",but let me add that,if physicists would be fully convinced that the mathematical infinite has (under certain circumstances,cosmology is one of them) physical meaning,then why would they search for theories whose equations would be infinite-free??

Daniel.

There's a difference between extensive quantities (space and time) and intensive quantities (everything else).

 

[bayan]

As infinite is described as having no end, how can universe be infinite when there is an end to it?

You may not be able to reach that end any time soon but it has an end which is expanding. if infinite is no end how do we have an infinite+1? wouldn't the new number take place as infinite, leaving the old one as a number therefore making no number such as infinite?

What I mean is that You can calculate or measure the length or magnitude of something at a given time while it is said to be infinite.

 

[hypermorphism]

You may not be able to reach that end any time soon but it has an end which is expanding.

Please provide a source for the statement that the universe (not just the observable universe) has an end.

if infinite is no end how do we have an infinite+1?

The ability to write a string does not give it meaning. The string 1/0, for example, is undefined in the reals. You must give precise definitions in order to talk about such objects in relation to other defined objects. In the extended complex numbers, infinite + 1 = infinite. Or if you define "infinite" to be a number greater than all other numbers, then infinite + 1 is undefined in your system, unless you further extend your system in the fashion of the ordinals.

What I mean is that You can calculate or measure the length or magnitude of something at a given time while it is said to be infinite.

You're talking about some finite object. An infinite object does not have this property.

 

[Uno Lee]

Infinity
1. The quality or condition of being infinite.
2. Unbounded space, time, or quantity.
3. An indefinitely large number or amount.
4. Mathematics. The limit that a function is said to approach at x = a when (x) is larger than any preassigned number for all x sufficiently near a.

5. a. A range in relation to an optical system, such as a camera lens, representing distances great enough that light rays reflected from objects within the range may be regarded as parallel.
b. A distance setting, as on a camera, beyond which the entire field is in focus.

Is infinity an undesirable quantity when it comes up in equations?

 

[russ_watters]

Is infinity an undesirable quantity when it comes up in equations?

Sometimes yes and sometimes no: the whole point of calculus is using infinity and division by zero to your advantage.

 

[Uno Lee]

Sometimes yes and sometimes no: the whole point of calculus is using infinity and division by zero to your advantage.

So, if one could rework an equation to get rid of a zero division or cancel out an infinity somehow, it would be preferred?

 

[Icebreaker]

For some reason, it made me think of this theorem.

 

[fuzzyfelt]

Yes, there are different "orders" of infinity. The word for this is cardinality.

In order to talk meaningfully about infinite sets of different cardinalities, we need a definition.

"Two infinite sets have the same cardinality if they can be put into one-to-one correspondence".

In the example you gave, the natural numbers and the even natural numbers have the same cardinality. This is because their is a one to one correspondence, i.e. for each n in the natural numbers there exists 2*n in the even numbers.

So the answer is yes, different orders of infinity exist; and no, you haven't found one.

The set of natural numbers, integers and fractions all have the same cardinality. The real numbers, however, belong to a "higher order" of infinity.

This really does inflict brain pain! This post provoked me to do some reading, have i got this right?- even positive integers, all positive integers and all rationals have the cardinality of Aleph null; all real numbers (rational and irrational), and all points on a continuum or plane or in a higher dimension have the cardinality of Aleph one; and subsets of these are Aleph 2 and so on? and so the subsets are higher orders, which i think gives a lovely concept that 'the whole is no bigger then some of its parts'. Can you say the whole can be smaller, or is that nonsense? And, as someone asked, where does imaginary numbers fit in? (And, btw, what is the basic meaning of imaginary numbers?)
Then, reading on, i think it is said that the continuum hypothesis cannot be dispoved and cannot be proved but that Woodin introduced some logic that if it were correct then ch wouldn't be? And if my understanding so far is on track, being very fond of thinking in terms of continuums, how should it be replaced? By restricting how divisible the continuum is, or saying that they don't make sense at all?
Thanks in advance for your thoughts, I'm going to take some panadol now and lie down!

 

[arildno]

Just a comment: Finite sets can also be assigned a cardinality ("Size"), the aleph numbers are the cardinalities of various INFINITE sets.

"..and subsets of these are Aleph 2 and so on? and so the subsets are higher orders, which i think gives a lovely concept that 'the whole is no bigger then some of its parts'."
This is dead wrong; a strict subset always has a cardinality less than or equal to the cardinality than the original set.

The concept of cardinality gives us a very neat and new interpretation of "infinity":
A set "A" is infinite if and only if there exists a bijection between A and a strict subset S of A.

 

[fuzzyfelt]

'Cantor accepted that the well-known pairing-off principle, used to determine if two finite sets are equal, is just as applicable to infinite sets. It followed that there really are just as many even positive integers as there are positive integers altogether. This was no paradox, he realized, but the defining property of infinite sets: the whole is no bigger than some of its parts. He went on to show that the set of all positive integers, 1, 2, 3, ..., contains precisely as many members—that is, has the same cardinal number or cardinality—as the set of all rational numbers (numbers that can be written in the form p/q, where p and q are integers). He called this infinite cardinal number aleph-null, "aleph" being the first letter of the Hebrew alphabet. He then demonstrated, using what has become known as Cantor's theorem, that there is a hierarchy of infinities of which aleph-null is the smallest. Essentially, he proved that the cardinal number of all the subsets—the different ways of arranging the elements—of a set of size aleph-null is a bigger form of infinity, which he called aleph-one. Similarly, the cardinality of the set of subsets of aleph-one is a still bigger infinity, known as aleph-two. And so on, indefinitely, leading to an infinite number of different infinities. '

thanks Arildno, this is what I read, I think i should have said 'sets of all' meaning infinite, can you see where else I went wrong?

 

[arildno]

Aah, I see where you went wrong:
" Essentially, he proved that the cardinal number of all the subsets—the different ways of arranging the elements—of a set of size aleph-null is a bigger form of infinity, which he called aleph-one. Similarly, the cardinality of the set of subsets of aleph-one is a still bigger infinity, known as aleph-two. And so on, indefinitely, leading to an infinite number of different infinities. '"

The first part is gibberish, I don't understand what is meant there (or that which I think I understand, is flat out wrong).


What IS true, is that for a given set S, the power set of S, called P, has always a greater cardinality than S itself.

The power set is the set of all subsets of S.

 

[fuzzyfelt]

thank you very much for setting me straight Arildno, my head does still hurt, but a little less now thanks for your help.