An actual infinite number of marbles

[honestrosewater]

Set theory is a good example. There are several versions of it, not just one, and each version is based on different axioms. Here is a link in case you're interested: http://mathworld.wolfram.com/SetTheory.html

Yep, I'm somewhat familiar with this. Do you still think there is confusion within each model?
If two different sets of axioms (and inference rules) produce the same set of theorems, why does it matter which of those theorems were used as axioms?

Because invalid axioms lead to contradictions. When you can use an axiom to prove that both a statement and its negative are true, you have an invalid axiom.

Not always, but we don't get much exposure to axioms that are known to be invalid. But as a simplistic example consider these two axioms:

- any number can be divided by any number
- zero is a number

You can use those axioms to prove that 2 = 3 (2x0 = 3x0). Now there's nothing wrong with the two axioms taken by themselves, they are invalid simply because they are not consistent with the other axioms involved in the proof (such as, for instance, the axiom that any number multiplied by zero equals zero).

Okay, we were just using different words for the same thing. What you call invalid, I call inconsistent.

I was talking about what the concept of infinity means in mathematics. It's not unlike the situation when mathematicians were faced with the square root of -1. They found a way around the problem, but they didn't know what it meant until imaginary numbers could be used to solve real problems. So we have a convenient way to deal with infinite quantities, but we haven't yet found a way to apply it to real problems. Because of that, some mathematicians believe the concept should be thrown out in favor of granular mathematics ("no infinitesimals")

Okay, I think I just misunderstood your original position, and you've cleared it up.

My mother language is not English, please forgive my misspellings. Hopefully the meaning should be clear from the context.

Don't worry, it wasn't misspelled. I'm just aware of them having slightly different meanings and wasn't sure what you intended. It's clear now.

 

[Hurkyl]

You can safely say that x - x = 0 without knowing what x is.

No, you cannot. Without knowing that there is a binary operation "-" that operates on objects of x's type, and that that type has an object labelled "0", you can't even utter "x - x = 0".

(Doing so would be tantamount to picking 5 random english letters, putting them together, and claiming you have a word)

Not always, but we don't get much exposure to axioms that are known to be invalid. But as a simplistic example consider these two axioms:

- any number can be divided by any number
- zero is a number

You can use those axioms to prove that 2 = 3 (2x0 = 3x0). Now there's nothing wrong with the two axioms taken by themselves, they are invalid simply because they are not consistent with the other axioms involved in the proof (such as, for instance, the axiom that any number multiplied by zero equals zero).

Actually, you're not quite there: having all numbers being equal is perfectly consistent. In fact, if you don't adopt the axiom that 0 is different from 1, such a structure is a field! (Called the zero field)

But I will admit I'm nitpicking at this one.

Because invalid axioms lead to contradictions.

If that is what you mean by "invalid", then you are patently wrong when you say:

Contrary to what you seem to imply, there is a lot of disagreement amongst mathematicians regarding ... the validity of some axioms

Or, at least I claim you're patently wrong: would you care to give an example of such a disagreement?

 

[Johann]

Without knowing that there is a binary operation "-" that operates on objects of x's type, and that that type has an object labelled "0", you can't even utter "x - x = 0".

So when people had no symbols, how did the first symbol get created? I very much doubt we got them from God, so we must necessarily have made them up.

(Doing so would be tantamount to picking 5 random english letters, putting them together, and claiming you have a word)

I'm confused. Certainly words are created by picking random letters and putting them together. What exactly do you mean?

Actually, you're not quite there: having all numbers being equal is perfectly consistent.

Consistent with what axioms?

would you care to give an example of such a disagreement?

You can read up on the controversy around the axiom of choice, which all mathematicians must accept but many do so grudginly. Many mathematicians believe the Banach-Tarski paradox implies the axiom of choice is not valid. I won't give you references because I'd have to google them up, and I'm sure you can do that by yourself.

 

[selfAdjoint]

So when people had no symbols, how did the first symbol get created? I very much doubt we got them from God, so we must necessarily have made them up.

Well, Chomsky, auteur of the generative grammar engine theory, used to teach that we have this mechanism for symbolic communication in our brains, which we evolved at some point (maybe pieces at a time over several species transitions). He now theorizes that all we really needed to evolve was a general recursive ability.

 

[Hurkyl]

I'm confused. Certainly words are created by picking random letters and putting them together. What exactly do you mean?

If I picked random letters and put them together, I could get something like qzrlabnt. This is clearly not a word. If the symbol x denotes something of a type upon which we have not defined a binary operation "-", then saying x - x is analogous to writing "qzrlabnt" in an English sentence.

Consistent with what axioms?

There are a lot of choices of axioms for which it will be consistent, but I already gave a specific example: the zero field. If you have not adopted the axiom that the additive and multiplicative identities must be unequal, then the zero field satisfies all of the axioms of a field.

You can read up on the controversy around the axiom of choice, which all mathematicians must accept but many do so grudginly. Many mathematicians believe the Banach-Tarski paradox implies the axiom of choice is not valid. I won't give you references because I'd have to google them up, and I'm sure you can do that by yourself.

You are inconsistent with your usage of "valid"! One of the great theorems of abstract set theory is that the axiom of choice is independent of the axioms of ZF (Zermelo-Fraenkel set theory). Among other things, this means:

If by using the axioms of Zermel-Fraenkel set theory together with the axiom of choice you are able to derive a contradiction, then you are able to derive a contradiction without using the axiom of choice.

In other words, using your terminology, if ZFC (ZF + Axiom of choice) was invalid, then ZF itself must be invalid.


As an interesting aside, if you happen to prefer to do things constructively, then the axiom of choice is actually the theorem of choice.

 

[Johann]

If I picked random letters and put them together, I could get something like qzrlabnt. This is clearly not a word.

When the first word was created, what did make it a word? As far as I can tell you are arguing that mathematical symbols have intrinsic meaning. That is nonsense.

But we have gotten way, way out of topic. You may start another thread if you want, but I think we should drop this discussion. It has nothing to do with infinity or marbles.

 

[Temporarily Blah]

In order to label the infinite number of marbles, you would have to start at 1, and keep going up a number to label the next one.

This is simply a disguised "will adding reach infinity?" problem, only you're labelling instead of adding.

Oh, and the answer is no, because you can always add 1 more number.

 

[Hurkyl]

When the first word was created, what did make it a word? As far as I can tell you are arguing that mathematical symbols have intrinsic meaning. That is nonsense.

No. Mathematical symbols mean exactly what they're defined to mean. Without knowing what x is, you cannot say anything about x - x because you have no clue if that string is defined or not.


(P.S. it is generally poor etiquette to have the last word in the same breath you're suggesting that the discussion be dropped)

 

[Johann]

Mathematical symbols mean exactly what they're defined to mean.

But symbols are defined in terms of other symbols, and the whole thing is completely undefined. It's not as simple as you're trying to make it, or perhaps I'm missing your point.

Without knowing what x is, you cannot say anything about x - x because you have no clue if that string is defined or not.

But x-x=0 is part of the definition of 'x'!

it is generally poor etiquette to have the last word in the same breath you're suggesting that the discussion be dropped

I was just trying to stay on topic, not to have the last word. We may continue the discussion if you wish.

 

[Hurkyl]

But x-x=0 is part of the definition of 'x'!

You said "You can safely say that x - x = 0 without knowing what x is.", so how can you possibly know that x-x=0 is part of the definition of 'x'?

There are lots of ways this statement can be wrong.

(1) There may be no such thing as "-" that operates on the things to which x may refer, and no such thing called "0" to which things can be tested for equality.

For example, x might be used to refer to a point in an abstract topological space.

(2) There may be a thing called "-", but no such thing as "0".

For example, x may be referring to a point on an elliptic curve. There is no such thing as "0" in this case, but we do have that x - x = ∞.

(3) There may be a thing called "0", but no such thing as "-".

For example, x may be referring to an element in a poset. There is no such thing as "-" for elements of a poset, but there is a thing called "0" that refers to the smallest element of the poset.

(4) There may be things called "0" and "-", but it's not true that x-x=0

For example, x might refer to a set of numbers, and you're using "-" on such things to refer to the pairwise difference. (i.e. A-B = {a-b | a in A, b in B}) You might be using "0" to refer to the empty set, or maybe the set containing zero, but either way, x - x = 0 is generally false.

(5) There may not be such a thing as "x".

I could easily be working in a theory whose alphabet doesn't even have the symbol "x", and thus you can't possibly be saying anything about "x", not even "x=x".

 

[Hurkyl]

(moderator hat on)

If the others would so desire, I could split the semantic discussion off into its own thread. It doesn't seem necessary to me, though, since it seems that the discussion has just found its way here.

(moderator hat off)

 

[Rade]

First I should distinguish between an actual infinite and a potential one. Aristotle once suggested the terms potential infinite and actual infinite. Roughly speaking, a potential infinite is a collection that grows towards infinity without limit, but never actually gets there. Take for instance a finite past starting from a beginning point. The universe gets older and older (1 billion years, 2 billion years...15 billion years) but no matter how far you go into the future, you’ll never actually reach a point where the universe is infinitely old. You can always add one more year. In contrast, an actual infinite is a collection that really is infinite.Here's my question: suppose there is an actual infinite number of marbles, each one numbered (1, 2, 3...). Will there be a marble labeled "infinity"?

In relation to the question asked, and the definition of "actual infinity" implied in the question, the answer is yes. Below is my argument why I hold this to be true.

Note that you define "potential infinite" as "never actually gets there". So, although you are unclear on this point in your post, logically you must then define "actual infinite" as "actually gets there".

But what does this mean, to either never actually get there vs actually get there ?

Consider that infinity has a boundary, but that in the case of a potential infinity, the boundary is not stationary, it keeps growing as the "collection" you refer to grows--thus it is clear that in such a condition the collection "never actually gets there".

Next, consider the same boundary but now do not allow it to change, keep it constant, and then, since marbles are an entity with mass, let us say the size of Planck's constant, then at some point in the future you will reach the limit of the "actual infinity" boundary condition where there is only as much space remaining that is the size of Planck's constant--then you can add only one last marble to this type of actual infinity--and so you can get out your pen and put the name on that last marble = "Infinity".

[Edit] It may also help to view "actual infinity" as a metaphor, whereas "potential infinity" is literal. That is, in order to grasp what Aristotle was trying to say with these two terms, one must take the view that "actual infinity" can be reduced to a "thing" with a never changing boundary condition. I do not know if there is a form of mathematics that describes such a metaphor for infinity ?

 

[Johann]

You said "You can safely say that x - x = 0 without knowing what x is.", so how can you possibly know that x-x=0 is part of the definition of 'x'?

If x-x=0 is true by definition, then it doesn't matter what 'x' is. I don't know if you see "x-x=0" as a well-defined mathematical operation or as an arbitrary sequence of symbols, but in any case my point is that there is not much of a difference.

There are lots of ways this statement can be wrong.

I wish we would concentrate on the meaning of infinity in mathematics. My claim was that the concept is not free of ambiguity. For instance, an infinite sum is something that can't be properly understood given our notion that someone or something must actually calculate each term. We just wave the magic wand and say "if this could be done, this would be the result", but we don't really address the issue of how it could be done. So when we apply infinite sums to solve problems like Zeno's paradox, we may end up with a nagging feeling that the solution is not really solving anything to anyone's satisfaction.

On a more fundamental level, I'm disputing the notion that mathematics makes perfect sense. It doesn't. A lot of it does, but not all of it.

I could easily be working in a theory whose alphabet doesn't even have the symbol "x", and thus you can't possibly be saying anything about "x", not even "x=x".

I do think this is way too abstract and doesn't have anything to do with much that is relevant. Surely you can argue that you can't explain quantum mechanics to a savage in the jungle because their language probably lacks the required concepts. I fail to see how that has any bearing on quantum mechanics itself. Likewise, you can come up with as many scenarios as you want in which basic algebra fails to apply, but again I fail to see what that has to do with basic algebra.

 

[Hurkyl]

If x-x=0 is true by definition, then it doesn't matter what 'x' is.

A true statement. (By virtue of the fact that the hypothesis is false)

For instance, an infinite sum is something that can't be properly understood ...

You have the mathematics entirely wrong. "if this could be done, this would be the result" may have been the motivation behind the modern concept of an infinite sum, but we've had a completely rigorous definition of what an infinite sum means in calculus for over a century. (And since, have devised other situations in which one might want to use something best described as an infinite sum, and have written precise definitions for all of those, too!)

And this is a general procedure in mathematics. From a vague, intuitive notion one distills the properties of interest, and then formulates a precise definition. Those who continue to focus on the vague, intuitive notions and neglect to see the way these notions are captured mathematically are doing the subject, and themselves, a disservice.

Likewise, you can come up with as many scenarios as you want in which basic algebra fails to apply, but again I fail to see what that has to do with basic algebra.

I do think our discussion is related: I'm trying to emphasize just how important precision and definition are.

You have been very sloppy, and have thus made incorrect statements, and have been refining what you meant to say, apparently without acknowledging the fact that what you did say was wrong. (You had meant to say something similar to "If x refers to a (real) number, then without knowing just what real number it is, we can say that x-x=0")

I think this attitude is also being applied to the case of the infinite: you look at mathematics in a sloppy manner, and thus you see sloppiness.

 

[Tisthammerw]

how can it be concrete? you have a set of marbles that is not finite? you have assigned them all labels? note that *you* need to state what your labelling is, not us.

I did, to some extent at least. It includes all natural numbers (1, 2, 3...). It seems though that if this is to be an actual infinite, the marbles would be all natural numbers plus a transfinite one. There would be a marble would be labeled ω, and to the right of it would be all of the natural numbers 1, 2, 3... at least that seems to be the best (perhaps only?) way for there to be an actual infinite number of marbles given a labeling scheme (excluding similar variants, like having ω and another marble labeled ω+1 etc.).

in any case, i can hardly think that this is a mathematical issue since you've not given a mathematical definition. you ideas seem more based upon the idea of counting things "in the real world", if so come back to me when you've got a collection of marbles that is not finite.

Uh, this is a collection of marbles that is not finite; it is an actual infinite. Perhaps you missed my first post?

oh, you're one of those "gettign to infinity but not reaching it" people. why didnt' you say earlier?

If you read my first post, you'd see that did say so earlier. I explicitly explained the difference between an actual infinite and a potential one.

as for potential v. actual infinity, well. let me put it this way (not, you understand, that this is a mathematcal opinion): the set of natural numbers i suppose to be an 'actual infinity' since it is infinite.

That sounds good on paper, but given the original labeling scheme (1, 2, 3, 4...) is this really an actual infinite or a potential one? The natural numbers go towards infinity without limit but never seem to actually get there. Hence the apparent paradox. So far it seems the only way to have an actual infinite would be the alternate labeling scheme I described. What do you think?

in any case, i can hardly think that this is a mathematical issue since you've not given a mathematical definition. you ideas seem more based upon the idea of counting things "in the real world",

True, this is in regards to what is metaphysically possible and not necessarily what is mathematically possible. Some mathematicians have claimed that while actual infinites are valid mathematically, they cannot exist physically. So perhaps this issue does fall into the realm of philosophy.

if so come back to me when you've got a collection of marbles that is not finite.

Again, perhaps you should read my first post.

 

[Tisthammerw]

If x-x=0 is true by definition, then it doesn't matter what 'x' is. I don't know if you see "x-x=0" as a well-defined mathematical operation or as an arbitrary sequence of symbols, but in any case my point is that there is not much of a difference.

The problem is that x-x=0 doesn't apply for all things. It works if x is a natural number, but not if x is infinity. You have to define what kind of entity (natural number, real number etc.) x represents before one can accept the statement as necessarily true.

The subtraction of infinites leads to contradictions. Suppose x represents infinity; more specifically an infinite number of marbles labeled 1, 2, 3, 4...

What happens if I take away all the marbles? Infinity - infinity = 0. Sounds good so far. But what happens if I take all the numbers except one? I take a way an infinite number of marbles, and here infinity - infinity = 1. What if I take all the numbers except two? I take a way an infinite number of marbles, and here infinity - infinity = 2. This can go all the way up to infinity. What if I take all the odd numbered marbles? I take a way an infinite number of marbles, and here infinity - infinity = infinity. I take the exact same amount of marbles each time and get different answers (note that mathematically, each is a countable infinite and has the exact same type of infinity, i.e. all the infinites I subtract in these examples are "equal" to each other). Incidentally, contradictions like these are why some mathematicians claim that actual infinites cannot exist in reality. A small albeit brilliant minority even claim that actual infinites shouldn't be used in mathematics.

On a more fundamental level, I'm disputing the notion that mathematics makes perfect sense. It doesn't. A lot of it does, but not all of it.

Indeed. See Hilbert's Hotel for another example.

 

[Hurkyl]

Tisthammerw: suppose that all of your marbles were labelled with either a natural number or the symbol ω, and that each label were used exactly once. I come along and take away the marble with the label ω. Now, all you have left are marbles labelled with natural numbers. Would you assert that you now have finitely many marbles?

The natural numbers go towards infinity without limit but never seem to actually get there. Hence the apparent paradox.

The paradox is apparent only because you insist on associating this fact with something almost entirely unrelated. (Actually, it is more appropriate to say that the natural numbers are unbounded. To speak about divergence towards infinity, you have to talk about a sequence. Of course, all sequences of natural numbers diverge to infinity, but that's beside the point)

The magnitude of the individual numbers within the set of natural numbers has absolutely no bearing on the cardinality of the set of all natural numbers. As a more obvious example, consider the set of all real numbers in the interval [0, 1]. This is also an obviously infinite set, but the numbers are all bounded! They don't even diverge to infinity!

 

[Royce]

Hurkyl, this is a bit off topic; and, goes back quite a bit to a previous discussion we had. I have two questions that still drive me nuts when I think about them.

If infinity is indeterminate how can it be a mathematical limit?

and

If the set of whole numbers is infinite how or why is it considered a closed set by definition?

 

[Tisthammerw]

Tisthammerw: suppose that all of your marbles were labelled with either a natural number or the symbol ω, and that each label were used exactly once. I come along and take away the marble with the label ω. Now, all you have left are marbles labelled with natural numbers. Would you assert that you now have finitely many marbles?

No.

The paradox is apparent only because you insist on associating this fact with something almost entirely unrelated.

What is "this fact"? Remember, I didn't say that the natural-number-marble-labeling-scheme causes the marbles to be of finite quantity, only that it might be a potential infinite instead of an actual one. Does an actual infinite require a ω here? It might be the case given the definition of a potential infinite and an actual infinite. (Note to non-mathematicians: ω is the infinity that comes “right after” all the natural numbers.) Unfortunately, the difference between cardinality and ordinality makes matters confusing.

The magnitude of the individual numbers within the set of natural numbers has absolutely no bearing on the cardinality of the set of all natural numbers.

Maybe so (at least in the case regarding the set of all natural numbers), but is the marble set (all labeled with natural numbers) a potential infinite or an actual one? I agree that the cardinality of set N is אo, but that is not the issue here. As I said, the difference between a cardinal infinite and an ordinal infinite makes matters confusing. Perhaps אo--when describing natural numbers--is actually a potential infinite whereas ω is invariably an actual one?

Of course, both the set of all natural numbers (set N) and set N + ω would both have the same mathematical "magnitude" of infinity: אo, even if one set is a potential infinite and the other is not.

 

[Tisthammerw]

Hurkyl, this is a bit off topic; and, goes back quite a bit to a previous discussion we had. I have two questions that still drive me nuts when I think about them.

If infinity is indeterminate how can it be a mathematical limit?

In calculus, the infinity ∞ really isn't an actual infinite but rather a potential one. Note that when denoting bounds, a parenthesis is always used instead of a bracket, e.g. [1, ∞) which means it includes 1 and goes towards infinity (though never actually getting there).

If the set of whole numbers is infinite how or why is it considered a closed set by definition?

I'm sure there's some proof that explains why, but I don't recall what it is.

 

[Hurkyl]

There are two major uses of the word "infinity" in calculus:

The first is simply as part of a phrase. When one says "this sequence diverges to infinity", it's really just a phrase -- there is no object called "infinity" which is referenced by this phrase. Similar things can be said for things like "the sum from n = 1 to infinity" et cetera.

However...

It was realized that one could formalize and make practical use of actual points at infinity (which is just another phrase, BTW). There is a topological space called the extended real line which consists of the real numbers with two additional honest-to-goodness points that we call +∞ and -∞. Then, we have honest-to-goodness intervals like [1, +∞] which consists of the real number 1, all real numbers bigger than 1, and the point +∞. (Which is the same as saying all extended real numbers that are no smaller than 1 and no larger than +∞)

This allows us to translate phrases with the word infinity in them into statements about these actual objects we call ±∞. For example, we have the function:

f(x) = x²

defined on the reals. Consider:

$$\lim_{x \rightarrow +\infty} f(x) = +\infty$$

Before we started using the extended real line, this was merely a formal equation which was used as a convenient and suggestive shorthand for what was really meant. However, when using the extended real number line, this is an honest-to-goodness equality -- ±∞ are just as good as any other point in the extended reals, and there is nothing special about them. (Aside from the fact they're the endpoints of the extended real line)

And, here, it makes exactly as much sense to say that f(±∞) = +∞ as it would to say that for the function g(x) = x²/x, g(0) = 0.


And, just to make it absolutely clear, these two additional points that lie on the extended real line, ±∞, have absolutely nothing to do with the concept of infinite sets.

If the set of whole numbers is infinite how or why is it considered a closed set by definition?

That can't be answered until you say what you mean by "closed". Some particular meanings might be:

The whole numbers are closed under incrementing, addition, and multiplication. (meaning that if we increment a whole number, or add or multiply two whole numbers, we get a whole number as the answer)

The whole numbers are not closed under things like subtraction or division. (Because these operations will give things that aren't whole numbers, or might not even be defined at all!)

Taking the whole numbers as a topological space (in any way you would like to do so), it would be true to say that the whole numbers are topologically closed within themselves. (Of course, such a statement is true for any topological space)

The whole numbers are closed under the "sup" operation on finite sets, becuase if we have a finite set of whole numbers, they have a least upper bound that is a whole number. (i.e. the maximum of the set)

The whole numbers are not closed under arbitrary sup's. For example, the sup of the entire set of whole numbers does not exist. However, within the ordinals, we have that the supremum (= least upper bound) of the whole numbers is ω. This is an instance where the supremum of a set exists, but the set does not have a maximum. This is quite analogous to the fact that the interval (0, 1) has a supremum, 1, although that interval has no maximum element.

emember, I didn't say that the natural-number-marble-labeling-scheme causes the marbles to be of finite quantity, only that it might be a potential infinite instead of an actual one.

Well, here are two problems you face:

(1) You have to explain how the act of removing a single member of an "actual infinity" would yield a mere "potential infinity". (Of course, you couldn't explain it without actually defining the terms)

(2) You agree that you would not be left with a quantity that is not finite. Therefore, by definition, that quantity is infinite. In the English usage of the term, it would be quite appropriate to say the quantity is actually infinite.



A lot of times when people talk about "potential infinity", they are referring to some sort of "changing" thing. For example, people like to talk about how natural numbers keep "growing" larger and larger, and I'm sure I've heard that called a "potential infinite".

The problem here is that there is nothing changing at all: each natural number is what it is, and the set of natural numbers is what it is, they are not dynamic, changing things.

What people seem to mean, I think, is that there is some sort of "process" where they "start" with the number one, then move onto the number two, then three, and so forth, and then confuse the set of identified numbers with the process, or even worse, speak about some sort of "end result". (Similar to some people like to think of 0.999~ as being some sort of process that starts with 0.9, then moves onto 0.99, and so forth, and then confuse the actual honest-to-goodness number 0.999~ with this process)

 

[BoardTracker]

Here are some variations on the theme:
(a) What is the last digit in the decimal representation of $\pi$
(b) A lightbulb is switched on for 1/2 minute. It is switched off for the next 1/4 minute and on again for the 1/8 minute following that and so on. At the end of one minute, is the light on or is it off?

Not really the same :)
I'll try to explain why I think that
(a) If $\pi$ has a finite number of digits, then there would be a last digit (while in the question originally brought here there is definite infinite number of items). Even if $\pi$ has no finite number of digits (which I am not sure if proven so... is it?) the question of "what is the last digit of an endless digit string" (which is a self contradicting question), is not the same as "is there a item numbered "infinite" in an infinite numbered items. Which as most people think here has a clear simple "yes/no" answer (which is no, btw).

(b) At the end on the minite the bulb will be off since it will be burnt ;) a bulb has a finite number of times at can be switched on/off. I bet if you tried your experiment, most bulbs would burn :D
Another difference with your (b) question is that there most certainly IS a state for the bulb at exactly 1:00 after the experiment starts. Wether it can be labeled on or off (assuming as I say it does not burn), is another question.
From a scientific pov, the bulb will actually be on :D since by the end of the minute the on/off switching will be infinitely fast that the coil will anyway be hot enough in the inbetween state to rule the bulb "on"

-

As to the orginal question..
You did hint that the numbers are integer positive (by giving the example of 1,2,3..). If so, there will be no "infinite" tagged marble, since "infinite" is either text/string OR a concept. Both not integer positive.
If you did not mean its integer positive, I would say that..
If there IS any logic in the tagging of the marbles, we need to know it in order to answer your question.
If there is NO logic and tags are totally random, there most definitely a marble tagged "infinite" since in an infinite number of random (all possible) events, one even will certainly be "infinite".
If you don't tell us what the logic behind the tagging, then the answer will be "dunno, start looking at each one and see" or... the more correct one "maybe"

 

[Johann]

(b) A lightbulb is switched on for 1/2 minute. It is switched off for the next 1/4 minute and on again for the 1/8 minute following that and so on. At the end of one minute, is the light on or is it off?

I think the answer must be that the bulb will be both on and off, or neither on nor off. And if you don't understand it, that's your problem, it doesn't mean mathematicians are capable of uttering nonsense

 

[Locrian]

No, the answer is that as you approach the one minute mark, you are flashing the bulb on and off so rapidly that it quickly burns out the filament. At the one minute mark the bulb will therefore be off.

 

[Temporarily Blah]

If there was a number labeled Infinity, you would not be able to comprehend it.

If there isn't one, there isn't.


We can't see infinity, so the question makes no sense.

 

[AKG]

Not really the same :)
I'll try to explain why I think that
(a) If $\pi$ has a finite number of digits, then there would be a last digit (while in the question originally brought here there is definite infinite number of items). Even if $\pi$ has no finite number of digits (which I am not sure if proven so... is it?) the question of "what is the last digit of an endless digit string" (which is a self contradicting question), is not the same as "is there a item numbered "infinite" in an infinite numbered items. Which as most people think here has a clear simple "yes/no" answer (which is no, btw).

This isn't the case. $\pi$ does have infinitely many digits, so it indeed makes no sense to ask of the last one. However, if you have an infinite number of items, then depending on the infinity, there may or may not be an item labelled with an infinity. The infinite set $\omega$ contains no "infinite" element, the infinite set $\omega + 1$ does.

(b) At the end on the minite the bulb will be off since it will be burnt ;) a bulb has a finite number of times at can be switched on/off. I bet if you tried your experiment, most bulbs would burn :D
Another difference with your (b) question is that there most certainly IS a state for the bulb at exactly 1:00 after the experiment starts. Wether it can be labeled on or off (assuming as I say it does not burn), is another question.
From a scientific pov, the bulb will actually be on :D since by the end of the minute the on/off switching will be infinitely fast that the coil will anyway be hot enough in the inbetween state to rule the bulb "on"

This really doesn't answer the question. No one is asking whether the bulb will be physically burnt out or whether it will be so hot that it will glow anyways. You're not supposed to be pedantic, you're supposed to try to see his actual point and respond to that. If we must, we can consider a function f : [0, 1] --> {0, 1} where f(x) = 1 iff there are some a and b in [0, 1] with a < b such that x is in [a, b) and such that the bulb was supposed to be on during the time interval from when a minutes had passed to before b minutes had passed. Yes, there is indeed a state for the bulb at the one minute mark, and the function f does indeed have a value f(1), but there is not enough information to determine the state or the value. If I say I have a function:

f : [0, 1] --> [0, 1]

with f(x) = x for x in [0, 1), does it then make sense to ask what f(1) is? Could you (Tide) tell me what f(1) is given only that f(x) = x for x in [0, 1)?

If there is NO logic and tags are totally random, there most definitely a marble tagged "infinite" since in an infinite number of random (all possible) events, one even will certainly be "infinite".
If you don't tell us what the logic behind the tagging, then the answer will be "dunno, start looking at each one and see" or... the more correct one "maybe"

This isn't right. If you randomly select an infinite number of labels, this doesn't mean that you will end up selecting all possible labels. You might pick all labels, and any given label can possibly be chosen, but there is absolutely no reason whatsoever to believe that every possible label will be chosen: "infinite" might never be chosen.

 

[DaveC426913]

Well, since an actual infinite number of marbles cannot be contained within our universe, the conditions are impossible, even in theory, to meet.

 

[AKG]

Why is it impossible to meet, even in theory? I am out of the loop when it comes to modern physics, but isn't it generally held that the geometry of space is flat? Given this, wouldn't the universe have infinite volume? Even if I'm wrong about the geometry, isn't it at least theoretically possible for the universe to be infinite in expanse, and thus large enough to "house" infinitely many marbles. Of course, there is the issue of where to find that much matter to build so many marbles, but it depends what we mean by "in theory." However, there is a far better way to get around all of these potential snags. Build the marbles so that each marble is half the size of the previous one. There will be infinitely many marbles, but they will only occupy a finite volume and assuming uniform density amongst all the marbles (well, all we require is that the density of the marbles doesn't grow geometrically from one marble to the next, which is a more-than-reasonable assumption), they will only require a finite amount of space. Actually, if we aren't too fussy, then we can take any slab of marble and look at it as a collection of infinitely many pieces of marble. The first piece will be the left half of the slab. The next piece will be the next quarter of the slab. The next piece will be the next eighth of the slab, etc. We don't have the precision to actually break the marble into all of these pieces (in fact, it's not even a matter of precision, we don't have the time to do it - although that brings up the question of supertasks), but whether we can cut out a piece representing the (2^{n[/sub] - 1)th (2n)th or not doesn't effect the fact that it is there. Even if a (2n)th of the slab is smaller than Planck length, it still exists. Since we can give real-number measurements to regions of space, the regions of space with such measurements exist, even if physical things cannot occupy that space. Of course, at this point, you'd be right to question whether such a region can be said to even be one of the infinite marbles any more. However, if you divide the region of space containing the slab up as prescribed, then although there may be empty ones, or ones that don't contain full atoms, there must be infinitely many that contain something, otherwise there is an upper bound to the order of the piece that contains something, and there are infinitely many pieces afterwards which would have to contain nothing, but all these contiguous pieces would have finite volume, so the total volume of the slab would be less than the total volume of the slab, and that's obviously a contradiction.}

 

[Antiphon]

I'm loosing my marbles.

Infinity can't exist except as a concept. The largest number doesn't exist
even as a concept.

Edit: If you doubt this, please name any infinite attribute of any extant
thing (mathematical constructions excluded because they are concepts.)

 

[Johann]

Infinity can't exist except as a concept.

It's interesting that some (many? most?) languages don't even have a word for "infinity". It took me quite a while to understand that the English words "infinity" and "infinite" do not refer to the same concept. And I'm still puzzled as to why people think "infinity" is not nonsense. I guess if you learn a word early in life you can never think of it as meaningless.

 

[AKG]

Well "infinity" is a noun and "infinite" is an adjective, what took you so long to understand that a noun and an adjective don't refer to the same concept? However "infinity" and "infinite" are very closely related. They don't "refer" to the same concept but they are strongly related to the same concept. A set whose cardinality is infinity is an infinite set. I'm absolutely puzzled as to why you think infinity is nonsense. Why would it be meaningless? We can speak about it meaningfully in so many ways. If you aren't able to do so, that doesn't make the word meaningless, it just means that you don't understand the meaning.

Infinity can't exist except as a concept. The largest number doesn't exist
even as a concept.

Edit: If you doubt this, please name any infinite attribute of any extant
thing (mathematical constructions excluded because they are concepts.)

What in the world is this? What's the difference between a number existing "as a concept" and just existing? Or existing as something else? When you say, "there is no largest number" I guess you're referring to something like the real numbers. Do you agree that the complex numbers are numbers? The complex numbers are not ordered, it is meaningless to talk about the largest complex number, or even to talk about whether one complex number is larger than another, but what does that have to do with anything? "Infinity" is not defined as "the largest number" so just because the largest real number doesn't exist, that has nothing at all to do with whether infinity exists. How many numbers are there? Infinity.

And naming an infinite attribute of an "extant thing" is irrelevant. I can't see how any of your sentences are relevant to the discussion, or even how anyone of your sentences is relevant to any of your other sentences.

 

[arman]

I think it'd be wise to first try and agree on a term for infinity itself. I think that arguments in this discussion are more likely to occur from semantic clashes rather than actual disagreements.

I like to think of infinity as a concept, and by that I mean that it's not actually a number. Infinity is often defined mathematically as

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

What I find interesting is that another common definition is

A quantity greater than any assignable quantity

Which implies that infinity is an unassignable quantity. By this definition no marble would be the infinity marble.

I don't see why people are taking this to be an argument of physics, I don't think it has anything to do with physics. Who are we to say whether an infinity of marbles would fit into the universe? To solve such a question we would firstly need to be equipped with enough data to understand the size of the universe, then we would need a definition for an infinite quantity that is properly integrated into our mathematical system. We don't have either of those things.

 

[AKG]

arman, you are not using the proper notion of infinity. We are talking about ordering and counting objects, so we should use the numbers that are specifically used to order and count, the ordinal numbers and the cardinal numbers. There are indeed infinite (or transfinite) ordinal and cardinal numbers. The transfinite number $\omega + 1 = \{1, 2, \dots , \omega\}$ contains infinitely many elements, and contains an infinitieth element, or rather, $\omega ^{th}$ element, namely $\omega$.

 

[Tisthammerw]

Well, here are two problems you face:

(1) You have to explain how the act of removing a single member of an "actual infinity" would yield a mere "potential infinity".

Simple. One arrangement approaches infinity but (apparently) never actually gets there. The other actually has a transfinite value (omega).

(2) You agree that you would not be left with a quantity that is not finite. Therefore, by definition, that quantity is infinite.

But then what kind of an infinite is it? A potential or an actual one?

A lot of times when people talk about "potential infinity", they are referring to some sort of "changing" thing. For example, people like to talk about how natural numbers keep "growing" larger and larger, and I'm sure I've heard that called a "potential infinite".

The problem here is that there is nothing changing at all: each natural number is what it is, and the set of natural numbers is what it is, they are not dynamic, changing things.

It may seem that way, but don't forget how I defined the terms "actual infinite" and "potential infinite" in my first post. The collection seems to grow towards infinity but never actually gets there--or does it?

 

[Hurkyl]

One arrangement approaches infinity but (apparently) never actually gets there.

The numbers in the arrangement approach infinity. Actually, that's not technically appropriate because you have not arranged the numbers in any sort of sequence. (Of course, any way you do so, that sequence would then approach infinity)

The correct term is unbounded. I've also heard it called unbounded, but finite for emphasis.


However, as I've emphasized above, this is referring to the numbers in the arrangement, not the quantity of marbles.


Something to emphase: you are talking about "approaching" infinity -- this means you are speaking about some sort of variation. This would be appropriate when talking about the labels on the marbles (at least once you've arranged them in a sequence), because you can say something like "the value of the sequence" varies as you "increase the index".


However, talking about the quantity of marbles "approaching" infinity is completely wrong. The quantity of marbles is fixed; it is a single, unchanging value. It makes absolutely no sense to speak of it "approaching" anything.

But then what kind of an infinite is it? A potential or an actual one?

Until you give an explicit definition of "potential infinity" and "actual infinity", this question has no answer.

I spoke to a friend of mine who was first a philosopher, then got his Ph.D in mathematics, and asked him if the terms "potential infinity" and "actual infinity" made any sense to him. The only sense he could make out of it was Aristotle's introduction of the term (I think it was Aristotle), which more or less coincides with the definition we use today. Saying that a thing X is potentially Y means exactly that: X has the potential to be Y. In other words, X could become Y, or it may be possible for X to be Y, or something along those lines.

Then, saying that X actualizes Y means that X has actually attained the quality Y.

According to this notion, it is blatently obvious that the quantity of marbles is a actually infinite. (From the very statement of the problem)

The labels on the marbles would be potentially infinite, meaning that it is possible for a marble to have a label denoting an infinite ordinal. A "label on a marble" is a type of logical variable, and the things to which it could refer may or may not be infinite, because it may refer to "1", "ω", or many other things.

However, any specific label can be classified. The label "1" is clearly not infinite. The label "ω" is clearly actually infinite.


However, we can also specify that every marble is labelled with a natural number. Then, the labels on the marbles are not potentially infinite, because we know that each and every one of them is a finite number. Of course, we still have an actually infinite quantity of marbles (and of labels!), by the statement of the problem.


Saying that we have a "potentially infinite" quantity of marbles would merely mean that we do not know if the quantity has a finite or an infinite value.