An actual infinite number of marbles

[Tisthammerw]

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"? I suspect so, but I'm uncertain. What do you guys think?

 

[TD]

No, because that would imply that you would 'reach' infinity 'all of the sudden'. After 'x' marble blocks, one would be labeled 'infinity'?
The thing with infinity is, you never do 'get' there. There are infinite natural numbers, but you can't say after x numbers, we label one 'infinity'. As long as you're labeling numbers, you can (as you said) add another one. Infinity is used to describe a quantity which isn't finite, so byond any boundary.

 

[Tide]

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?

 

[chronon]

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"? I suspect so, but I'm uncertain. What do you guys think?

No, as you have stated it the collection of marbles is infinite, but each is labelled with a finite number. However, there is nothing to stop you adding another marble and labelling it 'infinity' or \omega (just as if you have 2 marbles you can add one and label it 3). You can then go on adding marbles labelled \omega+1,\omega+2 ... This is essentially what Cantor did in creating the ordinal numbers. See http://mathworld.wolfram.com/OrdinalNumber.html for more information.

 

[Hurkyl]

suppose there is an actual infinite number of marbles, each one numbered (1, 2, 3...). Will there be a marble labeled "infinity"?

This one is very easy. You said the marbles are labelled with positive integers. "infinity" is not a positive integer, thus there is no marble labelled "infinity".

 

[Tisthammerw]

This one is very easy. You said the marbles are labelled with positive integers. "infinity" is not a positive integer, thus there is no marble labelled "infinity".

Well, I didn't say the marbles were labeled with only positive integers; only that they were numbered. Given an actual infinite number of marbles, would there be a transfinite (e.g. omega) marble?

 

[Tide]

can you actually ever say that you have an infinite amount of some kind of object? And if so, when?

I think that could occur only in an infinite and unbounded Universe. I'd rather not think about what would happen if they were all in one place! :)

 

[Tisthammerw]

No, because that would imply that you would 'reach' infinity 'all of the sudden'.

I agree that an actual infinite cannot be formed via successive finite addition, but this isn't quite the same. This is an actual infinite, with no regard to its origin. Given an actual infinite number of marbles, would there be a transfinite (e.g. omega) marble?

The thing with infinity is, you never do 'get' there.

With some infinites that's true. Calculus deals with limits and in those cases it’s often potential infinites (symbolized as ∞) but there are actual infinites in mathematics symbolized by e.g. ω. (Note: there is a small albeit brilliant minority who claim that actual infinites shouldn't be used in mathematics and that only potential infinites are legitimate mathematical entities.) So if there's an actual infinite number of marbles...

 

[Tisthammerw]

No, as you have stated it the collection of marbles is infinite, but each is labelled with a finite number. However, there is nothing to stop you adding another marble and labelling it 'infinity' or \omega

But wouldn't the latter be the example of what I'm talking about? An actual infinite number of marbles (as opposed to a potential one--growing towards infinity but never actually getting there)?

 

[robert Ihnot]

Thinking of the Calculus, take a simple example, like the limit as n goes to infinity of 1-1/n. Obviously the limit is 1.

Well then what value do we substitute for "n" to reach that limit? I guess we could, formally speaking, add on to the number system the limit point called infinity.

However is that point well ordered? Not at all since we have no value one less than infinity. However, as it has been pointed out, Cantor did add things like omega +1, omega +2, etc.

 

[chronon]

But wouldn't the latter be the example of what I'm talking about? An actual infinite number of marbles (as opposed to a potential one--growing towards infinity but never actually getting there)?

If you have an actual infinity (\aleph_0)of marbles then you have a choice of how you order them. If you choose to label each with a positive integer then in won't be possible to say that anyone marble is the last in the ordering. However, if you kept one back then you could still label all of the others with the positive integers and define the extra one as the last (i.e. the \omega^{th})

 

[Hurkyl]

Well, I didn't say the marbles were labeled with only positive integers; only that they were numbered. Given an actual infinite number of marbles, would there be a transfinite (e.g. omega) marble?

In this case, again you've said nothing about ω in your original post, so you cannot possibly argue that there is a marble labelled ω. (Nor could you argue there is no such marble)

Let me rephrase what you're trying to ask:

Suppose each positive integer is used to label a different marble in a collection of infinitely many marbles. Are there any marbles that do not have a positive integer as a label?

And the answer is that there is not enough information.

(Note that if the labels aren't placed "consecutively", it is also possible for every marble to be labelled with a positive integer, and there to be unused labels)

 

[matt grime]

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"? I suspect so, but I'm uncertain. What do you guys think?

in the standard interpretation of what you have written, no: all marbles are labelled by the natural nubmers. why are you talking about marbles anyway? just take the set of natural numbers. This is an infinite set without a 'maximal element'. There infinite countable sets that do have maximal elements. and no i won't accept that you chose marbles because you wanted to make this concrete; who has an infinite number of marbles on them?

 

[Tisthammerw]

In this case, again you've said nothing about ω in your original post, so you cannot possibly argue that there is a marble labelled ω. (Nor could you argue there is no such marble)

I mentioned ω indirectly, suggesting (albeit indirectly) that there was ω quantity of marbles by saying I was talking about an actual infinite. (See below regarding a potential infinite.)

Suppose each positive integer is used to label a different marble in a collection of infinitely many marbles. Are there any marbles that do not have a positive integer as a label?

And the answer is that there is not enough information.

Well, I did say it was an actual infinite instead of a potential one. In a potential infinite, there would be no transfinite marble. The numbers would just keep getting higher and higher, growing towards infinity but never actually getting there.

 

[Tisthammerw]

If you have an actual infinity (\aleph_0)of marbles then you have a choice of how you order them. If you choose to label each with a positive integer then in won't be possible to say that anyone marble is the last in the ordering. However, if you kept one back then you could still label all of the others with the positive integers and define the extra one as the last (i.e. the \omega^{th})

Hmm, makes sense. Methinks the last one more correctly represents an actual infinite (whereas the former seems to represent more of a potential one). What do you think?

 

[Tisthammerw]

in the standard interpretation of what you have written, no: all marbles are labelled by the natural nubmers.

If we were talking about a potential infinite (a collection that grows without limit towards infinity but never actually gets there) I would agree. However, in the example there is an actual infinite quantity of marbles--suggesting the possibility of a transfinite (e.g. ω) marble.

why are you talking about marbles anyway? just take the set of natural numbers. This is an infinite set without a 'maximal element'.

And there is also no transfinite element, hence what you’re proposing seems more like a potential infinite rather than an actual one. My example was about an actual infinite.

and no i won't accept that you chose marbles because you wanted to make this concrete

Too bad; I did.

 

[matt grime]

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. we can label an infinite countable set purely with a finite number on each marble, which is what you have implied with your notation, or we may choose not to and label them with some ordinals greater than w , depends on what you want to do with them.

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. soemthing that can be described by the natural numbers and is not bounded would be a potential infinity. consider for example the finite sets of marbles, none of these contains an infinite number of marbles, but there is no bound to the size of the sets.

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.

 

[matt grime]

oh, you're one of those "gettign to infinity but not reaching it" people. why didnt' you say earlier? incidentally, to show you why this is nonsense, i can relabel your potentially infinite set of marbles so that it becomes an actually infinite set of marbles. take the one labelled 1, and write w on it, now take all the others and replace n with n-1. makes you think... or at least it ought to.

 

[rashoumon]

This one is very easy. You said the marbles are labelled with positive integers. "infinity" is not a positive integer, thus there is no marble labelled "infinity".

hmm then if infinity is a negative number...supposedly that it increases...i don't think they would reach a stage of infinity as said, there is always 1 more infront of another

 

[Hurkyl]

hmm then if infinity is a negative number

There is no negative integer (or real number) called "infinity" either.

 

[Hurkyl]

I want to make this important point:

Counting is not the act of labelling each object with numbers, starting with the number 1, and then looking for the largest label used.

It's a tribute to how nice the finite is that the two methodologies are equivalent. (and that counting is equivalent to answering the question of "How many?")

A great many mistakes are made by hastily generalizing to the infinite our knowledge of the finite. Here's a quote from the preface of a textbook I recently checked out of the library:

These volumes deal almost exclusively with infinite-dimensional phenomena. Much of the intuition that the reader may have developed from contact with finite-dimensional algebra and geometry must be abandoned in this study. It will mislead as often as it guides. In its place, a new intuition about infinite-dimensional constructs must be cultivated.

-- Fundamentals of the Theory of Operator Algebras, Kadison & Ringrose

It's not about counting, but the point it makes is true in general.

 

[Learning Curve]

I've just been reading along and have a question. Actual infinite is not concrete in the sense you can just add another number ( potential), but with labels such as (n-1),(n-2)?

If anyone could expand on that I would greatly appreciate it.

~peace

 

[Hurkyl]

All right, I'm going to punt this off to philosophy, since so few seem interested in discussing mathematics.

 

[chronon]

If we were talking about a potential infinite (a collection that grows without limit towards infinity but never actually gets there) I would agree. However, in the example there is an actual infinite quantity of marbles--suggesting the possibility of a transfinite (e.g. ω) marble.

You seem to be confusing an actual infinity with an infinite set with a largest element (which somehow 'completes' the set). This is wrong. Indeed, the distinction between potential and actual infinities is more philosophical than mathematical. For instance the rationals in [0,1] have a largest element, but the rationals in (0,1) don't. Nothing to do with whether you consider these to be potential or actual infinities.

 

[Vossistarts]

You seem to be confusing an actual infinity with an infinite set with a largest element (which somehow 'completes' the set). This is wrong. Indeed, the distinction between potential and actual infinities is more philosophical than mathematical. For instance the rationals in [0,1] have a largest element, but the rationals in (0,1) don't. Nothing to do with whether you consider these to be potential or actual infinities.

or is it that with a potential infinate you accept that it goes on infinately and leave it at that and go about your business. As an immortal handling the actual infinate you devote an infinate amount of time assigning numbers to the ever increasing lot of marbles youre counting.

 

[Johann]

I think the best that can be said about infinity and marbles is that, by thinking about the former one easily loses the latter.

 

[Vossistarts]

Im really not qualified to comment on mathematics so maybe I shouldn't. Still, it would SEEM to me that the answer must lay in the origins of the two concepts of potential and absolute infinites. Maybe Potential imparts the quality to whatever your talking about that it may and for all practical purposes does or can continue on infinitely and allows for a sort of placeholder into count to say so.(But perhaps it might not?) With the absolute infinite it says that it does in fact go on infinitely. That says to me that nothing can be done about the fact that it goes on infinitely. You don't have to continue numbering on and on neither do you have to create a placeholder of sorts saying that it may or does go on infinitely such as "Infinity". You already know this because you've decided for whatever reason that its an absolute infinite. There is no need to call any point infinity. I imagine you can simply leave off wherever the count stops for the moment or the equation leads you in any given situation for the moment KNOWING that there is infinitely more beyond that. Absolutely.

Infinity is rather conceptual to begin with isn't it?

Did I repeat the same things 15 times there? eek!

 

[Hurkyl]

Well, the mathematics is fairly simple once you get accustomed to it: if you ask precise questions, you can use the precise definitions to give precise answers!

IMHO, the only reason people have trouble with the infinite is because they try to reason from some nebulous, unarticulated concept they have floating around in their brain.

 

[Johann]

Well, the mathematics is fairly simple once you get accustomed to it: if you ask precise questions, you can use the precise definitions to give precise answers!

This is actually funny, because different mathematicians seem to have different, equally precise definitions of what infinity is. I don't think infinity is a well-defined term in mathematics at all. It seems to be rather a nuisance actually.

the only reason people have trouble with the infinite is because they try to reason from some nebulous, unarticulated concept they have floating around in their brain.

Because that is what infinity is: a nebulous, unarticulated concept.

When I learned that two line segments of different sizes have exactly the same amount of points, I was quite shocked. It seemed to me back then that a two feet long rod must have twice as many points as a foot-long one, no matter how you define "point". I could never get a satisfying explanation for what to me seems like an arbitrary rule, and I can't think of a way to define "point" which makes both the rule true and the existence of real rods possible. (I can think of a way to define "point" which makes the rule true but meaningless, and I suspect that is the definition of "point" mathematicians use)

Mathematics is not as clear and unambiguous as some people claim it is.

 

[selfAdjoint]

This is actually funny, because different mathematicians seem to have different, equally precise definitions of what infinity is. I don't think infinity is a well-defined term in mathematics at all. It seems to be rather a nuisance actually.



Because that is what infinity is: a nebulous, unarticulated concept.

When I learned that two line segments of different sizes have exactly the same amount of points, I was quite shocked. It seemed to me back then that a two feet long rod must have twice as many points as a foot-long one, no matter how you define "point". I could never get a satisfying explanation for what to me seems like an arbitrary rule, and I can't think of a way to define "point" which makes both the rule true and the existence of real rods possible. (I can think of a way to define "point" which makes the rule true but meaningless, and I suspect that is the definition of "point" mathematicians use)

Mathematics is not as clear and unambiguous as some people claim it is.

The things mathematicians do with infinity are perfectly well-defined and follow from consistent axioms. You mustn't confuse working on different subjects within the concept infinity with disagreements. You can't form a sufficient basis for analysis of this question from popular accounts.

 

[Johann]

The things mathematicians do with infinity are perfectly well-defined and follow from consistent axioms.

But that is beside the point. You can safely say that x - x = 0 without knowing what x is. My point was that mathematicians don't know what infinity is, not that they don't know how to apply the concept.

You mustn't confuse working on different subjects within the concept infinity with disagreements.

Contrary to what you seem to imply, there is a lot of disagreement amongst mathematicians regarding the exact meaning of some concepts, the validity of some axioms, the relationship between mathematics and reality, even the nature of mathematics itself.

You can't form a sufficient basis for analysis of this question from popular accounts.

I'm not basing my analysis on "popular accounts", I'm basing it in four years of post-secondary education on the subject.

 

[honestrosewater]

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

Could you give some examples of these different definitions and disagreements?

Why are they having to question the validity of axioms? I mean, how are you using validity? I can only think of it being applied to proof-related things, and every definition that I've seen makes an axiom proof of itself. I'm not very experienced with math, which may be the problem, but it seems like you're generally talking about 'real world', non-mathematical, philosophical matters.

BTW, are you intentionally saying infinity instead of infinite or infinitely many or such?

 

[Johann]

Could you give some examples of these different definitions and disagreements?

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

Why are they having to question the validity of 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.

I mean, how are you using validity? I can only think of it being applied to proof-related things, and every definition that I've seen makes an axiom proof of itself.

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).

it seems like you're generally talking about 'real world', non-mathematical, philosophical matters.

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")

BTW, are you intentionally saying infinity instead of infinite or infinitely many or such?

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

 

[selfAdjoint]

The fact that there are various axiomatic systems for what is vaguely called set theory does not prove that anyone of them is unsatisfactory: they are not in competition. Mathematicians who are interested in mathematical logic, or proof theory, or model theory, will learn several, and undergraduate courses teach the first few mentioned at your Wolfram link, and their interesting relations with each other. This is a perfect example of trying to find support for a preconceived idea online. It seldom works.

 

[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.