Can infinite quantities exist? A philosophical approach

Seems like yet another confusion about the definition of "exist", which you conveniently don't give.

If you rephrase it as "Can infinite quantities be observed?" then the answer is negative. If you phrase it as "Can models with infinities in them fit the observations better then those without?", then the answer is affirmative. If you are interested in a metaphysical answer, such as "Do numbers exist?", then you have to be clear in what you mean by each term.

Moved to Discussion.

Reminds me of another question I read recently: "Has anyone really been far even as decided to use even go want to do look more like?" I may have better luck parsing your post if you chose to work on its formatting. Feedback you've received on LW in the past, to little avail. Avast! (?)

I'm also not sure about your apparently new concept of "brute distinguishability", my only association is "et tu, brute?" which of course is historically inaccurate.

Sorry, but I do not think this is a well written article. The formatting is strange and hard to read, and your points meander a lot. You also need to give coherent summaries a lot more. As written now, your post is hard to read, and I can't quite tell what points you are really trying to convey. Please take this as constructive criticism, and work to make your post better.

metacomment:

You seem to be getting a decent response, but the formatting screams "internet wingnut at his coming out party" to me.

I think most of this worrying is dissolved by better philosophy of mathematics.

Infinte sets can be proven to exist in ZF, that's just a consequence of the Axiom of Infinity. Drop the axiom, and you can't prove them to exist. You're perfectly welcome to work in ZF-Infinity if you like, but most mathematicians find ZF to be more interesting and more useful. I think the mistake is to think that one of these is the "true" axiomatization of set theory, and therefore there is a fact of the matter over whether "infinite sets exist". There are just the facts about what is implied by what axioms.

If you're worried about how we think about implication in logic without assuming set theory, perhaps even set theory with Infinity, then I agree that that's worrying, but that's not particularly an issue with infinity.

Then, on the other hand, you might wonder whether some physical thing, like the universe, is infinite. That's now a philosophy of science question about whether using infinite sets or somesuch in our physical theories is a good idea. Still pretty different.

Aside: your specific arguments are invalid.

• The indistinguishability argument, regardless of whether it's good in principle, is incorrect. For infinite X, X and X'=X \union {x} are distinguishable in ZF. For one thing, X' is a strict superset of X, so if you want a set (a "property") that contains X but not X', try the powerset of X. I'm not really sure what else you mean by "indistinguishability".
• In the relative frequency argument you do limits wrong. It can be the case that lim f(x) and lim g(x) are both undefined, but that lim f(x)/g(x) is fine.

First, I second the other's requests to define "exist".

Second, I don't understand the arguments.

But the identity of indistinguishables does apply to sets: indistinguishable sets are identical. Properties determine sets, so you can’t define a proper subset of brutely distinguishable things.

"Let A and B be brutely distinguishable points. Define the sets M and N as M = {A, B} and N = {B}. N is a proper subset of M."

What I have done wrong in the preceding quotation? It seems like something a mathematician could easily say.

To show that the existence of an actually existing infinite set leads to contradiction, assume the existence of an infinite set of brutely distinguishable points. Now another point pops into existence. The former and latter sets are indistinguishable, yet they aren’t identical. The proviso that the points themselves are indistinguishable allows the sets to be different yet indistinguishable when they’re infinite, proving they can’t be infinite.

1. You say that the points are brutely distinguishable and later you say that they are indistinguishable, which nevertheless you hold to be different properties.
2. Why are the sets indistinguishable? Although I don't particularly understand what predicates you allow for brutely distinguishable entities, it seems possible to have X = set of all brutely distinguishable points (from some class) and Y = set of all brutely distinguishable points except one. It is, of course, not a definition of Y unless you point out which of the points is missing (which you presumably can't), but even if you don't have a definition of Y, Y may still exist and be distinguishable from X by the property that X contains all the points while Y doesn't.
3. If the argument were true, haven't you just shown only that you can't define an infinite set of brutely distinguishable entities, rather than that infinite sets can't be defined at all?
4. What is your opinion about the set of all natural numbers? Is it finite or can't it be defined?
5. And how does the argument depend on infiniteness, after all? Assume there is a class of eleven brutely distinguishable points. Now, can you define a set containing seven of them? If you can't, since there is no property to distinguish those seven from the remaining four, doesn't it equally well prove that sets of cardinality seven don't exist?

The frequency of heads and of tails is then infinite, so the relative frequency is undefined.

The frequentist definition of probability says that probability is the limit of relative frequencies, which is the limit of (the number of occurrences divided by the number of trials), which is not equal to (limit of the number of occurences) divided by (limit of the number of trials). Note the positions of the brackets.

The sequence {2n/n} = {2/1, 4/2, 6/3, 8/4, ... } = {2, 2, 2, ...} has an obvious well defined limit, even if the limits of both {2n} and {n} are infinite.

A note about formatting: consider not copying a text from a word processor or a web browser directly to the LW post editor. The editor is "smart" and recognises the original font size and type and grey background color and whatever else and imports it to the post, which therefore looks ugly. I'd suggest copying to a Notepad/gedit-style editor first which kills the formatting and then to LW. (And emphasis is usually marked by italics, not red.)

To show that the existence of an actually existing infinite set leads to contradiction, assume the existence of an infinite set of brutely distinguishable points. Now another point pops into existence. The former and latter sets are indistinguishable, yet they aren’t identical. The proviso that the points themselves are indistinguishable allows the sets to be different yet indistinguishable when they’re infinite, proving they can’t be infinite.

Suppose I assigned each of the points different spacetime coordinates. Now another point pops into existence at a different spacetime coordinate. The two sets of points are distinguishable because one of them has a point at a spacetime coordinate that the other one doesn't. I don't see the contradiction here.

But if infinite quantities exist, then relative frequency should equal probability.

Um. Why?

In general, I agree with the other comments that this post is unclear and not well-written. In particular, I agree with shminux's comment that your definition of "exist" is unclear. You should taboo it.

As an abstract entity, infinity has been quite useful in describing the real world. Perhaps the question would be easier to answer if it were rephrased with "exist" tabooed.

My first thought upon reading this: You need to read Cantor, who showed that infinite sets can still have different cardinalities; it is entirely possible that some infinity N divided by some other infinity M is a finite quantity. You're trying to define infinity as a single thing; it's entirely possible for there to be multiple infinities.

Or, in other words, infinities are far, far messier than you're presenting them here, and your proofs are relatively naive in regard to the work that has already been done on the subject. I would suggest starting with Cantor; his is hardly the last word, but as old as it is, it is one of the best words. Kronecker, a contemporary of Cantor, is a decent read for the infinity-skeptic side of the equation, but didn't have much substantive to offer.

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My recollection is that The existence of infinite sets is neither proven nor contradicted by the basic operation available in set theory. You need an additional axiom to get them, and if you want it the additional axiom is consistent with the prior ones.

This means that if the above argument were formalized, there must be a step which effectively assumes no infinites. Likely in a definition of a term.

It's not clear to me that MWI has infinite worlds rather than merely 'a mindbogglingly huge number, so large one may reasonably approximate it as infinite for most purposes'.