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# Infinity And The Axiom Of Choice

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I am curious if there is any student of objectivism who knows about abstract math and metamathematics, and perhaps philosophy of math. For more than a year now I've been wrestling with the problem of what is a number, particularly, how does one account for the infinity of numbers. In a related question, how does one account for the very spooky Axiom of Choice? Both of these concepts seem to imply a kind of Platonism.

If anybody is familiar with this topic, I'd appreciate it. If anybody wants to look into it, the first link below is something I've already written on another (slightly dead) objectivist forum at MySpace that covers the question of infinity. The second link is my rough-and-ready discussion of the Axiom of Choice. But if you're not familiar with the philosophy of mathematics, it may be hard to provide insight.

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There really isn't an "infinity" of anything. Infinity is simply a concept of method NOT a metaphysical existant. The correct term is that the set of all numbers is unbounded not "infinite".

All the "spooky" Axiom of Choice means in reality is that to get meaningful answers one must input data drawn from reality. And that arbitrary non-reality based "choices" give one...gasp... arbitrary outputs.

Welcome to the forum.

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Well I tried that maneuver too, but the problem is you're left with a "method". So what is the ontological status of this method? Another thing that seems like a Platonic form? And why is it that mathematics today treats infinity like a true object, and gets quite useful results in doing so?

And I'm sorry, but I don't understand your answer to the Axiom of Choice. You don't put anything into it. It puts very defined, un-arbitrary results into the real number-line. You simply cannot produce an algorithm that will produce all real numbers. Which implies that some other function does it--an abstract function that sounds like a Platonic form.

Thank you and good to meet you.

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There just obviously isnt a well-ordering of the real numbers, therefore the Axiom of Choice cant possibly be valid

But on a slightly less flippant note, I'm ambivalent towards the AoC. Philosophically speaking, I agree with the constructivists - talking about mathematical objects 'existing' even though we havent managed to find them yet is nonsenscal, and sounds more like theology than science. But at the same time, some of the results which follow from the AoC are incredibly sexy, and the non-constructive proofs are often very elegant (the non-constructive proof for the existence of Hamel bases almost blew my mind the first time I saw it, and some the things you can do with the Baire Category theorem are very cool). Its annoying, because intuitively, I feel that some of the theorems which follow from the AoC _should_ be true (eg Baire Category, Hamel bases, countable union of countable sets being countable), but I also feel that some should be false (well-ordering/Vitali theorems are 2 that come to mind). Its really a case of deciding whether being able to prove some very nice theorems justifies bringing in a set of extremely dubious philosophical assumptions, along with some counterintuitive results. Personally I'm still undecided, although I tend to lean towards pragmatism here :/

The worst part about it is that it tends to leave me with a feeling of fundamental uncertainty. When you use the AoC to prove that X exists, I always find myself wanting to ask "yes, but does it ACTUALLY exist?". Would it actually be possible, for instance, to find a well ordering of the reals and write down an explicit definition of it? Well, the AoC tells us that there must be one 'somewhere', but what ultimate grounds do we have for believing the AoC is true? Similarly, the AoC tells us that we will never be able to find a vector space without a basis. But why cant we do this? What if we did? I dont think I could ever have the same degree of confidence in a result which required the AoC as I could in a result which was proved by more 'standard' methods, just because there always remains the possibility that the AoC itself might not be true (whatever 'true' means here).

Edited by Hal
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I havent done much set theory (other than the basics you need for doing actual math) so someone else might correct me on this, but I'm fairly sure that you dont need the AoC to show that the cardinality of the reals is greater than the cardinality of the rationals. I'm 95% certain that the proof you have given in this link, for instance, doesnt invoke the AoC at any stage. I'll try to explain why, although its pretty abstract and confusing.

Youre assuming, in order to deduce a contradiction, that you can find a 1-1 correspondence the reals and the natural numbers. Now, you want to construct a real number that isnt on the list, by selecting one decimal place from each of the listed numbers. But you dont need the AoC to do this, because you are giving an explicit rule for choosing the decimal places (youre saying "pick the nth decimal place from the nth number on the list"). You only need the AoC if youre trying to make a choice without an explicit rule (eg, if you wanted to say "pick a random decimal place from each real number"). To use a fantastic example given by Bertrand Russell, if you had infinitely many pairs of shoes, then you wouldnt need the Axiom of Choice to select one shoe from each pair, because you can give the explicit rule "always pick the left shoe". But if instead you had infinitely many pairs of socks, then you would need the AoC to select a sock from each pair, because the socks in a pair are indistinguishable hence you cant give an explicit rule for making the choice - youre now saying "just pick one - I dont care which". One of the key points here is that once you've chosen your choice set of socks/shoes, you _know_ which shoes are in the shoe set (it will be all the left ones), but youve no idea which particular socks are in the sock set, since the choice was arbitrary/random. This is why the AoC is fundamentally non-constructive - you cant explicitly specify the set you produce with it, all you can say that it exists. You dont know _which_ socks got chosen, you only know _that_ some got chosen.

And this lies at the heart of the controversy over the Axiom of Choice. If I have infinitely many sets of integers, then I can explicitly construct a set which contains one integer from each set, by giving the rule "choose the smallest integer from each set". And this doesnt need the AoC. Its an entirely constructive process, because I know which integer from each set will have been chosen (the smallest). But if I were choosing one real number from infinitely many sets of reals,I can no longer say "pick the smallest real number from each set", because there isnt guaranteed to be a smallest real number (eg, the set of all reals greater than 0 doesnt contain a smallest element). So if I'm not able to actually specify a way to make the choice, and I cant name a single element of the resulting choice set, why am I justified in assuming that such a chioce set exists? Well, this is where I invoke the AoC, and say that it just does. And again, like the sock example, I've no idea which particular real numbers will make up this set, because the choice was by definition arbitrary/random.

Note that this is intimately connected to the Well Ordering Theorem (which is equivalent to the axiom of choice) - if the reals can be well ordered, then I can specify an explicit rule for making the choice by saying "pick the minimal real number from each set, where 'minimal' is defined by the well ordering", and this is now guaranteed to exist. But here this just moves the problem back one level, because the Well Ordering Thereom is itself non-constructive - it just says that a well-ordering exists, without saying what this well-ordering is. So again, we have no idea which real numbers actually got chosen by the choice function (and you need the AoC or something equivalent to prove the well-ordering theorem is true).

edit: Here's a good clarification of what the AoC is and isnt saying. As it hints at various points, the controversy ultimately boils down to what 'exists' means in a mathematical context.

Edited by Hal
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First, to be clear, the Axiom of Choice (AC) only exists for dealing with infinite sets. You don't need AC for finite sets. If you reject infinite sets altogether, then AC isn't an issue, you can prove theorems to take its place.

AC is an *axiom* so it is really telling us something about how infinity behaves in the system. If it doesn't make sense you have to pause and think hard: Is my issue really just tantamount to me being concerned that I can't prove the axiom as a theorem?

I find it interesting that AC raises all the problems, but you don't hear much about the Axiom of Infinity (AI) (which, as I indicated above, is the thing that causes us to think we might need AC).

You should consider AI and AC as a paired set of axioms telling us that infinite sets exist and they behave a certain way. If you don't like infinity at all, get rid of both of them. If you think infinity ought to behave differently, introduce other axioms.

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I find it interesting that AC raises all the problems, but you don't hear much about the Axiom of Infinity (AI) (which, as I indicated above, is the thing that causes us to think we might need AC).

I think it comes down to intuitive plausibility. There are obviously infinitely many natural numbers (whether you want to define this in terms of potentiality or whatever, and yes I realise I'm glossing over the difficulties), so an axiom of infinity seems reasonable. But the axiom of choice (and its equivalents) have very little intuitive content. In a sense, there are independent grounds for accepting AI, but we only accept AC because we like the things we can do with it.

Edited by Hal
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I think it comes down to intuitive plausibility. There are obviously infinitely many natural numbers (whether you want to define this in terms of potentiality or whatever, and yes I realise I'm glossing over the difficulties), so an axiom of infinity seems reasonable. But the axiom of choice (and its equivalents) have very little intuitive content. In a sense, there are independent grounds for accepting AI, but we only accept AC because we like the things we can do with it.

The problem is intuitions start breaking down the moment you leave the natural numbers.

It starts with the rationals where any notion of successor is gone (since between any two rationals there is another rational, so you can't talk about the "next rational" after a rational A). And then it is all downhill from there.

Perhaps a case could be made for the naturals being the only intuitively plausible infinity, but we really want to work in the other substantially less intuitive infinities.

The axiom of infinity doesn't exist to get us to the set of all natural numbers, but rather to other infinities. In fact I suspect that if the set of all naturals was the only infinity you wanted to allow, you could get it without the axiom of infinity in some acceptable way (potential infinities...yadda yadda), or at least use a weaker axiom of infinity.

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I am curious if there is any student of objectivism who knows about abstract math and metamathematics, and perhaps philosophy of math. For more than a year now I've been wrestling with the problem of what is a number

I probably rambled on too much about the AoC so I'll be brief with this. The problem I have with saying 'mathematical objects are just concepts' is that it fails to explain what we actually do when we do mathematics. An example will hopefully clarify what I mean. There was a time when mathematicians wanted to know whether continuous nowhere-differentiable functions (CNDFs) existed (dont worry about what these things are, its just a random example). So, they spent quite a lot of time trying to find one. Now, these people were not looking to see whether the 'concept of a CNDF" existed. This wouldnt make sense; they already knew that the concept of a CNDF existed - they had this concept inside their skull! They knew what CNDF's were, and they could define them quite easily, they just didnt know whether any actually existed (compare to having having the concept of a unicorn, but not knowing if unicorns exist. We know the concept of a unicorn exists, but that is not the question here). The mathematicians were not asking a conceptual question, so saying that CNDF's 'only exist as concepts' seems to missethe point - the concept of a CNDF would still exist regardless of whether CNDF's had mathematical existence (the idea of a triplet of integers satisfying Fermat's Last Theorem exists, even though there is no such triplet).

Saying that the concept of '4' is even, would be like saying that my concept of 'unicorn' has a horn. But this is obviously absurd - the concept is inside my skull, probably represented as a neuronal pattern in my brain.

Edited by Hal
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I am curious if there is any student of objectivism who knows about abstract math and metamathematics, and perhaps philosophy of math. For more than a year now I've been wrestling with the problem of what is a number, particularly, how does one account for the infinity of numbers. In a related question, how does one account for the very spooky Axiom of Choice? Both of these concepts seem to imply a kind of Platonism.

If anybody is familiar with this topic, I'd appreciate it. If anybody wants to look into it, the first link below is something I've already written on another (slightly dead) objectivist forum at MySpace that covers the question of infinity. The second link is my rough-and-ready discussion of the Axiom of Choice. But if you're not familiar with the philosophy of mathematics, it may be hard to provide insight.

If there were a point at infinity, wouldn't it imply that the trichotomy law was false, and thus the axiom of choice was false?

Of course, infinity is not a boundary. It is a lack of a boundary. This has already been said.

The idea of an absolute infinity, IE one where you would go so high that you'd be low again, and the idea of the axiom of choice are mutually incompatible.

I don't really understand why. Maybe it's just in ZF that they are incompabile. What would happen if you allowed the axiom of choice AND the point at infinity?

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I dont think the point at infinity has anything to do with either trichotomy or the axiom of choice. In mathematics, the term 'infinity' is used in various different ways ("completed infinite sets", "infinite as a limiting concept", etc). The 'infinity' in question when doing set theoretic stuff like the axiom of choice involves the cardinal/ordinal numbers of sets. However the 'infinity' involved in the idea of 'point at infinity' is different - its a purely geometric/algebriac notion. All youre essentially doing is adding an 'extra' point to the real numbers (or complex plane) via an extension and introducing some new axioms for it, similar to what you do when youre (eg) constructing the hyperreal numbers, or defining a plane with 2 origins (if you havent studied any abstract algebra, the previous sentence probably wont make sense). And formally speaking, this doesnt have anything to do with transfinite ordinals/cardinals, other than that the word 'infinity' is used in both cases (and of course, we could choose to call it something else).

In other words, having a point at infinity is perfectly compatible with ZFC.

edit: oops, it isnt a field extension since R with the point at infinity isnt a field I've never actually seen the construction so I'm not sure what details are involved in practive (although I should have guessed that since it obviously isnt going to have a multiplicative inverse :/))

Edited by Hal
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Hal, are you sure we haven't found a mathematical object? What evidence is there that the Zermelo sets don't describe such an object? And were we not to have found it, that does not mean it cannot exist. Maybe it cannot exist for some other reason, but not that one.

And yes, you don't need the AoC to prove magnitudes of infinity, and in fact I did not use it to do that (as you'll note, the proof of magnitudes of infinity assumes the undecidability of the decimals in a real number, which distinguishes it from a rational). But the proof of magnitudes demonstrates the nature of the AoC--that there is some function that is not definable but which orders the real numbers, *is* the property that makes them larger.

"If you reject infinite sets altogether, then AC isn't an issue, you can prove theorems to take its place."

True enough, punk, but infinity is rather useful and it's hard to understand the quantity of natural numbers any other way.

"If it doesn't make sense you have to pause and think hard: Is my issue really just tantamount to me being concerned that I can't prove the axiom as a theorem?"

No, I have no problem about proving it. I have a problem about what it implies.

"but you don't hear much about the Axiom of Infinity (AI) (which, as I indicated above, is the thing that causes us to think we might need AC)."

I brought it up.

"Saying that the concept of '4' is even, would be like saying that my concept of 'unicorn' has a horn. But this is obviously absurd - the concept is inside my skull, probably represented as a neuronal pattern in my brain."

Well this runs into the inherent trouble/contradiction of reductionism that Kripke pointed out in On Sense and Reference.

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aleph-O-- What evidence is there for anyone to consider the existence of some "infinite form" except an abstract "mathematical discribition"? Do you understand what the term arbitrary means and when it apply's to a given subject?

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The only evidence I know of that implies the existence of genuine mathematical objects is the fact that every attempt to explain it appears Platonic, even when attempting to avoid it. I am here looking to find a sensible non-Platonic answer.

Arbitrary:

1. based on personal whim: based solely on personal wishes, feelings, or perceptions, rather than on objective facts, reasons, or principles

2. randomly chosen: chosen or determined at random

3. law not according to rule: based on the decision of a particular judge or court rather than accordance with any rule or law

4. authoritarian: with unlimited power

5. mathematics assigned no specific value: used to describe a constant that is not assigned a specific value

The Axiom of Choice does not produce arbitrary results--the reals that it assigns are the same ones each and every time, it is not dictated by any whim, there is a law that prescribes them you just can't define it. Do you know what the term 'arbitrary' means when applied to a subject?

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I mean arbitrary in the Objectivist sense. A thing can be true, false, or arbitrary. With the first two obviously being able to be construed from the facts of reality. While arbitrary means that there is no evidence whatsoever to support such a statement. I.e., there are purple people eaters on venus who control the fate of the universe OR there is a God that control the Universe, etc.

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No offense, but how was I supposed to know what non-standard dictionary you were using to define 'arbitrary'? Furthermore, in what book and on what page does Ayn define 'arbitrary' as truth-valueless? And why define it thus when 'senseless' is an already accepted word for statements that lack truth-value?

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No offense, but how was I supposed to know what non-standard dictionary you were using to define 'arbitrary'?

That's why he explained what he meant when there was some confusion. You may notice if you spend some time here that many of these folks use "non-standard" definitions, sometimes their own definitions when they think no dictionary objectively or accurately defines a concept. Of course, in doing so, it's useful if the user defines the word up front so as to avoid confusion. However, EC has been here for a while, and probably 99% of the users who participate here would have understood what he meant by the word. Please don't take it too personally just because there was some minor confusion.

Furthermore, in what book and on what page does Ayn define 'arbitrary' as truth-valueless?
OPAR, Chapter 5, page 163.

The Arbitrary as Neither True Nor False

Claims based on emotion are widespread today and are possible in any age. In the terminology of logic, such claims are "arbitrary," i.e., devoid of evidence. What is the rational response to such ideas, whether they are asserted by others or are a product of one's own fancy?

And why define it thus when 'senseless' is an already accepted word for statements that lack truth-value?

Perhaps because he prefers the word 'arbitrary'?

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"That's why he explained what he meant when there was some confusion."

I guess what I'm confused by is why he wrote the very condescending, "Do you understand what the term arbitrary means and when it apply's to a given subject?" As if there is one thing that 'arbitrary' might mean, and I got it wrong. But you're right, if there was no offense intended, none taken.

"You may notice if you spend some time here that many of these folks use "non-standard" definitions, sometimes their own definitions when they think no dictionary objectively or accurately defines a concept."

As I said, though, there already is a standard word that the logician community accepts for 'truth-valueless': 'senseless'.

"OPAR, Chapter 5, page 163."

Does Ayn define it anywhere? I've only read her works and Ominous Parallels. I know, I know, Piekoff is the intellectual heir, Ayn gave him permission to speak for the philosophy. All the same, on the website for the philosophy of Ayn Rand, it seems only reasonable that I should be responsible for being familiar with only what she wrote.

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I think the "arbitrary statements dont have truth values" thing is one of Peikoff's inventions, I havent read anything where Ayn Rand mentions it.

Edited by Hal
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I guess what I'm confused by is why he wrote the very condescending, "Do you understand what the term arbitrary means and when it apply's to a given subject?"

I think you are making the miscommunication of a term out to be more sinister than it was. However, if you are confused as to his intent, why not query of him whether or not he intended it to be "very condescending" rather than just assuming so?

Does Ayn define it anywhere?

I don't think that she does, but it still qualifies as "in the Objectivist sense" to which he refers based on the quote I provided. Why is it important to distinguish whether she said it or whether Peikoff said it for the purposes of Felix's usage when in either case it qualifies for "in the Objectivist sense"? His statement was not "Ayn Rand defined arbitrary as ..." Challenging his use of the word based on whether Ayn Rand defined it that way or not borders on attacking a strawman.

Would it be your contention that OPAR is not a valid or significant body of work with regards to the philosophy of Ayn Rand?

Edited by RationalCop
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What evidence is there for anyone to consider the existence of some "infinite form" except an abstract "mathematical discribition"? Do you understand what the term arbitrary means and when it apply's to a given subject?
That's a pretty big "except"; it is true that, qua mathematical concept, "the infinite" exists, namely the result of an unbounded process (or anything that can be put in 1-to-1 correspondence with the result of such a process). It is arbitrary to say that there infinitely many entities.

Anyhow, to A-null, although Rand does not use the exact words that Peikoff used in OPAR when he discussed truth values from an Objectivist perspective, it is obvious from reading ITOE that his explanation of the term correctly describes her usage.

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Cop, I'm not saying OPAR is invalid--maybe it is, maybe it isn't. I haven't read it. All things being equal, I'm just saying that I should be responsible for being familiar with only what Ayn Rand has wrote, no?

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aleph-- Ayn Rand named Peikoff as her intellectual heir and approved of what was written in OPAR, albeit in an indirect way. Therefore, what is included in OPAR is part of Objectivism. So if you want to understand Objectivism, the context of this forum, then yes, you are also "responsible" for what Peikoff wrote too.

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