What is /sci/'s opinion on the Axiom of Choice?

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Anonymous Fri Apr 19 21:57:39 2013 No.5696829 [Reply] [Original]

What is /sci/'s opinion on the Axiom of Choice?

>>	Anonymous Fri Apr 19 22:35:08 2013 No.5696872 It's independent from ZF, just accept it as an axiom and move on.

>>	Anonymous Fri Apr 19 23:13:38 2013 No.5696923 It's a victim of extensionality and excluded middle's wrongdoings.

>>	Anonymous Sat Apr 20 00:26:53 2013 No.5697023 >>5696829 AD is more believable than AC IMO but both are questionable

>>	Anonymous Sat Apr 20 00:41:54 2013 No.5697062 >>5697023 ZF + AD implies that ZF + AC is consistent, the "converse " is not true*.

>>	Anonymous Sat Apr 20 00:46:10 2013 No.5697073 >>5697062 so?

>>	Anonymous Sat Apr 20 00:56:24 2013 No.5697082 >>5697073 If you believe in AD, then you have to believe that AC doesn't have contradictions (again, the converse is not true) so, in a sense, AC is "more believable" than AD.

>>	Anonymous Sat Apr 20 01:05:24 2013 No.5697087 File: 42 KB, 250x250, mah-fravirt-gers.jpg [View same] [iqdb] [saucenao] [google] >>5696829 I use it but don't believe in it. My life is a joke.

>>	Anonymous Sat Apr 20 01:18:39 2013 No.5697099 >>5697082 I don't think I even implied that consistency had anything to do with the matter.

>>	Anonymous Sat Apr 20 01:50:25 2013 No.5697157 AC is about as "obviously" true as any supposed axiom can get, and yet it leads to results that seem just as "obviously" false. I think this spells trouble for modern set theory in general.

Anonymous Sat Apr 20 02:03:54 2013 No.5697187

>>5697157
the axiom of choice, for me, has two problems.

The first is that non-constructive anything is irritating as fuck. I really fucking hate it. But whatever.

The second is that I think it is too weak. By just saying "hey some choice function exists fuck it" you end up with indiscernables, which is why something like the continuum hypothesis is independent. Since you're not saying anything about choice, you're not able to, well, say enough about choice when it counts. For all ZF+AC can say, the cardinality of the continuum is like some fucking woodin cardinal or some shit. It literally could be fucking anywhere. What good does your sock-choosing magic get you?

Someone tells me, "I assume GCH." I respect that. Maybe I don't believe GCH. But the person isn't fucking around. Choice is pointlessly weak.

If you're investigating how little you need to reach some conclusion, AC occupies an interesting spot, but I wouldn't want to do math with it.

>>	Anonymous Sat Apr 20 02:08:23 2013 No.5697194 >>5696829 can someone explain in terms a filthy plebian would understand what is the axiom of choice?

>>	Anonymous Sat Apr 20 02:14:54 2013 No.5697205 >>5697194 wikipedia explains it very clearly, I don't think I could do a better job than its introduction

Anonymous Sat Apr 20 02:21:09 2013 No.5697212

>>5697194
Every surjection has a right inverse, you can extract a function from any entire relation, any infinite set A has a bijection with A^2, the nonempty product of a nonempty family of nonempty sets is nonempty. These are all classically equivalent to the axiom of choice.

>>5697187
>The first is that non-constructive anything is irritating as fuck. I really fucking hate it. But whatever.
In the BHK-interpretation of intuitionistic logic, the negation free formulations of choice are trivially true: "every surjection has a right inverse" and "you can extract a function from any entire relation" are the distributivity of dependent products over dependent sums, "the product of a family of inhabited sets is inhabited" is function application.
Countable choice and dependent choice are also true this way.

Anonymous Sat Apr 20 02:21:53 2013 No.5697214

>>5697205
ok i looked it up.
>In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that "the product of a collection of non-empty sets is non-empty"
i dont see how this could be false. you have two things and you combine them in a way that you don't lose any information, how could the end result be nothing?

>>	Anonymous Sat Apr 20 02:23:40 2013 No.5697216 >>5697212 Also, "a family of inhabited sets" is exactly the same as saying "the family has a choice function"

>>	Anonymous Sat Apr 20 02:23:54 2013 No.5697217 >>5697187 Math is full of non-constructive proofs. In particular, proof by contradiction is a non-constructive way of proving stuff. Could you explain your second problem? I don't think I understood it correctly

>>	Anonymous Sat Apr 20 03:28:24 2013 No.5697329 >>5697187 >The first is that non-constructive anything is irritating as fuck Pleb detected. Non-constructive proofs are the most beautiful.

Anonymous Sat Apr 20 03:29:23 2013 No.5697331

>>5697217
> could you explain your second problem
ZFC is just not strong enough to answer our questions. Either more axioms are needed, or stronger axioms are needed. I think choice, if it is right, is just too weak. In theorems, that kind of weakness means generality. But axioms play a different role, in that they define the world we're working in. There weakness is ambiguity.

Put plainly, if we can't answer CH then we just don't even know what the fuck we're talking about and we need to make our axioms have further reach.

>>	Anonymous Sat Apr 20 04:18:08 2013 No.5697411 >>5697329 > the most beautiful. All the nature of platonist-faggots

>>	Anonymous Sat Apr 20 04:43:24 2013 No.5697437 ITT: freshman philosophy majors who took a "philosophy of mathematics" elective

>>	Anonymous Sat Apr 20 04:46:24 2013 No.5697441 It is fine. We do maths for fun, and for that we need bases in vector spaces, maximal ideals in rings and compactness in infinite product spaces.

Anonymous Sat Apr 20 04:53:53 2013 No.5697452 [DELETED]

I personally think it's false. Sure it seems somewhat obvious at first, but that's obfuscating the fact that we're dealing with infinities. All results based on the axiom of choice I can think of right now all seem rather unintuitive, and I'm not gonna even discuss Banach-Tarski. For example Tychonoff's theorem: I really wouldn't expect an arbitrary product of compact spaces to be compact, I'd want to put a finiteness restriction on the product. Then, worst of all, there's the whole story about "predicting the future" using the axiom of choice:

http://www.google.ch/url?sa=t&rct=j&q=axiom%20of%20choice%20predicting%20the%20future&source=web&cd=1&ved=0CDAQFjAA&url=http%3A%2F%2Fwww.math.upenn.edu%2F~ted%2F203S10%2FReferences%2Fpeculiar.pdf&ei=qlVyUcDGH-2v7AaltYCYBA&usg=AFQjCNELa58VnmFjaeNwDH13a7Cx0UGBDg&cad=rja

You can read more about this issue here:

http://cornellmath.wordpress.com/2007/09/13/the-axiom-of-choice-is-wrong/

Be sure to read Tao's comment, as he gives a nice conceptual explanation of what the problem is.

Anonymous Sat Apr 20 04:55:38 2013 No.5697455

I personally think it's false. Sure it seems somewhat obvious at first, but that's obfuscating the fact that we're dealing with infinities. All results based on the axiom of choice I can think of right now all seem rather unintuitive, and I'm not gonna even discuss Banach-Tarski. For example Tychonoff's theorem: I really wouldn't expect an arbitrary product of compact spaces to be compact, I'd want to put a finiteness restriction on the product. Then, worst of all, there's the whole story about "predicting the future" using the axiom of choice:

www.math.upenn.edu/~ted/203S10/References/peculiar.pdf

You can read more about this issue here:

http://cornellmath.wordpress.com/2007/09/13/the-axiom-of-choice-is-wrong/

Be sure to read Tao's comment, as he gives a nice conceptual explanation of what the problem is.

Anonymous Sat Apr 20 05:38:09 2013 No.5697496

>>5697455
Choice gets always shit upon because of other axioms' fault, usually a too strict definition of equality or assuming you're omniscient, but for the Banach-Tarski paradox the fault is of point-set topology. It is false in pointfree topology such as locale theory, where Tychonoff's theorem is provable without choice (which people don't like because "choice is involved, here be dragons")

In the paper you linked they probably assume omniscience, use some undecidable property of real numbers or infinite sequences and then act surprised that they actually are omniscient like they assumed they are. It's unnerving.

Anonymous Sat Apr 20 05:42:09 2013 No.5697499

>>5697496
Regarding your last sentence: no. You should read the paper. Having "infinite memory" and being able to predict the future are two very different things. The axiom of choice makes the latter possible assuming the first, this is very problematic. And I repeat: I think Tychonoff's theorem is bad. Thus anything like pointfree topology where Tychonoff holds even without AC is even worse than point-set topology to me. Compactness is a finitary condition, infinite products shouldn't be compact.

Anonymous Sat Apr 20 06:27:10 2013 No.5697527

>>5697499
Definition 2.2 uses the well-ordering theorem. Show me a proof of it using only choice without excluded middle or extensionality. As I said, if you assume unreasonable assumptions, you'll get unreasonable results, it's the well-ordering theorem that's at fault there.

Honestly I don't see why products shouldn't preserve compactness, either.

Anonymous Sat Apr 20 06:32:11 2013 No.5697528

Such controversies all over again not only on 4chan and not only troll-style makes me feel mathematics is quite far away from being complete and solid

Anyways we do know which parts of it really work for the real world regardless of foundations and philosophy

>>	Anonymous Sat Apr 20 06:41:23 2013 No.5697534 Doesn't the banach tarski theorem essentially say that something of volume 1 = something of volume 2 ? Seems pretty retarded. It's about as much of a "paradox" as zeno's paradox.

>>	Anonymous Sat Apr 20 07:13:08 2013 No.5697573 >>5697534 That's the thing about paradoxes. They always turn out not to be actually paradoxical. The value of paradoxes is that they challenge your paradigm.

>>	Anonymous Sat Apr 20 07:15:23 2013 No.5697575 >>5697573 And I don't mean paradoxes turn out to be true or false. It tells you that by failure of explanation your paradigm is false or insufficient.

Anonymous Sat Apr 20 08:04:06 2013 No.5697617

>>5697527
You aren't listening. You said that in my paper the probably assume omniscience and then conclude omniscience. That is not what they do. They assume you can well-order any set and then conclude that you can predict the future. Are you that daft?

And if you don't see an issue with arbitrary products of compact spaces being compact then you haven't really understood the point of compactness. The principal (and in fact only) reason why compact spaces are interesting is because they're inherently finitary: every open cover has a FINITE subcover. The point is that this finiteness can be exploited in proofs (and always is when compactness is a necessary condition). I think I make a fair point when I say that I see no reason why finitary properties (like the one just described for compact spaces) should be preserved for infinite operations (like infinite products in this case).

>>	Anonymous Sat Apr 20 08:11:51 2013 No.5697632 I don't see the problem. A set is non-empty if it has at least one element. There's your axiom of choice. By using my free will I can choose that element.

>>	Anonymous Sat Apr 20 08:26:06 2013 No.5697642 File: 24 KB, 574x478, shiggy diggy piggy.jpg [View same] [iqdb] [saucenao] [google] >infinite sets

>>	Anonymous Sat Apr 20 08:51:52 2013 No.5697668 >>5697632 what if you can't specify which element it is you want to choose?

Anonymous Sat Apr 20 08:58:52 2013 No.5697678 [DELETED]

>>5697668
Let \left(X_i\right)_{i\in \mathcal{I}}[/spoiler] a collection of non-empty sets with \mathcal{I}[/spoiler] being an index set. Since each X_i \neq\emptyset[/spoiler] there exists an element a_i \in X_i[/spoiler]. No axiom of choice needed here, just the definition of non-empty. By definition of cartesian product (a_i)_{i\in\mathcal{I}}\in\prod_{i\in\mathcal{I}}X_i[/spoiler]. This proves \prod_{i\in\mathcal{I}} \neq\emptyset[/spoiler] and thus the axiom of choice. QED

>>	Anonymous Sat Apr 20 09:00:08 2013 No.5697682 >>5697678 Last one should be <span class="math">\prod_{i\in\mathcal{I}} X_i \neq\emptyset[/spoiler]

>>	Anonymous Sat Apr 20 09:44:07 2013 No.5697714 >>5697678 > just the definition of non-empty You didn't use the definition of non-empty. You used the axiom of choice.

>>	Anonymous Sat Apr 20 09:47:52 2013 No.5697718 >>5697714 The definition of non-empty says the set contains at least one element. That's what I used. As I proved, it's equivalent to the axiom of choice.

Anonymous Sat Apr 20 09:55:07 2013 No.5697734

>>5697718
>Definition of non-empty
If X[/spoiler] is a non-empty set, then X[/spoiler] is not the empty set.
>Axiom of choice
If X[/spoiler] is a non-empty set, then there is a a \in X[/spoiler].

Learn the difference.

>>	Anonymous Sat Apr 20 09:59:07 2013 No.5697739 >>5697642 Finite math is called 'Numerical Methods'. Go be an engineer.

>>	Anonymous Sat Apr 20 10:06:07 2013 No.5697746 >>5697739 mathematics isn't only about farting around with philosophy. the part that's actually impressive is the problem solving.

>>	Anonymous Sat Apr 20 10:09:06 2013 No.5697750 >>5697746 TO YOU

>>	Anonymous Sat Apr 20 10:09:22 2013 No.5697751 >>5697734 Then the definition of set union or intersection only make sense by implicitly using the axiom of choice because they are talking about the elements of the sets.

>>	Anonymous Sat Apr 20 10:20:36 2013 No.5697761 >>5697746 +1 >>5697739 -1 to carma

Anonymous Sat Apr 20 10:34:23 2013 No.5697771

>>5697734
That's absolute nonsense. You can choose an element out of a set without the axiom of choice. The point at which the axiom of choice is used in the above so called proof is when he talks about the definition of the cartesian product. If you have an arbitrary collection of sets, yes you can choose an element out of each one. But saying that the collection of all those elements is in fact an element of the product of all the sets is precisely the axiom of choice.

>>	Anonymous Sat Apr 20 10:53:07 2013 No.5697790 >>5697761 Go help Pythagoras throw people off boats.

>>	Anonymous Sat Apr 20 11:24:52 2013 No.5697827 >>5697331 Do you realize that you can't have a recursive definable, complete and consistent list of axioms of mathematics?

Anonymous Sat Apr 20 12:31:06 2013 No.5697912 [DELETED]

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>>5697214
>In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that "the product of a collection of non-empty sets is non-empty"

I consider this formulation polemic.
You have a bunch of other definitions in the background together with a large universe, this is why this happens.
The construction of the product is a set (per definition, so it IS a set in any case) whose elements are such and such. In your set theory without choice, "the product of non-empty sets results to be an empty set" is true, but only because the defintion requires you to collect the elements the set is supposed to contain, and then, without the axiom of choice, you can't find any. Hence the definition returns the empty set.

To say
"In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that 'the product of a collection of non-empty sets is non-empty' "
subliminally suggests that if you negate the axiom, then the theory is more well behaved - yes, but only because you robbed it it's power.

As far as opionions go, I' currently in a melancholy mood, telling me to not care too much about the man made desire to have a single framework for foundation.
That is I learn towards computation and complexity considerations now.

Anonymous Sat Apr 20 12:34:51 2013 No.5697918
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>>5697214
>In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that "the product of a collection of non-empty sets is non-empty"

I consider this formulation polemic.
You have a bunch of other definitions in the background together with a large universe, this is why this happens.
The construction of the product is a set (per definition, so it IS a set in any case) whose elements are such and such. In your set theory without choice, "the product of non-empty sets results to be an empty set" is true, but only because the defintion requires you to collect the elements the set is supposed to contain, and then, without the axiom of choice, you can't find any. Hence the definition returns the empty set.

...fuck, I don't want to take position here, really.

As far as opionions go, I' currently in a melancholy mood, telling me to not care too much about the man made desire to have a single framework for foundation.
That is I learn towards computation and complexity considerations now.

>>	Anonymous Sat Apr 20 12:53:21 2013 No.5697929 >>5697771 Why doesn't it work with the definition of cartesian product alone?

>>	Anonymous Sat Apr 20 13:04:52 2013 No.5697944 >>5697929 Because the index set could be infinite and then the cartesian product isn't so well defined.

>>	Anonymous Sat Apr 20 13:06:23 2013 No.5697946 >>5697944 Why isn't it well defined for an infinite index set?

>>	Anonymous Sat Apr 20 13:08:07 2013 No.5697950 >>5697946 http://en.wikipedia.org/wiki/Cartesian_product Read the infinite products bit

>>	Anonymous Sat Apr 20 13:13:22 2013 No.5697955 >>5697950 Thanks. Where can I find an example for >Even if each of the X_i is nonempty, the Cartesian product may be empty in general.

>>	Anonymous Sat Apr 20 13:16:22 2013 No.5697958 >>5697955 If someone could find an example then the axiom of choice would definately be false and wouldnt be used. It's just saying that theres nothing in the definition of the Cartesian product that says it cant happen

Anonymous Sat Apr 20 13:26:21 2013 No.5697970

>>5697955
That's precisely the point of the axiom of choice. From the ZF axioms and the definition of Cartesian product you can't prove that an arbitrary product of non-empty sets is non-empty. You can't however construct a counterexample either. Thus you can either assume that non-empty products are non-empty, or you can assume that there are infinite products of non-empty sets that are empty. The former is called the axiom of choice and is assumed by many mathematicians. However, it leads to some rather unintuitive results (Banach-Tarski, Tychonoff, predicting the future...etc.)

>>	Anonymous Sat Apr 20 13:31:06 2013 No.5697983 >>5697970 ^ This guy it explains it better than me

Anonymous Sat Apr 20 13:34:38 2013 No.5697989 [DELETED]

>>5697970
But the wiki article defines the infinite cartesian product as the set of all maps from the index set into the union of all the indexed sets, so that the projection thing commutes. All I have to do to show non-emptiness is to construct one map satisfying this property. Since each set X_i is non-empty, there's an a_i in X_i. Map the element i of the index set on a_i by defining f(i) = a_i.
QED

Anonymous Sat Apr 20 13:44:06 2013 No.5698006

>>5697617
>They assume you can well-order any set and then conclude that you can predict the future.
I said "unreasonable assumptions" and "the well-ordering theorem is at fault". Why is choice taking the fault of the well-ordering theorem here?

>And if you don't see an issue with arbitrary products of compact spaces being compact then you haven't really understood the point of compactness.
But the point is that the compactness of infinite spaces is being also used computationally to do searches in finite time in infinite spaces such as the Cantor space:
http://math.andrej.com/2007/09/28/seemingly-impossible-functional-programs/
http://math.andrej.com/2008/11/21/a-haskell-monad-for-infinite-search-in-finite-time/
You can deny that infinite products preserve compactness, but then you'd need to explain this in some other way, or show an input for which this doesn't halt. Moreover, it makes intuitive sense that infinite spaces such as the circle and the interval are compact, and the fact that they are compact and infinite is much more useful than finite and compact.

Anonymous Sat Apr 20 15:47:21 2013 No.5698239

The reason I accept AoC is because I think the statement "an infinite product of nonempty sets is nonempty" should be true. Sure it implies some weird things: for example, Banach-Tarski (which is because AoC implies the existence of unmeasurable sets), but there's a lot of really fucked up shit anyway, especially in real analysis and topology (even without AoC). The thing is that the problems AoC causes are really only theoretical. We can't split things into unmeasurable sets in real life, so Banach-Tarski really isn't a paradox IRL. That being said, I can see why some people don't accept it.

>>	Anonymous Sat Apr 20 16:25:21 2013 No.5698315 >>5697827 yes You can't have a complete "English language" either but we still invent new words.

>>	Anonymous Sat Apr 20 16:26:51 2013 No.5698318 How is Tychonoff's theorem unintuitive?

>>	Anonymous Sat Apr 20 16:41:06 2013 No.5698342 >>5697062 but AD and AC are inconsistent

Anonymous Sat Apr 20 23:23:30 2013 No.5699116

>>5697617
>I think I make a fair point when I say that I see no reason why finitary properties (like the one just described for compact spaces) should be preserved for infinite operations (like infinite products in this case).

Whose topology is defined in a finitary way.

>>5698342
Obviously, AC and AD are incompatible. However, even if AD is incompatible with AC, it can decide stuff about ZFC as a list of axioms. In particular, from ZF + AD you can deduce the existense of an inner model of ZFC (obviously this set can't model AD) and therefore ZF + AD deduce the consistency of ZFC.

I'm sageing this because I'm not sure if anyone is still interested in this topic.

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