Axiom Of Choice?

[GrandMinnow]

Personally (aside from whatever Objectivism might say about the matter; and I am not aware of any authoritative Objectivist literature that explicity addresses such things as the axiom of choice), I don't find it required to allow only constructive mathematics. One may regard adopting the set theoretic axioms (including infinity and choice) as a means for study of those abstract mathematical structures in which said axioms happen to hold. To work with such axioms it is not required to take such axioms as assertions that must hold in all mathematical situations, but rather only that the axioms provide a context of study of those mathematical situations in which the axioms do hold.

[Eiuol]

 Resorting to terminology that is not even in the language of set theory is not considered a viable solution since the solution mathematicians demand is that of a formulation in the language of set theory.

Can my statement be phrased with better terminology then?

[GrandMinnow]

I don't think so. It is the very issue at hand that there is not in general a way to distinguish a next "choice". 

 

Aside from a technical explanation, Russell himself gave the problem in anecdotal form (so this is not a technical description but rather a non-technical visualization of the problem):

 

Suppose we have an infinite set of pairs of shoes and we wish to choose one member from each pair. In this case, we could say, "Choose the right shoe" and this is fine. But then suppose we have an infinite set of pairs of socks...oops...there's not a way to go through all the pairs to specify which sock to choose.

 

And it's even worse really. (I'm using "laundry" here just as illustration; of course, set theory does not itself mention such things as laundry). Imagine that we have infinitely many piles of laundry (and some of these piles are themselves infinite). From each pile we want to choose exactly one piece of laundry. It is proven that with the axioms of ZF there is no general description of a "way" to do this. 

Edited June 24, 2013 by GrandMinnow

[GrandMinnow]

Or, paraphrasing the language of set theory:

 

From the axioms of ZF there is no proof of this statement:

 

For all S, if the empty set is not a member of S then there is a function C such that the domain of C is S and for all x in S we have C(x) in x.

Edited June 24, 2013 by GrandMinnow

[aleph_1]

Personally (aside from whatever Objectivism might say about the matter; and I am not aware of any authoritative Objectivist literature that explicity addresses such things as the axiom of choice), I don't find it required to allow only constructive mathematics. One may regard adopting the set theoretic axioms (including infinity and choice) as a means for study of those abstract mathematical structures in which said axioms happen to hold. To work with such axioms it is not required to take such axioms as assertions that must hold in all mathematical situations, but rather only that the axioms provide a context of study of those mathematical situations in which the axioms do hold.

Similarly, one mignt adopt the hypotheses of Spinoza in order to study that particular philosophy. It seems to me that while it is important to study even non-reducible philosophies, it is also important to pursue the consequences of reduction wherever that leads.

Concerning AoC and WOTh, if you do not allow for theories that introduce non-reducible distinctions, then these principles are relatively unimportant. If you allow for the axiom of infinity, then many non-reducible distinctions are introduced into the theory

[GrandMinnow]

What you say does seem to fit your general argument. As far as I can tell, the axiom of infinity does not meet your criterion of reducibility to the extent I understand your notion of reducibility. But even more fundamentally, as far as I can tell, formal axiomatics such as these are quite different from Objectivist notions. It seems to me that there's not even much sense in talking about this formal mathematics in context of Objectivist requirements.

[Eiuol]

Suppose we have an infinite set of pairs of shoes and we wish to choose one member from each pair. In this case, we could say, "Choose the right shoe" and this is fine. But then suppose we have an infinite set of pairs of socks...oops...there's not a way to go through all the pairs to specify which sock to choose.

 

And it's even worse really. (I'm using "laundry" here just as illustration; of course, set theory does not itself mention such things as laundry). Imagine that we have infinitely many piles of laundry (and some of these piles are themselves infinite). From each pile we want to choose exactly one piece of laundry. It is proven that with the axioms of ZF there is no general description of a "way" to do this. 

Why couldn't you just as well say choose the left sock? I've heard that example from Russel before, but I didn't understand why you can't just say "pick any member". Another way to approach my question is: what are the limitations of choice functions in general. As far as I've seen, this all depends on how the choice function must behave, such as it must have a distinct end to its process, which would be problematic for any infinite set of course. My thinking is that as long as there is something to select, then any arbitrary selection is sufficient. On the other hand, I probably need to understand ZF axioms first. I still don't notice though what's bad about my reasoning.

[aleph_1]

What you say does seem to fit your general argument. As far as I can tell, the axiom of infinity does not meet your criterion of reducibility to the extent I understand your notion of reducibility. But even more fundamentally, as far as I can tell, formal axiomatics such as these are quite different from Objectivist notions. It seems to me that there's not even much sense in talking about this formal mathematics in context of Objectivist requirements.

 

I guess that I should add that once one has made the choice to accept the Infinity Axiom, the Axiom of Choice just carries forward the selection procedure that works for finite sets. Why not just well-order the set and choose the least element. Of course, well-ordering the complex plane is something I don't know how to do but the Axiom of Choice implies is possible. This was the source of my difficulty. Mine was the Post Hoc Ergo Propter Hoc fallacy. My philosophical difficulties are resolved not by rejecting an otherwise valid selection procedure but by rejection of the Infinity Axiom.

 

I have still not quite resolved that Oism and axiomatic notions at the foundations of math are incompatible. Oism has its own axiomatics. To the extent that mathematics is pure rationalism I accept your view. I might even be persuaded to argue that pure rationalism is sometimes necessary in that it allows for a diversity of models and hence we may become less frame-bound. However, this is far afield of the Axiom of Choice.

[GrandMinnow]

"Why couldn't you just as well say choose the left sock?"  [Eiuol]

 

Just socks. No left or right.

 

"My thinking is that as long as there is something to select, then any arbitrary selection is sufficient." [Eiuol]

 

Then in that regard you agree with the axiom of choice. Roughly speaking,  "arbitrary selection is sufficient" is the axiom of choice. 

 

Lets's step back first:

 

In formal set theory, we only can prove things that the axioms allow us to prove and everything proven is stated and proven in the formal language.

 

Now,

 

For all S, if the empty set is not a member of S then there is a function C such that the domain of C is S and for all x in S we have C(x) in x

 

is pretty much how, in the formal language, we express the more informal version you mention: "arbitrary selection is sufficient".

 

But the ZF (a formal set theory) axioms do NOT allow us to prove:

 

For all S, if the empty set is not a member of S then there is a function C such that the domain of C is S and for all x in S we have C(x) in x.

 

So we have to take the above statement as an additional axiom, which we call 'the axiom of choice". In other words, ZF does not allow us to prove that "arbitrary selection is sufficient", so whoever thinks arbitrary selection is sufficient will add to ZF the axiom of choice.

Edited June 26, 2013 by GrandMinnow

[GrandMinnow]

"well-ordering the complex plane is something I don't know how to do but the Axiom of Choice implies is possible" [aleph_1]

 

Even the set of real numbers. The axiom of choice entails that there is a well ordering of the set of real numbers. But also it happens that in ZFC there is no definition of a particular such well ordering.

 

"I have still not quite resolved that Oism and axiomatic notions at the foundations of math are incompatible. Oism has its own axiomatics."  [aleph_1]

 

The notion of 'axiom' in Objectivism is quite different as the notion of 'axiom' in modern mathematical logic. I don't think Objectivism is entirely hostile to formalizations such as found in ordinary mathematics; otherwise Objectivism would reject computer languages, and I don't think Objectivism does reject computer languages. However, I suspect (and only suspect, since I can't speak for Objectivism) that formalizations such as ZFC (which provides a formalization of analysis) would be (are?) seen by Objectivism as irrational  -  as either platonism or nominalism. In any case, I am not aware of any Objectivist proposal for a formal (formal in the manner of a recursive syntax) arithmetic, analysis, and other branches of mathematics other than computer languages. The only Objectivist I know of that works in mathematical logic and set theory is Steven Simpson. In my personal conversation with him he admitted that Objectivism has a lot of work to do in order to provide such foundational explanations. Moreover, as to Objectivist posters in this particular forum, aside from the late Steven Speicher, most response to mathematical logic and set theory is hostile.

Edited June 26, 2013 by GrandMinnow