Bookman

If you're going to distinguish between the two, what is the justification for treating the AC as a logical conjecture and not a mathematical one?

I guess the part that bothers me from the OP is this:
I don't see that what is needed as a solution for AC is a "mechanism outside logic"; rather, a better formation of the set of axioms of set theory. Several have been proposed: Theory of Types, Stratification, Zermelo. None of these is as elegant as most mathematicians would like, and yet none require a change to nor make any challenges to the fundamentals of logic.

Bookman

John Page

*Humbly blushing* Great! What was the question to which you refer?

Luiseach

I don't want to derail your thread...

...but to clarify: I've been trying to decide if there is a difference between 'truth' and 'knowledge' (a real difference I mean, not just semantic).

For me, the word 'truth' has connotations that make me uneasy...it's vague and religious in some sense.

I prefer the term 'facts' and/or 'knowledge.' But maybe I'm just being fussy about words.

But I've been trying for ages to determine what truth is.

Your statement that truth depends on the system we're employing somehow helped me answer this question that's been nagging me. I had the following thought:

Truth is a product of a given system's assessment of the chosen assemblage of facts.

That sounds a bit wordy....sorry.

I'm afraid I'm not overly conversant with formal logic (although my husband is a philosopher and teaches it), so I can't translate what I'm saying into symbols...

John Page

Because it deals with set theory and the logical relations between entities and types rather than mathematical operations. However, I'm not opposed to other distinctions between the two.
IMO, Types and stratification imply a pre-existing order within reality that doesn't really exist, its all in the mind/brain.

As for the "mechanism outside logic", it seems to me that AC requires a function or arbitrary choice, independent of the system of logic/math being considered, to determine what the result is. For me, either there is reason or there is not. I do accept, though, that for math there are certain issues that do not matter.

Cheers, John

Bookman

As I said, they are inelegant solutions. I'm not enamored with them either.

I think I'd put it differently.

It seems to me that AC requires the existence of a function (representative of an arbitrary choice) which may not be able to be determined within the formal system under consideration.

The existence of such a function is accepted as an axiom; the failure of the function to be able to be determined (under certain specific conditions) isn't surprising or alarming, nor is it a threat to logic or mathematics, in my view.

John Page

Lu:

LOL - if (emoticons are symbols or words are symbols) then the above is false!
I have spent much time pondering this also. I think there is a difference between personal and public (intersubjective) versions of the truth and knowledge. My mental model is of direct experience giving rise to knowledge that becomes increasingly certain through repeated experience and/or comparison with the reports of others' experiences, attaining the status of fact. A fact is taken to be "a true statement about reality" with facts becoming more objective through use of scientific methods etc.

Cheers, John

John Page

Yes, it must be an axiom because its not justified by any other means than its supposed self-evidence. AC remains an anathema for me because set membership is predicated upon a fit with the qualities required for a member of that set. Why then should anything need to be chosen?

IMO the issue arises because of a confusion between a thing (set of qualities) and thing-in-itself. There may be a set of three sheep and if all we want is a sheep it doesn't matter which one we choose. On the other hand, all the sheep are individuals with separate identities. I think one needs to distinguish between the a) real, individual, they've even got names, sheep, b) flock of sheep and c) the axiomatic concept of a sheep that allows the mind to perceive the real instances sheep and place them in the mental category of sheep which, when intersected with what is being perceived right now, mentally derives a flock of sheep that I can describe to you as such.

If the above is accurate, the "choosing function" in the AC is a process that enables us to decide "these are all sheep, which one would you like". In reality, no choosing function is required - it happened a priori in determining (not choosing) which parts of reality were sheep in the first place.

Cheers, John

Primal

Well then is the above contradictory or tautologous?

Ultimately your reasoning, as well as mine must either end somewhere or go off forever, so do you end in contradiction or tautology?

Also how is your claim proven? I consider mine deduced via self-evident axioms, how do you hold yours up?

Also I am not saying truth is relative but that the truth value of a claim is relative to what axioms hold it up. I believe the axioms themselves are absolute.

And I don't see any other way out of this, as to follow any other system seems to simply lead to some sort of irrationalism.

Luiseach

<---------blush

Yep...this makes good sense to me. Thanks again. :-)

Bookman

I don't interpret it this way.

Farmers Abel, Baker, and Charlie each have a set (flock?) of sheep. The fact that we have classified these "parts of reality" as sheep and identified them as members of the sets "Abel's Sheep" (A), "Baker's Sheep" (B), and "Charlie's Sheep" (C) has nothing to do with the axiom of choice.

Furthermore, mundane sets such as these present no problem for the Axiom of Choice and do not lead to Russell's paradox nor any variant thereof. To wit, in the example provided the AC merely asserts that there exists a set of sheep which consists of one member of each non-empty farmer's-flock; that there exists a function f by which we can "choose" some member of A, B, C such that the resultant set S satisfies the condition of the axiom of choice.

In real-world examples, several such functions present themselves: let f(x) yield the cheapest, the fattest, the eldest and so on. Sets of real things that can be ordered present a host of candidate functions for f(x) and upon these sets the acceptance of the axiom of choice seems trivial. Although we've both referred to the choosing function as arbitrary, that is correct only in the sense that in the absence of a specific f(x) one can not determine which member a of set A will be chosen. The function itself is deterministic: f(x) operating on A will always yield the same member a; that is the nature of a function.

Bookman