Freethought & Rationalism Archive

John Page · 05-16-2003, 12:20 PM

Background

Quote:

Axiom of Choice. "Given any set S of mutually disjoint nonempty sets, there is a set C containing a single member from each element of S. C can thus be thought of as the result of "choosing" a representative from each set in S. Hence the name. " or
"For any set X there is a function f, with domain X\(0), so that f(x) is a member of x for every nonempty x in X. "

Quote:

However, if C is the collection of all nonempty subsets of the real line, it is not clear how to find a suitable function f. In fact, no one has ever found a suitable function f for this collection C, and there are convincing model-theortic arguments that no one ever will. (Of course, to prove this requires a precise definition of "find," etc.)

Then from Russell and Eric Schecter's Axiom of Choice site:

Quote:

"To choose one sock from each of infinitely many pairs of socks requires the Axiom of Choice, but for shoes the Axiom is not needed."
The idea is that the two socks in a pair are identical in appearance, and so we must make an arbitrary choice if we wish to choose one of them. For shoes, we can use an explicit algorithm -- e.g., "always choose the left shoe." Why does Russell's statement mention infinitely many pairs? Well, if we only have finitely many pairs of socks, then AC is not needed -- we can choose one member of each pair using the definition of "nonempty," and we can repeat an operation finitely many times using the rules of formal logic (not discussed here).

OP question 1. Assuming the AC is logically valid, it implies a mechanism outside logic is necessary for set theory to operate on the set of all non-empty sets. It seems to me that if AC is considered logically valid, the system of logic of which it forms part is not a closed system (in that its results cannot be fully derived from its axioms) and incoherence makes an entrance.
OP Question 2. If, to keep coherence, the answer to question 1 is that the AC is invalid, other solutions require that the set of all sets is not a member of itself. True or False?

Cheers, John

Big Spoon · 05-22-2003, 04:52 AM

Axioms are self-evident truths, do they need to be logically valid?

Anyway, set theory and all that logic stuff are third year courses, I'll reply to this after I've taken them!

John Page · 05-22-2003, 05:33 AM

Quote:

Originally posted by Big Spoon
Axioms are self-evident truths, do they need to be logically valid?

Self-evident to some is not to others. There seems to be a general opinion that the should be consistent with the system of which the form the foundation.

Witt · 05-22-2003, 07:27 AM

Big Spoon: Axioms are self-evident truths, do they need to be logically valid?

IMO,
Self-evidence seems to be a dubious concept.
Axioms are beliefs of the system that uses them.
They are assumed theorems.
They cannot be logically valid within that system.
Although, they may well be theorems of another system.

They are undecidable by the system that claims them.

Witt

pariah · 05-22-2003, 03:56 PM

axioms are just generally accpeted truths...

Primal · 05-22-2003, 09:47 PM

Quote:

Self-evident to some is not to others. There seems to be a general opinion that the should be consistent with the system of which the form the foundation.

Well now that's either begging the question or non sequitur.

Either you are saying something cannot be self-evident, because it cannot be. "For some and not others" assuming its relative/arbitrary from the onset.

Or you are saying that mere disagreement constitutes a refutation. Which it does not.

For example lets say I state 1 plus 1 equals 2. And another says "No that equals four". Does his mere disagreement disprove my case?

That's simply an argument from incredulity.

In any event we are stuck with some sort of axiom ultimately, that or infinitism, which is absurd. And you can either consider these axioms matters of reason or faith. If you consider them matters of reason: then they must be self-evident.

Dominus Paradoxum · 05-22-2003, 10:15 PM

Blanshard, in The Nature of Thought, actually makes some good arguements against self-evidence.

John Page · 05-23-2003, 05:25 AM

Quote:

Originally posted by Primal
In any event we are stuck with some sort of axiom ultimately, that or infinitism, which is absurd. And you can either consider these axioms matters of reason or faith. If you consider them matters of reason: then they must be self-evident.

Well, that's either begging the question or a step on the road to relativism, where one's POV at any given moment is bound by one's axioms. Is this self-evident?

Cheers, John

Primal · 05-27-2003, 11:43 AM

Quote:

Well, that's either begging the question or a step on the road to relativism, where one's POV at any given moment is bound by one's axioms. Is this self-evident?

I'm not sure actually though it stands to reason that we must either start somewhere or go off into infinity.

And this only becomes an epistemic relativism if we believe all axioms are equal, however such a belief itself must be assumed and is inconsistent with those axioms or rationality or logic.

John Page · 05-27-2003, 02:11 PM

Quote:

Originally posted by Primal
And this only becomes an epistemic relativism if we believe all axioms are equal, however such a belief itself must be assumed and is inconsistent with those axioms or rationality or logic.

Would you agree it appears that reality is the judge of the relative effectiveness of a set of axioms? The problem then becomes one of pinning down reality?

Cheers, John

05-22-2003, 07:27 AM	#4
Witt Banned Join Date: May 2003 Location: Toronto Canada Posts: 1,263	self-evidence Big Spoon: Axioms are self-evident truths, do they need to be logically valid? IMO, Self-evidence seems to be a dubious concept. Axioms are beliefs of the system that uses them. They are assumed theorems. They cannot be logically valid within that system. Although, they may well be theorems of another system. They are undecidable by the system that claims them. Witt

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05-22-2003, 04:52 AM	#2
Big Spoon Junior Member Join Date: Feb 2003 Location: UK Posts: 74	Axioms are self-evident truths, do they need to be logically valid? Anyway, set theory and all that logic stuff are third year courses, I'll reply to this after I've taken them!

05-22-2003, 03:56 PM	#5
pariah Banned Join Date: Mar 2003 Location: الرياض Posts: 6,456	axioms are just generally accpeted truths...

05-22-2003, 10:15 PM	#7
Dominus Paradoxum Veteran Member Join Date: Nov 2002 Location: California Posts: 1,000	Blanshard, in The Nature of Thought, actually makes some good arguements against self-evidence.

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