Philosophy question

[quinn]

Well, which is it?

[usmhot]

There are proofs, in a number of different forms, that were develoiped by a number of different people - most notably Godel and Turing - that in any sufficiently rich formal system - mathematical, logical, computational, etc - there are statements that can be made that can neither be proven nor disproven.

So, the question you're posing has already been answered - the answer being the third option you give.

[invictus33]

The first two options can't be true, so that only leaves the third option. Saying all things can be proven is thrown out the window when questions of irreducible complexity show up. By proving that nothing can be proven defeats itself and can never be a true statement. So that just leaves you with option #3.

[drudman]

The answer is number two.

All claims are synthetic propositions. Note that all sentences that purport to be claims about "God" or whatever metaphysical entity are not real claims, they are non-senses.

No synthetic propositions can be proved with 100% certainty.

All analytic propositions are true. "No claims can be proven" is an analytic proposition, so it escapes the pitfall of contradiction that another poster believed it had.

[Darryl_P]

Can I not claim a yet-to-be-proven but provable statement about prime numbers for example? If that wouldn't count as a claim then a lot of mathematicians don't know what the word "claim" means.

[drudman]

Mathematical propositions are analytic.

[Piers]

Proof is only meaningful inside a model defined by a set of axioms. For claims within such a system option three applies.

For a claim that concerns a real world phenomenon then it is necessary to encompass the claim within such a model for the concept of proof to have meaning. However as the mapping between model and reality is subjective I expect any claim can be proved or disproved depending on the framework in which you construct the proof. So option one will apply.

[kpux]

Number Three. In any axiom system there are statements that can neither be proven or disproven, ala Godel.

Taking an example from set theory, the cardinality of the set of all rational numbers (numbers that can be expressed as fractions, eg. integer/integer) is infinite. So is the set of real numbers (the rationals plus numbers like pi, e, the square root of 2, etc...). However, they do not have the same cardinality. There are more real numbers than rationals. For simplicity, call the cardinality of the rationals x1 and the cardinality of the reals x2.

The Continuum Hypothesis says that there is no set with cardinality say, x3, such that x1 < x3 < x2. Unfortunately, this is neither true nor false. I think it was Godel who proved that assuming the Continuum Hypothesis is true in conjunction with the normal axioims of set theory creates a perfectly logically consistent system. Then some other guy in the 60's proved that assuming the negation of the Continuum Hypothesis did the same thing.

So, the Continuum Hypothesis is neither provable nor disprovable.

[quinn]

[ QUOTE ]

So, the Continuum Hypothesis is neither provable nor disprovable.

[/ QUOTE ]

Axiom 1: The Continuum Hypothesis is right

Conclusion: The Continuum Hypothesis is right, by Axiom 1

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Since you can use false axioms, everything is provable.

[kpux]

That isn't one of the axioms of set theory.