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Can someone give me a proof that,

No true claim can derive a contradiction in a consistent system of axioms,

With out using a proof by contradiction

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How do you define "a consistent system of axioms"? –  Daniel Fischer Jul 15 '13 at 14:03
    
A list of axioms, that can't derive two contradictory statements –  user758575 Jul 15 '13 at 14:05
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Okay, and what's the definition of a "true claim"? (I'm not trying to be a jerk here, just trying to ascertain what's what.) –  Daniel Fischer Jul 15 '13 at 14:07
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True and false are semantic properties of sentences in a given structure (or a class of those). Deriving contradictions and consistency are syntactical properties. –  Asaf Karagila Jul 15 '13 at 14:09
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Please let's remain civil, @RGB. –  vadim123 Jul 15 '13 at 14:31
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2 Answers

There is no proof of the sort asked for in the question because the assertion to be proved isn't correct. A consistent system of axioms may well contain a false axiom. The negation of that false axiom would be a true statement which, in the system under consideration, leads to a contradiction.

A trivial example of a consistent system of axioms containing a false axiom is the system consisting of the single axiom $2+2=5$. As long as you don't have other axioms available to deduce the correct facts of arithmetic, $2+2=5$ is consistent.

A more reasonable but more complicated example would be to take the usual ZFC axiom system for set theory (whose consistency I take for granted) and adjoin to it the false axiom that expresses "there is a formal proof of a contradiction in ZFC" (i.e., it expresses "ZFC is inconsistent"). Despite the false axiom, this system is consistent, by Gödel's second incompleteness theorem.

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"No true claim can derive a contradiction in a consistent system of axioms."

What does this mean? That given a consistent set of axioms as background assumptions, and presumably some logic to do some derivations with, there is no derivation that leads from a true claim to a contradiction??

But that is just false. Take the single axiom $P$ [if you like, "the moon is made of green cheese"] and let standard propositional logic be your derivation system. Then we have a trivially consistent set of axioms. But, equally trivially, the proposition $\neg P$ (given the background axiom) leads immediately to contradiction. Yet $\neg P$ [on the given interpretation] is true!

The point is a straightforward one: if the axioms are themselves unsound, not true on a given interpretation, then you have a theory which can lead from a true additional premiss to a false conclusion, even a necessarily false one.

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