Surely an axiom stating that the previous axioms don’t lead to an internal contradiction isn’t going to make new theorems true; in fact it will either make your axioms inconsistent or have no effect. Presumably 0=1 directly contradicts some other axiom (e.g. with PA, 1 = S(0) and by definition is non-zero).

Indeed, I can prove that axioms of this type won’t lead to new theorems becoming available because you need never make use of an axiom y that states “axiom x is not violated” in a proof since you already have axiom x to call upon. It follows that all axioms of the form “axiom x is not violated” need never be used in a proof.

This is very different from making an unprovable but true thing into an axiom. (Consider the axiom of choice, which can be assumed to be true, assumed to be false, or avoided altogether, and which leads to different theorems being provable.)

Perhaps this entire line of reasoning is spurious. A set of axioms should be irreducible before proceeding, and an axiom stating that the preceding axioms are consistent is by its nature eliminated.

]]>he figered out that there is NO consistent system.

‘Nuf said. 🙂

inquisitor #33: I wish it didn’t affect me so much, but the fact that someone as *contentedly* ignorant as you are exists is going to depress me for the rest of the day.

Why don’t you go read Wikipedia if you want to understand what it is that Gödel “figered out.” (I’ll give you two hints: (1) it involves the *completeness* of formal systems, not just their consistency; (2) it’s been the implicit starting point, known to everyone here, for the entire conversation on this thread.)

If wiki is to be trused it seemed likely he didn’t have a choice:

“In his memoirs Turing wrote that he was disappointed about the reception of this 1936 paper and that only two people had reacted – these being Heinrich Scholz and Richard Bevan Braithwaite.”

http://en.wikipedia.org/wiki/Alan_Turing

So at the time Turing was working on his thesis, he probably was just a bright grad student who had done some competent work before he got to grad school that nobody really read. Probably not that different from alot of his Princetion classmates.

]]>he figered out that there is NO consistent system.

‘Nuf said. 🙂

]]>Paragraph 3: I think you mean “can prove that M does not halt.”

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