The logic of Kurt Godel

His point was that mathematics is not a logical tautology.

It was? How do you figure that?

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incompleteness theoremsPrincipia Mathematica und verwandter Systeme" (called in English "On formally undecidable propositions of Principia Mathematica and related systems"). In that article, he proved for any computable axiomatic system that is powerful enough to describe the arithmetic of the natural numbers (e.g. the Peano axioms or ZFC), that:

1. If the system is consistent, it cannot be complete.
   
2. The consistency of the axioms cannot be proven within the system.
   

These theorems ended a half-century of attempts, beginning with the work of Frege and culminating in Principia Mathematica and Hilbert's formalism, to find a set of axioms sufficient for all mathematics. The incompleteness theorems also imply that not all mathematical questions are computable.

As I said, his point was that brains are useful.

Well, he fairly convincingly shows the logical unsustainability of mathematical axioms when examined only in mathematical terms.

Did he write that?

In other words, we need a brain to do it.

Im not following you at all.

Godel just showed the basic foundational problem of epistemology. The mathematical axioms cannot be shown to be true by the very system itself --that would be circular. We need an outside reference point to do that, but obviously we dont have one. So instead we just accept them to be self-evidentially true.

So, what does Godel's theorem rest on?

In other words, we need a brain to do it.

We also need a brain in order to be irrationally paranoid or to create abstract art. So I don't see how this illuminates Godel's thoughts at all.

The best way to understand the importance of Godel is to read what Bertrand Russell wrote about him. Godel singlehandedly destroyed Russell's obsession with proving that all mathematics was reducible to logic. It rocked Russell's world, and he wrote about it in correspondance.

Why do you need to wax poetic about what Godel means to you when his work speaks for itself?

Why if we need a brain for other things does not the fact that we need a brain to prove some theorems illuminate Godel's thought.

Mathematicians before Godel did not know that there were theorems that could not be proved.

Because it's so generic a statement that it lacks explanatory power.

No, that's not correct. The implication of Godel was that NO theorem could be proved.

Here is what Bertrand Russell said regarding Godel:

NO theorem could be proved.

The implication of Godel was that NO theorem could be proved.

Huh? That's simply false. Wherever did you get such an idea?

What you say is just nonsense.

If a given set of axioms leads to a contradiction, it is clear that at least one of the axioms must be false. Does this apply to school-boys' arithmetic, and, if so, can we believe anything that we were taught in youth? Are we to think that 2 + 2 is not 4, but 4.001?

What makes you think that the unprovable theory will be "the final theory"?

From Bertrand Russell, as I've quoted, who said that Godel threw all mathematical axioms into doubt, including 2+2=4. And Russell was not exactly a slouch -- he can be considered the most important mathematical logician in modern history, so I tend to respect his opinion.

How about what Bertrand Russell says?

Because if we reach the final theory we must have also reached the understanding of the primordial sublime Uncaused Cause. Could that be theorized? I think not!

But Godel is not talking about axioms.

L by the Leibnizian[20] idea that, rather than the universe being "small," that is, one with the minimum number of sets, it is more natural to think of the set theoretic universe as being as large as possible.[21]This idea would be reflected in his interest in maximality principles, i.e., principles which are meant to capture the intuitive idea that the universe of set theory is maximal in the sense that nothing can be added; and in his conviction that maximality principles would eventually settle statements like the CHA (i.e., V = L

incompleteness theorems, published in 1931 when he was 25 years of age, one year after finishing his doctorate at the University of Vienna. The more famous incompleteness theorem states that for any self-consistent recursive axiomatic system powerful enough to describe the arithmetic of the natural numbers (Peano arithmetic, which codes formal expressions as natural numbers.
He also showed that the continuum hypothesis cannot be disproved from the accepted axioms of set theory, if those axioms are consistent. He made important contributions to proof theory by clarifying the connections between classical logic, intuitionistic logic, and modal logic.

Is Godel using the term "axiom" loosely or strictly here, then?



[/INDENT]And I suppose the intro in Wikipedia needs to be revised, then:

Strictly.

About Wiki I have not read it, but I would not be surprised.

Ok. Well so much then for Godel "not talking about axioms", as you said.

Ok. Prove it.