Separating Truth from Proof in Mathematics. Comment about Godel\'s Proof, or God

Question

Separating Truth from Proof in Mathematics. Comment about Godel's Proof, or Godel's Incompleteness Theorem. This is the most important development in mathematical logic in the twentieth century. More than 130 words please.

As an example of something in mathematics that is likely true but not provable, (as well as easily to understand, consider Goldbach's conjecture: That every even number greater than two is the sum of two primes. It has been around for more than two hundred years, it has been verified for billions and billions of even numbers, but there is no satisfactory proof of the conjecture. Mathematicians have started to hold that Goldbach's conjecture is true but not provable.

Explanation / Answer

Gödel's incompleteness theorems are two theorems of mathematical logic that demonstrate the inherent limitations of every formal axiomatic system containing basic arithmetic. These results, published by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of mathematics. The theorems are widely, but not universally, interpreted as showing that Hilbert's program to find a complete and consistent set of axioms for all mathematics is impossible

The Intuitionists argue that all mathematics can be stated operationally, and as such, for all intents and purposes, all mathematical symbols other than the glyphs we use to name the natural numbers, are nothing more than names for functions (sets of operations).

However, the intuitionist ('recursive') solution causes a problem in that the excluded middle is impermissible - but without it, much of mathematics because much more difficult, and harder to prove. So with that constraint on the excluded middle, the higher truth requirement of computational and constructivist, intuitionist logic has been deemed not useful for departmental mathematicians.

So under the ZFC+AC and 'spontaneous platonic imaginary' creation of sets, we obtain the ability to do mathematicsthatinclude both double negation and the excluded middle.  

This 'trick' separates Pure math in one discipline and Scientific math, Computational mathematics, and philosophical realismintodifferent discipline, each with different standards of truth. In fact, technically speaking, mathematics is absent truth (correspondence) and relies entirely on proof. ie: there are no true statements in pure mathematics.

IF ANYONE KNOWS --->> It does not appear that Brouwer or any of his followers understood why their method failed and the set method succeeded. But even if they failed, I am trying to figure out if the Formalists understood their 'hack' and why it worked.  

And lastly, if anyone at all understood how Intuitionist, constructivist, and computational logic could be improved to solve the problem of retaining correspondence (truth) while also retaining the excluded middle (even if it was burdensome).

I'm going to assume that you mean unprovable under the usual axioms of set theory, assuming those axioms are consistent: Zermelo-Fraenkel, with the Axiom of Choice (ZFC).

I'm also going to assume that you want statements whose negation is known to be unprovable. After all, it's not very exciting to observe that 1+1=3 (operating in the integers) is unprovable; ofcourse it's unprovable, because its negation is provable and we're assuming that ZFC is consistent.

The Continuum Hypothesis is probably the most famous example. Informally, it says that there is no set X that lies between the natural numbers and the real numbers in size. A bit more formally, there is no set X such that 1) the natural numbers map into X by a one-to-one function and 2) X maps into the real numbers by a one-to-one function. Gödel proved that the Continuum Hypothesis is consistent with ZFC, and Cohen proved that the negation of the Continuum Hypothesis is also consistent with ZFC.

The Collatz conjecture may be unprovable. There are similar-looking statements that are known to be unprovable. They say the same thing about always terminating at 1, but they use functions that are different from n/2 and 3n+1.

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