Austrian-born mathematician, logician, and philosopher Kurt Gödel created in 1931 one of the most stunning intellectual achievements in history. His shocking incompleteness theorems, published when he was just 25, proved that within any axiomatic mathematical system there are propositions that cannot be proved or disproved from the axioms within the system. Such a system cannot be both complete and consistent.
The understanding of Gödel’s proof requires advanced knowledge of symbolic logic, as well as Hilbert and Peano mathematics. Hilbert’s Program was a proposal by German mathematician David Hilbert to ground all existing theories to a finite, complete set of axioms, and provide a proof that these axioms were consistent. Ultimately, the consistency of all of mathematics could be reduced to basic arithmetic. Gödel’s 1931 paper proved that Hilbert’s Program is unattainable.
The book Gödel’s Proof by Ernest Nagel and James Newman provides a readable and accessible explanation of the main ideas and broad implications of Gödel's discovery.
Mathematicians, scholars and non-specialist readers are invited to offer their interpretations of Gödel's theory.
Hey Goedel was correct. You can't prove everything. Ask Donald Trump
People should know/read about Gödel's proof, but might also benefit of reading the seven main lines from the Tractatus Logicus by Wittgenstein. The sub-sentences are harder to read and might appear confusing. Especially the last one might sometimes be relevant: "Whereof one cannot speak, thereof one must be silent.". Again Donald Trump might have been better off if he would have understood the sentence.
The idea is you have to go out of the system in order to prove some results.
Like the Fermat problem needed things like Eliptic curves also, foreign to just
plain integer arithmetic.
That is if you mix up a bunch of rules, there are also unexpected side effects which might be true, but cannot be proven. Some results are random (prime numbers)
Many other problems are still open.
The idea that the incompleteness of a or any 'n' series, physically in math is an understatement it defines dimensional reality... and also Peano arithmetic...
Zachary Knutson could you explain on it 'defines dimensional reality'? I do not really gett the idea you are trying to bring accross.
The solution of Fermat’s last theorem is really important for mathematics as it supports the Langland Program. The Langland Program looks at integrating all fields of mathematics. It may be vey important for statistics as well. Statistics used to be shown as formulas and geometry, but it is advancing with/into Graph representations as well.
I have used all three to understand issues in IRT models, but with a more fundamental mathematical basis, we could more easily switch between the different representations.
It would greatly benefit science in general, but statistics in particular may benefit when the Langland program is more developed or even be completed.
To understand more, read Singh Fermat's Last Theorem (2002). I found it very inspiring.
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