Post by Khong Dong
which should be a *summary* of why Gödel's 1931 incompleteness paper is
an invalid argument (1st Incompleteness statement anyway).
There are more reasons than those detailed in the summary, but if you'd like to correctly, technically counter argue the summary as it is, you could of course
make yourself be heard.
Comprehensive reply from Alex Eustis, Ph.D. Mathematics, University of
California, San Diego (2013)
No, that's not possible. Even without fully understanding the substance
of your complaint, I can surmise that you're wrong, and with as much
certainty as it is ever possible to have about anything.
If all of Gödel's results happen to be complete bunk because he wrote <
instead of >
or something like that, then there is absolutely no chance that nearly a
century's worth of scrutiny by the whole worldwide Mathematical
community somehow missed this — and yet you, some random dude with a
B.A. in Mathematics and an inflated ego, are the first person to notice
this error. I mean, how arrogant is that?
This paper was a breakthrough — it more or less kicked off the entire
study of Incompleteness, Independence, provability logic, and so forth.
But just as we can do Galois Theory without having to refer back to
Galois’ original notebooks that he frantically scribbled the night
before his fatal duel, we can today do Incompleteness Theory and prove
the independence of various statements from certain axiomatic systems
without directly citing Gödel's 1931 paper.
For instance, according to Wikipedia,
Stephen Cole Kleene (1943) presented a proof of Gödel's
incompleteness theorem using basic results of computability theory.
Franzén (2005, p. 73) explains how Matiyasevich's solution to
Hilbert's 10th problem can be used to obtain a proof to Gödel's first
Smorynski (1977, p. 842) shows how the existence of inseparable
sets can be used to prove the first incompleteness theorem.
Chaitin's incompleteness theorem gives a different method of
producing independent sentences, based on Kolmogorov complexity.
So even if you happen to be right (which you aren't), and there is some
fundamental, uncorrectable error with Gödel numbering (which there
isn't) — can you find an ingenius rebuttal for all these proofs, and
more? Can you single-handedly overturn a century of Mathematics?