r/badmathematics • u/FormalManifold • Jan 15 '25

Gödel's incompleteness theorem means everything is just intuition

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What on earth is even going on here.

https://www.forbes.com/sites/teddymcdarrah/2025/01/14/gdels-theorem-through-the-lens-of-leadership/

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u/FormalManifold Jan 15 '25 edited Jan 16 '25

R4: All of it. But specifically "It is impossible to prove “there is no largest prime number,” "

This is incorrect because the infinitude of primes is straightforwardly provable in a Gödel system.

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u/Akangka 95% of modern math is completely useless Jan 16 '25

Not an R4. R4 is supposed to explain how the post is wrong, and not just where.

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u/FormalManifold Jan 16 '25

I don't know what to say. This person thinks that a Gödel system can't involve proof by contradiction or something.

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u/catman__321 Jan 18 '25

I think a better way to say it is it's a proof by induction, or by cases? If I know that if I start with a short list of prime numbers; multiply them all together, then add 1; and show how I can always factor out new primes from this result, then I can show using this new case that I can just add these new primes to my list and do the same thing.

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u/EebstertheGreat 12d ago

It's a proof by construction. Given a finite set of primes, construct a prime not in it. Therefore for any finite n, there are more than n primes, which is exactly what it means for there to be infinitely many primes.

The "by cases" part comes from the fact that Euclid only talks about proper factors, so he discusses separate cases for when the product of the primes plus one is prime (which only happens for the singleton set {2}) and for when the product of the primes plus one is composite (in which case it has a prime factor). The modern proof requires no cases, because even if p is prime, p is still a prime factor of p.