Seonglae ChoPeano axioms or ZFC apply only to sufficiently complex and meaningful Axiomatic system, and both can be avoided if there is only a single element or only two truth values.
If you have a finite axioms in Axiomatic system, no matter how much you build and establish theories within that system, it can be like the continuum hypothesis where it is not inconsistent whether this is true or false.
In a consistent mathematical system, there must exist at least one proposition that cannot be proven. No matter how meticulously an Axiomatic system is designed to avoid this, it cannot be avoided. This is proven by the incompleteness theorem. The set of propositions provable in first-order logic precisely has a model. In other words, there necessarily exists a case where the truth defined by proof theory and the truth defined by model theory coincide.
2nd Gödel’s incompleteness theorem
A sufficiently complex axiomatic mathematical system cannot prove its own consistency. In other words, when a mathematical system tries to prove that it is not contradictory, it becomes incomplete.
Gödel’s Incompleteness Theorems
Gödel’s two incompleteness theorems are among the most
important results in modern logic, and have deep implications for
various issues. They concern the limits of provability in formal
axiomatic theories. The first incompleteness theorem states that in
any consistent formal system \(F\) within which a certain amount of
arithmetic can be carried out, there are statements of the language of
\(F\) which can neither be proved nor disproved in \(F\).
According to the second incompleteness theorem, such a formal system
cannot prove that the system itself is consistent (assuming it is
indeed consistent). These results have had a great impact on the
philosophy of mathematics and logic. There have been attempts to apply
the results also in other areas of philosophy such as the philosophy
of mind, but these attempted applications are more controversial. The
present entry surveys the two incompleteness theorems and various
issues surrounding them. (See also the entry on
Kurt Gödel
for a discussion of the incompleteness theorems that contextualizes
them within a broader discussion of his mathematical and philosophical
work.)
https://plato.stanford.edu/entries/goedel-incompleteness/
Gödel's incompleteness theorems
Gödel's incompleteness theorems are two theorems of mathematical logic that are concerned with the limits of provability in formal axiomatic theories. 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.
https://en.wikipedia.org/wiki/Gödel's_incompleteness_theorems
Zarathu Blog: 괴델의 불완전성 정리
김진섭 대표는 5월 30일(목) 제주대학교 경영정보학과 산업·직무 특화 전문가 특강에 참석, 괴델(Kurt Gödel)의 불완전성 정리가 나온 배경을 소개하고 증명의 핵심 아이디어를 수학과 메타수학(meta-mathematics), 괴델수(Gödel number), 그리고 메타수학의 수학화 3가지로 나누어 설명할 예정입니다. 정리한 슬라이드와 강의록을 미리 공유합니다. 초청해주신 현정석 교수님께 감사드립니다. 18/19 세기 미적분학, 해석학의 발전으로, 수학은 점점 기존의 직관과 상식에서 벗어나 추상화되면서 많은 문제점들이 생겼다.
https://blog.zarathu.com/posts/2019-05-20-godelincompleteness/
괴델의 불완전성 정리
괴델의 불완전성 정리에 대한 많은 책들이 시중에 출판되어 있다. 허나 대부분이 역사와 증명의 의미에 치중되어 있고 실제 증명의 아이디어를 설명한 책은 거의 없으며, 있다고 하더라도 외국 서적을 번역하면서 난해한 표현이 많이 나와 이해하기가 어렵다. 이에 본 글에서는 불완전성 정리 증명의 핵심적인 아이디어를 크게 수학(mathematics)과 메타수학(meta-mathematics), 괴델수(Gödel number), 메타수학의 수학화 의 3가지로 나누어 설명하겠다.
https://jinseob2kim.github.io/godel.html
당신이 수학을 모르는 이유. (feat. 불완전성의 정리)
1400만 조회수를 기록한 영상!거짓이라는 것이 모두 다 증명 될 수는 없습니다. 이 사실은 무한대를 재조명하였고, 세계 대전을 단축 시켰고, 현대 컴퓨터의 발명으로 이어졌습니다.학창 시절 때 대체 수학이 어디에 쓸모 있을지 의구심을 가졌던 기억이 납니다. 수학계의 여러 이야기들을...
https://www.youtube.com/watch?v=oippSXvxUlw
