Demonstrated in an alternative way, that the Theorems of Gödel are true, and hold not only for some special mathematical problems but in general (for any kind of statement in any kind of system/situation). As applications: Hilbert’s Second Problem Solved. Agnosticism is solved. The burden of Disproof is given to atheists. Andrew Wiles’s proof of Fermat’s Last Theorem (which is a hypothesis) uses unproven hypothesis-es of set theory (not the axioms of set theory), thus, the proof is debunked.
Proof of the Second Incompleteness Theorem
The set of axioms produces statements. Some are decidable, some are undecidable. To prove in full range the consistency of mathematics is to prove the validity of all statements, including undecidable ones. Latter to do is impossible by definition. Thus, it is not possible to prove, that mathematics is consistent.
Another way to prove the Gödel’s Second Theorem:
- Axioms are defined as undecidable things.
- Such things are true.
- Thus, axioms are true, and, thus, the set of axioms are without self-contradiction, i.e. consistent.
Thus, a consistent set of axioms can not be proven.
The axioms are defined not as assumptions, but as undecidable but obvious things. Indeed, some axioms can be logically demonstrated [thus, gaining the status of theorems or facts].
Application to Fermat’s Last Theorem
Colin McLarty: „This paper explores the set theoretic assumptions used in the current published proof of Fermat's Last Theorem, how these assumptions figure in the methods Wiles uses, and the currently known prospects for a proof using weaker assumptions.“
What Does it Take to Prove Fermat's Last Theorem? Grothendieck and the Logic of Number Theory | Bulletin of Symbolic Logic | Cambridge Core
Such assumptions are not axioms, because they are not obvious things. Secondly, the Proof of Fermat’s Theorem is outside the axioms of algebra, because it supposed to use axioms of the set theory. Therefore, within the algebra the Fermat’s theorem is still neither proven, nor disproven. It is a strong candidate then for an undecidable statement of algebra [therefore the Hilbert’s Second Problem, which is talking about algebra axioms, is becoming solved through my arguments above]. Conclusion: Fermat’s Hypothesis was proven by another hypothesis-es („assumptions“), thus there is no proof of Fermat’s statement even in the set theory.
Application to Agnosticism
Agnostics are making one claim: God is not decidable. But if one can neither prove nor disprove God, then God exists.
Application to Gnostic Atheism
The fact to accept: if one can neither prove nor disprove God, then God exists. Hereby because Gnostic Atheists hope for absence God, then God could be disproven. Because God could be disproven, then it is wrong to assign Burden of Disproof exclusively to theists. In such a case the atheists must accept, that God satisfies Popper’s Falsifiability criterion, thus the God is scientific.
More in the viXra:
Wiles Has not Proven the Fermat’s Last Theorem, viXra.org e-Print archive, viXra:2005.0209
