Quotations that help to explain the philosophical consequences of Gödel’s Theorems

>> A logical system is always based on axioms.
>> A logical system cannot prove the consistency of its own axioms and hence cannot prove whether itself is true.
>> No logical system will ever be able to prove everything.
>> Truth cannot be reached by logic.
>> Truth is singular (Truth is One).
https://www.abarim-publications.com/KurtGodel.html

Thus the simplest way to understand Gödel 1931 Incompleteness is to understand that it is nothing more than the mistake of trying to correctly determine the truth value of an expression of language that is not a truth bearer.

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The key philosophical implication of Gödel’s 1931 incompleteness theorem is with the conclusion of its isomorphic cousin the Tarski Undefinability theorem: The notion of [analytic] truth itself can never be completely and consistently formalized within a single formal system.
This prevents Davidson’s truth conditional semantics from being anchored, thus greatly limiting artificial intelligence research.

P. Olcott; Quora

[…] according to Gödel’s incompleteness theorem every (sufficiently powerful) axiomatic system has undecidable formulas; and so a final axiomatization of mathematics is impossible.
en.wikipedia.org/wiki/Mathematics; 11/15/2007

One would normally define a ‘religion’ as system of ideas that contained statements that cannot be logically or observationally demonstrated … Gödel’s theorem, not only demonstrates  that mathematics is a religion, but shows that mathematics is the only religion that proves itself to be one.
J. D. Barrow; The World Within the World; 1988

Gödel’s theorem […] has been taken to imply that you’ll never entirely understand yourself, since your mind, like any other closed system, can only be sure of what it knows about itself by relying on what it knows about itself.
William Denton; Gödel’s Incompleteness Theorem

[…] no axiomatic system can ever be proved to be fully coherent and consistent from within its own rules and postulates.

Gödel’s theorem is a proof of impossibility.

Gödel’s work demonstrated the futility of Principia’s mission […] namely, the mission of formulating a mathematical-logical system that permits demonstration of its own completeness and consistency.
David Gorman et al; Gödel’s Theorem; Proceedings of the Modern Language Association; v110; n5; 1995

But we are not angels, who view the universe from the outside. Instead, we and our models are both part of the universe we are describing. Thus a physical theory is self-referencing, like in Gödel’s theorem. One might therefore expect it to be either inconsistent, or incomplete. The theories we have so far, are both inconsistent, and incomplete.
Stephen Hawking; Gödel and the End of Physics

While it is possible that Gödel may have been inspired in his researches by the antinomy of the Liar, he chose to deal with the concept of demonstrability rather than with the notion of truth. The sentence constructed by Gödel, for examination of its curious consequences, was one that asserted of itself (by implication) that it could not be demonstrated, that is, that it could not be derived from the primitive propositions and definitions of the language of which it was a part, by applying the rules of inference of that language. The difficulties which confront us when we utter such a sentence are obvious at a glance. If such a sentence can be demonstrated, then it cannot be demonstrated, since this is what it says of itself. If, on the other hand, it can be refuted, then it can be demonstrated, since this is the contradictory of what it says of itself. The conclusion emerges that this sentence which proclaims its own indemonstrability, can neither be demonstrated nor refuted within the limits of the language to which it belongs, that it is, in technical parlance, ‘undecidable’ in that language. It can at best be decided in another higher language which has certain novel terms and notions at its disposal. The sentence considered by Gödel does not, however, directly say of itself that it is not demonstrable, but only that a certain number, which is uniquely correlated with the sentence in question, is characterized by a numerical property which corresponds to indemonstrability. It therefore raises the issue of undecidability in the arithmetical as well as in the linguistic realm.
J. Findlay; Goedelian Sentences: A Non-Numerical Approach; Mind; 1942; pp259-60

In term of Gödel’s paper, it was not possible to come up with an axiomatization of mathematics that would have its proof of consistency within itself.

Then it follows that there can be no final physical theory which would be necessarily true at least in its mathematical part.

[…] expectations about a final form of physics are illusory, a fact which physicists do not seem to know.

[…] Gödel’s theorem denies us the possibility of constructing a complete, consistent description of physical reality.

It does not mean at all the end of Physics. It means only the death knell on endeavours that aim at a final theory according to which the physical world is what it is and cannot be anything else.

Gödel’s theorem means, among other things, that physicists who aim at reading God’s mind will not succeed, because they cannot read their own minds in the first place.

Stanley L. Jaki; A Late Awakening to Gödel in Physics

Gödel’s Second Incompleteness Theorem states that no sufficiently strong consistent mathematical theory can prove its own consistency.
Thomas Jech; On Gödel’s Second Incompleteness Theorem; Proceedings of the American Mathematical Society; v12; n1; 1994

In 1931, Kurt Gödel (1906-1978), in his First Incompleteness Theorem, proved that any classical self-consistent formal language capable of expressing arithmetic must also contain sentences of arithmetic that cannot be derived within that system, and hence that the propositions expressed by those sentences could not be proven true (or false) within that system. Thus the concept of truth transcends the concept of proof in classical formal languages. This is a remarkable, precise insight into the nature of truth.
But when the Austrian logician Kurt Gödel turned his mind to this in 1931, he revealed a hole at the heart of our conception of numbers. His “incompleteness theorems” showed that arithmetic can never have truly solid foundations. Whatever axioms are used, there will always be gaps. There will always be facts about numbers which cannot be deduced from our chosen axioms.
By Richard Elwes 7:50AM GMT 09 Nov 2010

The point of Gödel’s theorem is in the fact that it is possible to construct propositions meaningful within a system yet undemonstrable in that system.
Louis O. Kattsoff; The Place of Logic in a World of Fact; Philosophy and Phenomenological Research; v10; n1; 1949

Gödel’s theorem, which shows the incompleteness of any attempt to systematize arithmetic, is perhaps the single most famous and important result in logic in this or any century. It is also one of the most difficult to understand. Recently, however, George Boolos has found a much simpler proof of the incompleteness theorem. The proofs of Gödel and Boolos are each modelled on famous, but different, paradoxes. Gödel makes use of the Liar Paradox (What I am now saying is false — a paradox because if true then it’s false, and if false then it’s true.) He formulates a sentence of arithmetic which says something like ‘I am not provable’, and sure enough it turns out to be unprovable, so it would seem to be a true but unprovable sentence of arithmetic, making the system of axioms incomplete. Boolos’s proof is modelled on the Berry Paradox (The least integer not namable in fewer than nineteen syllables — a paradox because I just named that integer in eighteen syllables.) Even though his proof is much simpler than Gödel’s, it still requires a fair bit of work.
James R. Brown; Philosophy of Mathematics; 1999; p71

Gödel’s Ontological Axioms
Axiom 1: If a property is positive, then its negation is not positive.
Axiom 2: Any property entailed by – i.e., strictly implied by – a positive property is positive.
Axiom 3: The property of being God-like is positive.
Axiom 4: If a property is positive, then it is necessarily positive.
Axiom 5: Necessary existence is positive. (cf. Descartes: existence is a perfection)
Axiom 6: If a property P is positive, then being necessarily P is positive.
Daniel Bonevac; https://www.youtube.com/watch?v=LS3S2FNqxV4