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My friend claimed that 'Any type of formal system is inconsistent' to support his another claim 'reason for each moral judgement will be different, situationally we can change ( for eg; consent will work in homosexuality, but not in incest)', is it a true statement? I can understand that 'Any type formal system in which arithmetic has a role cannot prove its own consistency' in the light of 'Incompleteness theorem'. Can we generalise the so called fact to something like logical argumentations (set of premises and conclusions) in fields of 'ethics', 'politics', etc.?

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    No application of [GIT}(plato.stanford.edu/entries/goedel-incompleteness) to Ethics at all. In addition, the statement "Any type of formal system is inconsistent" is simply WRONG. May 25, 2022 at 7:14
  • Sir, Can you suggest some consistent system other than mathematical ?
    – Messi Lio
    May 25, 2022 at 7:17
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    But GIT applies to mathematics. And you for sure can imagine "toy systems" with simple axioms that are consistent. Consider the following simple "Axiomatic theory of dogs" with the single axiom: "There is a dog". I think it is consistent. May 25, 2022 at 7:20
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    @MauroALLEGRANZA, yes, the claim that there are no consistent systems is just a bizarre misunderstanding. For that matter "There is not a dog" is consistent too. Consistency is a pretty weak condition satisfied by lots of sets of propositions. May 25, 2022 at 7:33

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The statement your friend made is a poor misuse of logical jargon. Godel's incompleteness theorem states that any formal, axiomatized system of logic that can talk about the natural numbers will always allow for self-referential statements to be made such that there exists some statement within the language of the system but is not provable from that system's axioms because of its self-referential nature. If this statement is to be taken as a new axiom of the system, it is effectively a new system, therefore creating the necessary existence of an equivalently problematic statement. This, by definition, makes the statement incomplete, hence the name "incompleteness theorem". I think this may be what your friend was trying to say, although I hesitate to see how this and the meaning of what he said may be confused.

Godel's incompleteness theorems also proved that it's impossible for a system to prove its own consistency. If a logic is consistent, it is therefore not possible to prove both a statement and its negation to be true. The uncertainty of consistency is in no way identical to inconsistency, as your friend claimed.

The rather complex issue of translating ethics and politics into formal logic is not entirely relevant to Godel's work. The issue with real-life disagreements, generally speaking, is that people tend to operate under different sets of axioms. Take for example the debate between a theist and an atheist. The theist's axioms contains the statement "God exists" along with some characteristics of that god, presumably, whereas the atheist's axioms will include the statement "God does not exist". Therefore they are coming to conclusions originating from nonidentical sets of axioms, and there for are using two nonidentical logics. So to try and logically analyze and compare their arguments doesn't make any sense, you can't mix together two different systems of logic. It is perfectly conceivable that the theist's argument is just as valid as the atheist's despite the contradictions of their conclusions. This at least concerns why analyzing opposing arguments concerning real life issues with physical manifestations isn't sensical.

However, concerning the argument of an individual without comparison to others, there is more to be gained as well as more to be said about Godel's theorems. Assuming this person has a describable set of axioms and does not make logical errors within this system, then it is quite conceivable that the system may be formalized. Once formalized, and considering human can count, it is reasonable to assume this system can operate on the naturals. Therefore it is subject to Godel's incompleteness theorems. What does this mean though, in actuality? How does this manifest itself in the real world? Godel's theorem claims the existence of true statements which can not be proven to be true. Maybe under one person's axioms the statement "cats ought to eat human flesh" is true but cannot be proven to be so. It's inconceivable and arguably irrelevant really the possibilities. The reason I call it irrelevant is because this lies in the realm of humans and human biases and behavior. Humans will not posit some unjustifiable statement to be true. They themselves will not believe it nor will anyone else, and because of the nature of axiomatic logic and the difference in axioms all humans seem to have, it wouldn't even apply to anyone else, necessarily. I realize there are exceptions, namely the case where somebody uses illogical justification to claim a statement true, although I find the chances of this being one of these sought out unprovable but true statements quite low, and my point on irrelevance to other humans still stands.

Alongside all this it's important to note that not human has a well-defined set of axioms that they can ever completely state in explicit logic. Humans are simply too human to understand exactly what the foundations of their reason is, if it's consistent, complete, or contradictory. Interestingly enough, there are ways to find and talk about (from my understanding, this specifically is not something I'm well-educated in) unprovable but true statements in computer science. I've heard from others in my studies that in some areas of the field it's actually quite a significant and well-understood issue. So theoretically, if humans could have a well-defined set of axioms for their reason, these person-specific Godel statements may be better understood. Although like I said, that's not possible.

Your question touched on a lot of interesting concepts in the field of logic and I hope they served you well in your learning. With that said, it is clear to me that you and your friend have minimal education in the field of logic and I would highly encourage you to study more as to better understand these grandiose topics you're trying to critique.

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    "because of its self-referential nature". Not always correct: self-referentiality was only a "technique". Later researches have found ""natural" example of a true statement about the integers that could be stated in the language of arithmetic, but not proved in Peano arithmetic" without use of self-referentiality. See Paris–Harrington theorem May 25, 2022 at 7:47
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    [GIT]( proves that - under the condition of the theorem: formalized system, consistency, etc - there is an undecidable statement, i.e. a statement G such that both G and not-G are unprovable form the axioms of the system. Due to the fact that both a re "syntactically correct" formulas, and thus meaningful, our natural expectation is that one of them (and only one, due to consistency) must be TRUE in the "natural" interpretation of the system. May 25, 2022 at 7:50

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