Will you be able to shed any light, on this matter? — Aryamoy Mitra
Have I significantly misapprehended the argument, — Aryamoy Mitra
Interpreting this sentence, is harder than accruing a mastery over all of Mathematics. — Aryamoy Mitra
Have I significantly misapprehended the argument — Aryamoy Mitra
'By the way, in case you'd like to know: yes, it can be proved that if it can be
proved that it can't be proved that two plus two is five, then it can be proved
that two plus two is five.' — Aryamoy Mitra
I want to see something like this: AxAy(x + y = y + x) in a Godel sentence — TheMadFool
What's a meta-language for F? Does [a meta-language for F] concern or describe the language of F? — Aryamoy Mitra
I think that the incompleteness results have an effect on a wide range of things not just in the set theoretic realm and with the foundations of mathematics. We just don't want to make or are ignorant about the link to the incompleteness results.
I think the classic example of something being true but unprovable is a game theoretic situation where it's easy to show that a correct solution exists, yet there seems to be no way to get there. The existence of a correct solution can be shown...based on mathematics — ssu
Godel's second incompleteness theorem explained in words of one syllable — bongo fury
. I do, nonetheless, intend to learn the language of formal logic — Aryamoy Mitra
Have I significantly misapprehended the argument,
— Aryamoy Mitra
At (5) and (6), yes. — bongo fury
'By the way, in case you'd like to know: yes, it can be proved that if it can be
proved that it can't be proved that two plus two is five, then it can be proved
that two plus two is five.'
— Aryamoy Mitra
No — fishfry
As a clarity, are you refuting the original exposition? This passage, for instance, was word-for-word sourced from another, non-technical resource. — Aryamoy Mitra
By the way, in case you'd like to know: yes, it can be proved that if it can be proved that it can't be proved that two plus two is five, then it can be proved that two plus two is five.' — Some Unknown Entity
Gentzen's consistency proof is a result of proof theory in mathematical logic, published by Gerhard Gentzen in 1936. It shows that the Peano axioms of first-order arithmetic do not contain a contradiction (i.e. are "consistent"), as long as a certain other system used in the proof does not contain any contradictions either. This other system, today called "primitive recursive arithmetic with the additional principle of quantifier-free transfinite induction up to the ordinal ", is neither weaker nor stronger than the system of Peano axioms. Gentzen argued that it avoids the questionable modes of inference contained in Peano arithmetic and that its consistency is therefore less controversial.
I did see the italics but I did not see a link to the source. So I couldn't tell if you were quoting someone else or quoting yourself from some other publication or forum, or just separating out your ideas into quoted form. — fishfry
I believe what they mean is this. For definiteness let's take Peano arithmetic (PA). By Gödel's second incompleteness theorem, PA can not prove its own consistency. That means PA can not prove that it can't prove that 2 + 2 = 5. Agreed so far? Then if PA can prove that it can't prove that 2 + 2 = 5, then PA must have proved its own consistency, which it can only do if it's inconsistent; and if it's inconsistent, then it can prove that 2 + 2 = 5. — fishfry
The point is that proof and consistency are relative to given axiom systems. It's true that PA can't prove its own consistency; but we CAN prove the consistency of PA by other means.
So, to sum this all up: Using ZF (which at least I understand, as opposed to Gentzen's proof, which I don't) I can indeed prove that PA is consistent, and that PA can't prove that 2 + 2 = 5, and that PA can prove that it can't prove that 2 + 2 = 5.
I can always do this as long as I'm willing to go outside PA. And this is true in general. Just because some given system can't prove its own consistency doesn't mean we can't prove its consistency. — fishfry
Thank you, for expanding. PA can't show its own consistency, but PA can be proved consistent outside itself (with other axioms) - and that's a generality that may hold for other arithmetic systems; is that the crux of the argument? — Aryamoy Mitra
Yes, you got the point exactly. I would say that the issue has more than just a superficial relation, but that is just my personal view about the subject.As Nash demonstrated fixed-point theory is useful in game theory. Brouwer's fixed-point theorem was proven indirectly, with no simple path to its value, and this distressed Brouwer, who later turned to intuitionism. Proving a math object exists indirectly, but without a process for its construction, is still proving a theorem. This sort of thing has a superficial relation to Godel's works, but I don't think it's what he had in mind. Others here, with more knowledge of the matter can correct me if I'm in error. — jgill
my complaint is that [Boolos] did not distinguish between "PA can prove ..." and "It can be proved ..." — fishfry
Cantor's proof that there are more reals than natural numbers. The issue here is that the reductio ad absurdum proof [...] — ssu
My guess is that he means ZFC, — TonesInDeepFreeze
while there is obviously is a correct solution: because fixed point theorem proves that there exists a correct solution — ssu
Isn't he just ensuring that what 2 + 2 is equal to is being discussed with respect to a system big enough for the second theorem to apply? — bongo fury
Boolos says that he means proved by "the aid of the whole of math". My guess is that he means ZFC, which is ordinarily understood to provide an axiomatization for mathematics. So, as far as I can tell, he's talking about the second incompleteness theorem for ZFC. — TonesInDeepFreeze
what he wrote was wrong.
— fishfry
How? — bongo fury
Consistency and provability are always relative to a given axiom system. — fishfry
By omitting the fact that we are always working in a particular axiomatic system, the essence of the matter is ignored. — fishfry
First of all, when I say "proved", what I will mean is "proved with the aid of the whole of math".
the maximal (consistent) extension or union of all the systems you mention. — bongo fury
Haha. No? Not a plausible reading of "the whole of math"? — bongo fury
the maximal (consistent) extension or union of all the systems you mention. — bongo fury
No worries. — bongo fury
what Boolos wrote was wrong. — fishfry
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