It is direct conclusion from Löb's theorem.
Löb's theorem:
Substitute P with False statement:
But is equivalent , i.e.
I.e., if provable that it's impossible to prove false statement, then false statements are provable. We have reached contradiction, Q.E.D.
I am not sure that the statement □False or "It is provable that False" means anything.
Basically, you have that the word False and a false statement are not the same thing. Therefore, it is not generally the case that one can make a statement of the form "X is false" without X already being either the word False or another statement of the form "X is false."
What this implies is that one cannot in general prove that "False" (just the bare, basic statement that False).
□(¬□False)→□False.
But □False →False.
Thus □(¬□False)→¬□False.
This seems consistent to me. We ...
A system asserting its own soundness: For all T, (□T -> T)
Löb's theorem:
From □T -> T, it follows □(□T -> T). (necessitation rule in provability logic).
From □(□T -> T), by Löb's theorem it follows that □T.
Therefore: any statement T is provable (including false ones).
Or rather: since for any statement the system has proven both the statement and its negation (as the argument applies to any statement), the system is inconsistent.
I don't see how it proves that for X it has proven both the statement and its negation.
I do see how it can be concluded that for any positive statement T, T is provable.
However, if T is a negative statement, then it is provable that not T.
The system isn't inconsistent that way. If it was, we'd have that Löb's theorem itself is false (at least according to PA-like proof logic!).
How do you interpret "soundness"? It's being used to mean that a proof of X implies X, for any statement X in the formal language of the theory. And yes, Löb's Theorem directly shows that PA cannot prove its own soundness for any set of statements save a subset of its own theorems.
I consider myself to be a reasoner at least as powerful as PA, since I am using myself plus PA plus Löb's Theorem to reason about systems at least as powerful as myself (which encompasses everything I've thus far described).
I consider myself "sound" if I believe myself to be trustworthy / reliable enough to believe what I believe Löb's Theorem's says, which is something to do with self-trust and the ability to believe that certain propositions are true, as long as I can prove that proving them true means that they are true.
It's sometimes thought or said that the second incompleteness theorem shows that formal systems such as PA are in some sense "incapable of self-reflection", and their incapability to prove their own consistency reflects this lacuna in their "cognitive capabilities" or something along those lines.
Löb's theorem highlights how this is a much deeper phenomenon: it's not just that these formal systems lack abilities, or formalizations of relevant mathematical insight, or whatever; no, it's that it's formally impossible for any consistent system of the relevant sort to even baldly assert their own consistency, on pain of inconsistency!
That is, we can for instance construct, using the diagonal lemma, a sentence A such that A is, provably in PA, equivalent to "PA + A is consistent" -- thus creating, in a sense a theory that just asserts its own consistency, in the form of a random assertion about Diophantine equations or however we carry out the arithmetization of syntax, for no particular legible reason whatever -- but we then find that PA + A is inconsistent! (This is an immediate consequence of the second incompleteness theorem.)
I recommend Torkel Franzén's excellent "Inexhaustibility -- a non-exhaustive treatment" for anyone pondering these matters.
I'm not sure I understand what you're interested in, but can say a few concrete things.
We might hope that PA can "learn things" by looking at what a copy of itself can prove. We might furthermore expect that it can see that a copy of itself will only prove true sentences.
Naively this should be possible. Outside of PA we can see that PA and its copy are isomorphic. Can we then formalize this inside PA?
In the direct attempt to do so, we construct our inner copy , where is a statement that says "there exists a proof of in the inner copy of PA".
But Löb's Theorem rules out formalizing self-trust this way. The statement means "there are no ways to prove falsehood in the inner copy of PA". But if PA could prove that, Löb's Theorem turns it directly into a proof of !
This doesn't AFAICT mean self-trust of the form "I trust myself not to prove false things" is impossible, just that this approach fails, and you have to be very careful about deferral.
I don't think that PA being able to prove that you cannot prove falsehood means that you can prove falsehood from the theorem. If you look at my response to quetzal_rainbow's answer, a simple substitution of false for X returns "it is provable that (it is not provable that False) implies (it is not provable that False)."
Hm I think your substitution isn't right, and the more correct one is "it is provable that (it is not provable that False) implies (it is provable that False)", ala .
I'm again not following well, but here are some other thoughts that might be relevant:
It's provable for any that , i.e. from "False" anything follows. This is how we give "False" grounding: if the system proves False, then it proves everything, and distinguishes nothing.
There are two levels at which we can apply Löb's Theorem, which I'll call "outer Löb's Theorem" and "inner Löb's Theorem".
So from quetzal_rainbow's answer, I don't have an assumption of self-trust, I only have the substitution of false. I believe making this substitution is fine, but that we prove that "it is not provable that False" from this. Also fine.
From ZT5's answer, he asserts that self-trust is equivalent to the additional assertion "For all T, it is provable that T implies T." I don't have a problem with that assertion, only with the claim that this means that both T and not T are provable (it says that if T is provable, then T. But if T is provable, that doesn't mean that not T is provable).
From both answers, and yours, I gleam that it is either believed that Löb's Theorem by itself is inconsistent, or that it with the additional assumption of "self trust" (not yet formalized to my personal satisfaction yet) is inconsistent.
Depending on the answer, we apparently are not sure if self trust is an additional assumption on top of the theorem or one that is contained in the theorem itself as a conditional.
Saying some more things, Löb's Theorem is a true statement: whenever talks about an inner theory at least as powerful as e.g. PA, the theorem shows you how to prove .
This means you cannot prove , or similarly that .
is one way we can attempt to formalize "self-trust" as "I never prove false things". So indeed the problem is with this formalization: it doesn't work, you can't prove it.
This doesn't mean we can't formalize self-trust a different way, but it shows the direct way is broken.
It might be that the formalization that Lob's Theorem is based in isn't powerful enough to deal with this problem yet.
However, it's got to be powerful enough to deal with the self-trust issue, since on some level, it is about the self-trust issue.
Lob's Theorem is ostensibly about PA or a system at least as powerful as PA. So it itself must be working within such a system.
If that system isn't capable of self-trust, then we can't trust Lob's Theorem. What I believe you're arguing and that the Sequences are arguing essentially amounts to that. What I'm not satisfied with is the Sequences' level of clarity at articulating whether or not Lob's Theorem (and by extension PA and systems at least as powerful as it) are adequate enough to be trusted such that we can use it to formalize self-trust in general. The Sequences are quite ambiguous about this IMO - they don't even state the problem as if this issue were indeterminate - they say only that Lob's Theorem itself means that we can't formalize self-trust. That amounts to essentially saying that "We trust Lob's Theorem completely, and PA, etc., which states that we can't trust Lob's Theorem completely nor PA, etc."
IMO - from my own analysis of the problem, which I choose to trust for deep and very well-thought out reasons [heh] - Lob's Theorem is compelling enough to trust, which says we can trust it and PA-or-greater systems enough to believe in things that it proves. This is the common-sense, intuitive way of interpreting what it means, which I note that importantly means the same thing as having self-trust and trust in the system you're using.
I am looking for some understanding into why this claim is made.
As far as I can tell, Löb's Theorem does not directly make such an assertion.
Reading the Cartoon's Guide to Löb's Theorem, it appears that this assertion is made on the basis of the reasoning that Löb's Theorem itself can't prove negations, that is, statements such as "1 + 3 /= 5."
This is a statement that [due to the presence of negations in it] itself can't be proven within PA.
Now it seems that it is being argued that the inability to do this is a bad thing [that is, being able to prove that we can't prove PA sound with respect to any 'important' class of statements].
I think this is actually a very critical question and I have some ideas for what the central crux is here, but I'd be interested in seeing some answers before delving into that.