No matter how many articles I read on this subject I cannot comprehend how it proves what it proves. I do well with words and rhetorics, philosophy and science – but as soon as you add numbers my mind goes blank. Not very helpful when those fields often rely on equations and models for explanations and proof. I can somewhat understand equations if explained in a simple or cohesive way – but if at all possible analogies or just word-centric explanations would be very helpful.
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The essential idea is Quine’s paradox, which is just an upgrade of Liar’s paradox.
Liar’s paradox said “this sentence is false”. This paradox relies on self-reference (the sentence talk about itself). Every formal system of logic in math don’t allow self-reference.
Quine’s paradox said ” ‘yields falsehood when preceded by its quotation’ yields falsehood when preceded by its quotation”. This sentence does not refer to itself, but the quotation of itself. It said that if you take the string of letter “yields falsehood when preceded by its quotation”, put that at the end of a sentence, then put the entire string in the quote, then put it at the beginning of the sentence, to form a sentence, then that sentence if false. This is a clever way for a sentence to talk about a different statement that looks exactly like itself.
So if you have a logical system of string manipulation, you can potentially perform Quine’s paradox. If you do Quine’s paradox literally, you end up with Tarski’s undefinability of truth: since the paradox cannot happen, the only way out is that there are no ways to define “true” or “false” in the system.
Instead of of doing Quine’s paradox literally, we use it as a recipe to change any self-reference paradox into a paradox that does not use self-reference.
For example, we have the following Curry’s paradox: “if this sentence is true, then Santa is real”. If the sentence is false, then the implication “this sentence is true” -> “Santa is real” is false, but the only way the implication can fail is if the hypothesis is true and the conclusion is false, so the sentence is true, a contradiction. So the sentence must be true, then it is concluded that Santa is real.
We can use that same recipe for Quine’s paradox to upgrade this into Lob’s paradox: ” ‘implies Santa is real when preceded by its own quotation’ implies Santa is real when preceded by its own quotation”. By a similar argument to above, this sentence must be true so Santa is real.
But as I mentioned above, what we discovered is that “true” and “false” cannot be defined by the system (to prevent Quine’s paradox), so we can’t actually talk about whether the sentence is true or false. However, what we could talk about is whether the sentence is provable or not. This is because provable is a much more mechanical condition: a proof must exist, and a proof is a string satisfying all the law of logical inference.
So we have the following version of Lob’s paradox: ” ‘when preceded by its own quotation, if provable implies P’ when preceded by its own quotation, if provable implies P” (where P is an arbitrary statement). If the sentence is provable, then it is also provable that the existence of the proof of this sentence implies P (because that’s what the sentence said). But if there is a proof that the sentence is provable implies P, then if there is a proof that the sentence is provable, there is also a proof of P. Combining the 2 statement above, if the sentence is provable, and there is a proof that the sentence is provable, then there is a proof of P. But if there is a proof of the sentence, then of course there is a proof that the sentence is provable, so we can eliminate a redundancy above: if the sentence is provable, then there is a proof for P. Now we make a jump, an additional assumption! We assume that we have a proof that a proof of P implies P is true. Then if the sentence is provable, then P is true. But that’s exactly what the original sentence said. So the sentence is provable, so P is true.
That’s Lob’s theorem: if “P is provable then P” is provable then P. For any P. This implies Godel’s incompleteness theorem. Let P be the statement “the system is inconsistent”. So we have if “‘the system is inconsistent’ then the system is inconsistent” is provable, then the system is inconsistent. So if the system is consistent, “‘the system is inconsistent’ then the system is inconsistent” is not provable, but that statement is equivalent to saying that the system is consistent. So the claim that the “system is consistent” is not provable.
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