Gödel used a modified version of the liar paradox, replacing "this sentence is false" with "this sentence is not provable" — Wikipedia

Yes, there is an analogy and similarity, but also the modification used makes a great difference too.

Yes. Again, note the 'roughly speaking'. I think that is what our expert contributors are emphasising. But I do agree there is a broader, underlying point. It all seems to revolve around the issue of self-reference or reflexivity.

I can't recall learning of anything in classical analysis (complex or real) affected by incompleteness, but that doesn't mean much since I have been out to pasture for many years and my memory is imperfect. If one of you comes across something please post. :chin:

reflexivity — Wayfarer

Correct but it's not just self-reference, it's also negation of some kind. The self-referential sentence, "I exist" doesn't create problems like the self-referential negation, "I don't exist."

Let's look at the liar sentence L = this sentence (L itself) is false.

According to most books, the logic is as below,

Option 1
1. If L is true then L is false (the liar sentence)
2. L is true (assume)
3. L is false (1, 2 modus ponens)
4. L is true and L is false (2, 3 together, contradiction)
Ergo,
5. L is false (2 - 4 reductio ad absurdum)

Option 2
6. If L is false then L is true (the liar sentence)
7. L is false (assume)
8. L is true (6, 7 modus ponens)
9. L is true and L is false (7, 8 together, contradiction)
Ergo,
10. L is true (7 - 9 reductio ad absurdum)

Now, what's interesting is,

1. If L is true then L is false = L is false or L is false = L is false
That means the argument in option 1 becomes,
11. L is false = If L is true then L is false
12. L is true (assume)
13. L is true and L is false (11, 12 together, contradiction)
Ergo,
14. L is false (12 - 13 reductio ad absurdum)

However, notice line 11 (premise) = line 14 (conclusion). In other words, what was to be proved was assumed beforehand among the premises. The argument is circular which simply means 14. L is false is unwarranted.

Similarly, revisiting option 2, the statement 6. If L is false then L is true = L is true or L is true = L is true

The argument for option 2 then becomes,
15. L is true = If L is false then L is true
16. L is false (assume)
17. L is true and L is false (15, 16 together, contradiction)
Ergo,
18. L is true (16 - 17 reductio ad absurdum)

Notice here again that line 15 (premise) = line 18 (conclusion). Put simply, the conclusion has been assumed in the premises. Circular argument, which means we're not justified in concluding 18. L is true.


What does this all mean? We can arbitrarily assign a truth value to the liar sentence (true/false) but that's where it all stops - all logic beyond that is going to be circular and useless. Since the alleged contradiction of the liar paradox can only occur after an inference which begins with an assumption of a truth value for the liar sentence, and since, as explained above, all such arguments are circular, we're no longer justified to infer anything at all (that includes any further truth value for the liar sentence) from the initial assigned truth value for the liar paradox. Ergo, there being no inferrable truth value, there can be no contradiction. In short, the liar paradox doesn't entail a contradiction at all.

"[...] a self-referential sentence which “says of itself” [...] Such figures of speech may be heuristically useful, but they are also easily misleading and suggest too much." - https://plato.stanford.edu/entries/goedel-incompleteness/#DiaSelRef

Read more there for more explanation.

On the other hand, see page 44 of Franzen's'Godel's Theorem: An Incomplete Guide To Its Use And Abuse' for a different take on the matter.

In any case, in whatever sense we may reasonably say the proof uses "self-reference", the proof is not vitiated by it, as the proof can be carried out in finitistic combinatorial arithmetic.

If L is true then L is false = L is false or L is false = L is false — TheMadFool

The equal sign there and the ones following it are not syntactical. Perhaps you mean the biconditional. If so, rewrite to see whether your argument still holds up,

My disagreement with Pfhorrest seemed to perhaps hinge on his use of the term "definitely true". He hasn't responded to say whether he would claim that "There's never a statement in any given language that is both true according to the rules of that language and also not provable in that language, because to be true according to the rules of a language just is to be provable in that language." (The statement he made leaving out the word "definite"). — Janus

Leaving out the "definitely"s completely changes the meaning, so no, I wouldn't claim that modified sentence.

We cannot know for sure ("definitely", or "certainly") that some proposition is true, without in the process having proven it, so we cannot know for sure that any given proposition is true-but-unprovable, because to be sure of the first part we would have to violate that second part.

It might in some principled way remain the case that something or another could be true but not provable, but we could never say for sure that we had an example of that, because in somehow saying for sure that something was true, we just would be proving it.

we cannot know for sure that any given proposition is true-but-unprovable, because to be sure of the first part we would have to violate that second part. — Pfhorrest

It is unprovable in the system being discussed. It is provably true in the mathematics used to discuss that system.

It might in some principled way remain the case that something or another could be true but not provable, but we could never say for sure that we had an example of that — Pfhorrest

We have a sure example. It's as sure as finitistic combinatorial arithmetic. It's a quite complicated calculation, but it is still a finite calculation.

I too will plead, as at this time, especially as I am rusty in the subject, I wouldn't be able to marshal enough knowledge to explicate the details of the mathematics mentioned there. — TonesInDeepFreeze

I'm back and I've read the texts you referenced. They were very interesting but not very helpful in answering my questions. I don't need to go any further with this except to say I was surprised to see that some of the philosophical claims associated with Godel's theorum are similar to the mystical/philosophical claims associated with quantum mechanics. These in particular struck me:

proves that Mechanism is false, that is, that minds cannot be explained as machines.

They all insist that Gödel’s theorems imply that the human mind infinitely surpasses the power of any finite machine or formal system.

I'm going to leave it here. I appreciate your help. This was fun.

That's somewhat helpful, thanks.I'd need to investigate deeper into the subject to really get what you're saying I suspect.

It is unprovable in the system being discussed. It is provably true in the mathematics used to discuss that system. — TonesInDeepFreeze

Right, which is what I said a few posts back in that exchange. It's only in the meta-language, not the object language, that we can assign a truth value to the proposition, and in the meta-language we are also able to prove that proposition. The meta-language will itself also contain propositions which it cannot prove, and to which it cannot assign truth values; and in yet another language being used to discuss that first meta-language as an object language, a truth-value can be assigned to the unprovable statements of the first meta-language, but in that even higher-order language those statements can also be proved. You never have both a definite truth value and unprovability on the same level. "True but unprovable" only works when you mix levels: it's true, according to the higher-level system we're using to discuss the lower-level system, and unprovable, according to that lower-level system; but that lower-level system has no idea whether or not it's true (because it's unprovable), and in the higher-level system it's provable (which is how it can be known true).

Leaving out the "definitely"s completely changes the meaning, so no, I wouldn't claim that modified sentence.

We cannot know for sure ("definitely", or "certainly") that some proposition is true, without in the process having proven it, so we cannot know for sure that any given proposition is true-but-unprovable, because to be sure of the first part we would have to violate that second part.

It might in some principled way remain the case that something or another could be true but not provable, but we could never say for sure that we had an example of that, because in somehow saying for sure that something was true, we just would be proving it. — Pfhorrest

I agree that we cannot know for sure that some proposition is true without having proven it, and thus it would seem to follow that we cannot know that any given proposition is "true-but-unprovable". But the truth of any proposition is not dependent on our having proven it, but only our knowing of the truth of it is.

If I understood mathematics much better than I do I might be able to offer an opinion as to whether Godel has proved that there are true propositions within mathematics that cannot be mathematically proven.

Perhaps the answer to that last question is no] it seems: is this from the SEP is correct and if I have correctly interpreted its meaning:

"A common misunderstanding is to interpret Gödel’s first theorem as showing that there are truths that cannot be proved. This is, however, incorrect, for the incompleteness theorem does not deal with provability in any absolute sense, but only concerns derivability in some particular formal system or another. For any statement A unprovable in a particular formal system F, there are, trivially, other formal systems in which A is provable (take A as an axiom). On the other hand, there is the extremely powerful standard axiom system of Zermelo-Fraenkel set theory (denoted as ZF, or, with the axiom of choice, ZFC; see the section on the axioms of ZFC in the entry on set theory), which is more than sufficient for the derivation of all ordinary mathematics. Now there are, by Gödel’s first theorem, arithmetical truths that are not provable even in ZFC. Proving them would thus require a formal system that incorporates methods going beyond ZFC. There is thus a sense in which such truths are not provable using today’s “ordinary” mathematical methods and axioms, nor can they be proved in a way that mathematicians would today regard as unproblematic and conclusive.".

Is there any proof that such a "formal system that incorporates methods going beyond ZFC." will or even could be found? In light of all this what you said about there always being a meta-level wherein the unprovable truths within a system can be proven seems questionable.

Even if there were always a meta-level capable of (non-trivially) proving any true mathematical proposition, then there would still seem to be a kind of infinite regress of incomplete systems. Would this
fact render all such proofs non-exhaustive and/ or trivial I wonder (in my mathematical ignorance)?

The general idea you expressed is okay, but I suggest some clarifications and context (much of which you likely know already).

We are concerned not just with the object-language and a meta-language, but the object-theory and a meta-theory.

With a meta-theory, there a models of the object-theory. Per those models, sentences of the object-language have truth values. So the Godel-sentence is not provable in the object-language but it in a meta-theory, we prove that the Godel-sentence is true in the standard model for the language of arithmetic. Also, as you touched on, in the meta-theory, we prove the embedding of the Godel-sentence into the language of the meta-theory (which is tantamount to proving that the sentence is true in the standard model). That is a formal account of the matter. And in a more modern context than Godel's own context, if we want to be formal, then that is the account we most likely would adopt.

Godel himself did not refer to models. Godel's account is that the Godel-sentence is true per arithmetic, without having to specify a formal notion of 'truth'. And we should find this instructive. It seems to me that for sentences of arithmetic, especially ones for which a computation exists to determine whether it holds or not, we are on quite firm ground "epistemologically" to say, without quibbling about formality, that the sentence is true when we can compute that it does hold.

that lower-level system has no idea whether or not it's true (because it's unprovable) — Pfhorrest

There is an even more fundamental reason that the object-theory does not yield a determination of truth. That is that the object-language does not have a truth predicate. There's a subtle difference: A theorem of the object-theory is true in any model of the theory, so in that sense one would say that the object-theory does determine the truth of certain sentences. But the object-theory does not itself have a theorem that the sentence is true in models of the theory (or else, the object-theory would be inconsistent per Tarski's theorem).

I might be able to offer an opinion as to whether Godel has proved that there are true propositions within mathematics that cannot be mathematically proven. — Janus

It depends on the definition of 'mathematically proven'.

Is there any proof that such a "formal system that incorporates methods going beyond ZFC." will or even could be found? — Janus

It is easy to see that there are theories that are proper extensions of ZFC . But that doesn't settle the question of whether those theories are within what we consider to be justifiable mathematics, or even the question of what it means to be justifiable mathematics.

what you [Pfhorrest] said about there always being a meta-level wherein the unprovable truths within a system can be proven seems questionable. — Janus

It is the case that there is an infinite escalation of theories, each proving arithmetical truthts not provable in the lower theories.

Would this fact render all such proofs non-exhaustive and/ or trivial — Janus

The theories are not exhaustive, indeed. But I don't see why that would make the proofs trivial.

It seems to me that for sentences of arithmetic, especially ones for which a computation exists to determine whether it holds or not, we are on quite firm ground "epistemologically" to say, without quibbling about formality, that the sentence is true when we can compute that it does hold. — TonesInDeepFreeze

What gets to me, and maybe you can clarify, is how it could be that we can “compute that it does hold” and yet not have, at some level or another, thereby “proven” it. If we are taking for granted (as proven, in some sense or another, or else just assumed) that arithmetic works the way we usually use it, and an arithmetical operation yields a certain output, have we not consequently proven (or assumed) that output as part and parcel of having proven (or assumed) that arithmetic works in such a way as that?

What gets to me, and maybe you can clarify, is how it could be that we can “compute that it does hold” and yet not have, at some level or another, thereby “proven” — Pfhorrest

Doesn’t it mean that there must always be some assumptions? Something that is assumed to be so, but can’t be proven to be so? Isn’t that what the issue is about?

I'm pretty sure that's not what Godel is on about at all.

But on that unrelated topic, I am vociferously opposed to justificationism, the usual kind of rationalism (contra critical rationalism), which says that you should reject everything that can't be proven conclusively "from the ground up", because per Agrippa's / Munchausen's trilemma that is inherently impossible. Instead, as a critical rationalist, I think it's fine (and necessary) to run with whatever assumptions you're inclined to, until they can be disproven.

Godel's about whether there are things that are true but aren't provable. And I don't see how we can ever do better than "Maybe? I suppose it's always technically possible, but we can never be sure whether or not there are". Because to sure, we would have to be sure that something was unprovable, and also be sure that it was true -- and I don't see how we could "be sure that it was true" without, in doing so, proving it, and so showing it to be not-unprovable.

As I understand it, Godel only shows something about the relationship between a formal system and statements in it: that some systems can't prove some things they're capable of talking about either way, even though we can know, through in a proof made in a higher-level system, that those things are true.

Godel's about whether there are things that are true but aren't provable. — Pfhorrest

Isn’t that what I said?

You said things that are assumed but can't be proven. By the nature of assumptions, we don't know whether or not they are true. Whether or not it's okay to believe things that are only assumptions and not proven is a different question from whether or not there definitely are things that are true but not provable.

I’m afraid to say that you’re splitting hairs.

All philosophical progress is made by splitting the right hairs.

I seem to recall reading somewhere that Gödel was a mathematical Platonist. Are you suggesting that Gödel's incompleteness theorem would be trivially true on a formalist understanding of mathematics because to be true in a language just is to be proven in that language? — Janus

I wasn't specifically referring to Gödel's theorem, but using that example, a strictly formal reading of the first incompleteness result would be like this quote in the Wiki article:

Any consistent formal system F within which a certain amount of elementary arithmetic can be carried out is incomplete; i.e., there are statements of the language of F which can neither be proved nor disproved in F. — Raatikainen 2015

Here the statement is not said to be either true or false; if pressed, an anti-realist* might say that (a) the question of truth is meaningless outside the context of a particular formal system, and (b) in the context of system F the Gödel statement is neither true nor false, since it can be neither proved nor disproved in that system.

On the other hand, you have, no doubt, heard paraphrases to the effect that the Gödel statement is "true but unprovable." Such readings lean on a realist/Platonist understanding of mathematical truth. They would appeal to the structure of the Gödel statement, which states an arithmetical truth.

This is a very crude and clipped summary. Like I said, the question of truth in mathematics and its relation to provability has been investigated and debated at great length. Just searching for works with the words "truth" and "provability" or "proof" in the title will net you several pages of results on Google Scholar.

* I will withdraw the label "formalism" and use instead the more vague "anti-realism" or "anti-Platonism."

There are different versions of formalism, and it is not the case that in general formalism regards truth to be just provability. — TonesInDeepFreeze

Sextus Empiricus against the dogmatist's criteria of truth:

At the end of Sextus’ discussion in PH II, he clearly signals, as one would expect, that he suspends judgment on whether there are criteria of truth:

You must realize that it is not our intention to assert that standards of truth are unreal (that would be dogmatic); rather, since the Dogmatists seem plausibly to have established that there is a standard of truth, we have set up plausible-seeming arguments in opposition to them, affirming neither that they are true nor that they are more plausible than those on the contrary side, but concluding to suspension of judgement because of the apparently equal plausibility of these arguments and those produced by the Dogmatists. (PH II 79; cf. M VII 444) — Stanford Encyclopedia of Philosophy

According to Chisholm, there are only three responses to the Problem of the Criterion: particularism, methodism, and skepticism. The particularist assumes an answer to (1) and then uses that to answer (2), whereas the methodist assumes an answer to (2) and then uses that to answer (1). The skeptic claims that you cannot answer (1) without first having an answer to (2) and you cannot answer (2) without first having an answer to (1), and so you cannot answer either. Chisholm claims that, unfortunately, regardless of which of these responses to the Problem of the Criterion we adopt we are forced to beg the question. It will be worth examining each of the responses to the Problem of the Criterion that Chisholm considers and how each begs the question against the others. — Internet Encyclopedia of Philosophy

how it could be that we can “compute that it does hold” and yet not have, at some level or another, thereby “proven” it. — Pfhorrest

Because the reckoning itself is not necessarily in a formal context, so it is not formal proof, though it could be.

Something that is assumed to be so, but can’t be proven to be so? Isn’t that what the issue is about? — Wayfarer

I don't know what issue you mean when you ask what the issue is about. But for incompleteness, it's not just a matter of having to assume things to prove things.

Godel's about whether there are things that are true but aren't provable. — Pfhorrest

That is exactly the most salient oversimplification that causes misunderstanding.

You know the following, but it bears emphasizing:

There is no mathematical statement that isn't provable. That is, for any mathematical statement (even a self-contradiction) there are systems that prove the statement.

Godel's theorem is that for any given system S of a certain kind there are statements F in the language for S that such that S proves neither F nor ~F.

It's a matter of quantifier order:

Godel: For any system S of a certain kind, there exist statements undecided by S.

False: There exist statements F such that for any system S of a certain kind, F is undecided

Godel only shows something about the relationship between a formal system and statements in it: that some systems can't prove some things they're capable of talking about either way, even though we can know, through in a proof made in a higher-level system, that those things are true. — Pfhorrest

That seems to me to be a reasonable summary.

I’m afraid to say that you’re [Pfhorrest] splitting hairs. — Wayfarer

I don't think he is. The distinction he's making is very important.

Any consistent formal system F within which a certain amount of elementary arithmetic can be carried out is incomplete; i.e., there are statements of the language of F which can neither be proved nor disproved in F.
— Raatikainen 2015

Here the statement is not said to be either true or false [...] in the context of system F the Gödel statement is neither true nor false, since it can be neither proved nor disproved in that system. — SophistiCat

Yes, the theorem itself, as you quoted it, does not mention truth. But from the theorem, we do go on to remark that the undecided sentence is true.

And the statement is neither true or false in the system on an even more fundamental basis than that it is undecided by the system:

There is an even more fundamental reason that the object-theory does not yield a determination of truth. That is that the object-language does not have a truth predicate. There's a subtle difference: A theorem of the object-theory is true in any model of the theory, so in that sense one would say that the object-theory does determine the truth of certain sentences. But the object-theory does not itself have a theorem that the sentence is true in models of the theory (or else, the object-theory would be inconsistent per Tarski's theorem). — TonesInDeepFreeze

realist/Platonist — SophistiCat

When we're talking about plain arithmetical truths, I don't know why we would have to go down the road of wondering about realism. I mean, non-realists still recognize the truth of arithmetical statements.

It is unprovable in the system being discussed. It is provably true in the mathematics used to discuss that system. — TonesInDeepFreeze

I should qualify that remark and others I made along the same lines.

We prove (though not in the object system) that the Godel-sentence is true on the assumption that the object-system is consistent.. That qualification might be regarded as implicit in my remarks, but it is best for me to make it explicit.

we are on quite firm ground "epistemologically" to say, without quibbling about formality, that the sentence is true when we can compute that it does hold. — TonesInDeepFreeze

I should put that remark on hold. I need to figure out whether saying that we have a "computation" is correct.