In response to the above remarks of Daryl McCullough, let me say what my instincts are. In general they’re pretty similar to Matt’s: I am uneasy about any notion of mathematical truth that smacks of metaphysics. However, I have degrees of unease, and I’m not sure I can justify them. I can see that I would have a hard time defending the view that there is no fact of the matter as to whether every even natural number is the sum of two primes (jokes about 2 being a counterexample apart), but if I go one level deeper in the quantification then my attitude changes.

For example, is there a fact of the matter about whether the decimal expansion of contains infinitely many 0s? The difference here is that an imaginary thought experiment (in which, say, you are punished for your earthly sins by being put in a cell and told to watch as the digits of run past you for ever) never comes to an end, regardless of what you observe at any finite stage. The analogous Goldbach verification probably wouldn’t come to an end either, but one can at least entertain the possibility that an even integer might come along and spring a huge surprise.

One might argue as follows. If I concede that there is a fact of the matter as to whether contains at least one 0 after the nth digit, then surely I am now reduced to asking whether some factual statement holds (namely not having any more 0s after the nth digit) for at least one And now I am back with a statement of the Goldbach type. But I don’t buy that, because in the Goldbach situation one can check statement in finite time and here one cannot: it seems to me to be genuinely worse.

If we deepen the quantification further, by asking whether the proportion of 0s in the decimal expansion of tends to 1/10, then all these concerns only increase. The imaginary thought experiment becomes wilder and wilder.

In fact, I have some sympathy with ultrafinitism: I’m tempted to say that there may be no fact of the matter as to whether the proportion of 0s in the first Ackermann(10,10) digits of is under 20 percent. Of course, if someone came up with a proof, then I would change my tune completely. But what if it was a statement that had no proof short enough to write down?

I do have one argument on the other side, which is that although we may not have a proof, we do at least have evidence that we can look at, such as how the digits of behave. The fact that we take this evidence seriously could be taken as showing that we do have some concept of the truth or falsity of statements about the long-term behaviour: it’s just that to justify our beliefs we use inductive rather than deductive methods. But this is to use the word “truth” in a somewhat different way.

Hmm if the same problems exist with any other sensible way of expressing it, I guess I don’t really have a good reason to be bothered by the version with “true”.

Harumph! It still causes an unconformable feeling in the back of my head.

I should get back to doing maths where I’m far enough from the foundations that I can just assume that everything works. If I don’t prove stuff about those bounded derivations… it doesn’t really matter.

Matt, my immediate facetious reaction to the first one of your quotes that I have copied is that both Gödel and Tarski wrote in German so the relevant phrase would be “ Deutsches Wort Wahrheit

Strictly speaking, I think that would be “wahr” rather than “Wahrheit”. In any case I was referring to popular treatments in English.

When we talk about truth in a formal language we are talking about a concept of that formal language and not a concept in English or Deutsch. In formal languages the concepts that we use are defined exactly and specifically for that language and although we my use a concept in English that is homophone to the concept in our formal language we should not be deluded into thinking that they mean the same. …
…you are confusing talking about a formal language informally in a meta-language, in this case English, and using that formal language itself.

Well I knew I wasn’t using the formal language (I’m not a total moron). I do confess that I wasn’t totally aware that there was a technical usage of “true” in formal logic. But meh, using a technical term, which sounds like a common term in the language you are writing in, without defining it sounds to me an unforgivable sin.

You are probably right that this isn’t the best forum to discuss this but no theorem about formal systems proves anything about the correct use of the English word “true”. What constitutes “truth in mathematics” is an open discussion amongst philosophers and I don’t think any of them mean things like a Tarski truth function.

Now most mathematicians are not set theorists or logicians. If we say “the closed graph theorem is true” we mean “in our fixed set theory (presumably ZFC) there is a proof of the closed graph theorem”. Most (IME) would hold that the question of whether the continuum hypothesis is true has no real meaning; “it’s undecidable” would be the whole answer. If asked specifically about different systems we would say “Hahn-Banach is true in ZFC” but not in “ZF (NOT C)”. “True” is not in this set up a formal concept in maths itself much less a formula within a theory, but why should it be?

Matt, my immediate facetious reaction to the first one of your quotes that I have copied is that both Gödel and Tarski wrote in German so the relevant phrase would be “ Deutsches Wort Wahrheit” however here you have the answer to your question in the second quote that I have copied.

When we talk about truth in a formal language we are talking about a concept of that formal language and not a concept in English or Deutsch. In formal languages the concepts that we use are defined exactly and specifically for that language and although we my use a concept in English that is homophone to the concept in our formal language we should not be deluded into thinking that they mean the same. As far as Gödel Theorem goes, it was thought before he published that provability and truth were equivalent concepts for formal languages, his theorem shows that they are not for any formal language “rich” (Reichhaltig) enough to contain Peano arithmetic. There are formal languages where it is the case that provability and truth are equivalent, First Order Predicate Logic for example, as was in fact proved by Gödel in his doctoral thesis. In the second quote of yours that I copied you are confusing talking about a formal language informally in a meta-language, in this case English, and using that formal language itself.

That at least is my reading of you problem! Thony C.

My comment is agnostic about the philosophical question of whether statements of arithmetic have absolute truth values. I’m saying that you don’t need to solve that question to understand what “true but unprovable” means.

My complaint about rephrasing Godel’s result as saying that the Godel sentence G can neither be proved nor disproved is that, to me, the ontological status of “G cannot be proved” is *exactly* the same as the status of “G is true”. To say that G is true is to say that there does not exist a natural number satisfying such and such a property. To say that G cannot be proved is to say that there does not exist a proof satisfying such and such a property. I don’t see how the latter makes any less ontological commitment than the former.

I think TMCM questions wasn’t “Is “Does every Goodstein sequence terminate?” the same as “Is it true that every Goodstein sequence terminates?”” . I think his question was “Does every Goodstein sequence terminate?”

I think it’s a fairly interesting question if you are asserting that arithmetic statements have absolute truth values (which I think you were).

That’s how I’m taking “true” in the part where I say “I can sort of see it”. But as I say and (if I understood rightly) G.D. agreed this does rely on the statement having the semantic content about numbers and not just being a “meaningless”, uninterpreted formula (as I understand many respectable people would argue it was). It also relies on the formula still having this meaning when we look at it from “outside” as opposed to only meaning that relative to the system around it.

Let me repeat again, I’m not asserting that this is wrong. Just that if it is right it relies on deep subtleties of the philosophy of mathematics that are apparently not settled amongst those who specialise in this stuff (see of what G.D.. wrote about formalists and intuitionists different problems with it). Thus, I think the language of truth it is best avoided if you are writing about a theorem like this in, say, New Scientist. Writing “can neither be proved or disproved” doesn’t seem a great loss in readability. .

Yes, to me, the question “Is it true that every Goodstein sequence terminates?” means exactly the same as the question “Does every Goodstein sequence terminate?” The use of “Is it true that” is eliminable.

In Godel’s theorem, Godel constructs a sentence G such that (under some additional assumptions about the soundness of PA, Peano’s theory of arithmetic) G is true, but PA doesn’t prove G. The use of “is true” here is in principle eliminable, but in practice, eliminating it would make the claim humanly incomprehensible. Instead of saying

G is true but PA cannot prove G

one could spell out G in all its gory details and write

forall natural numbers x, blah blah blah,
but PA cannot prove forall natural numbers x, blah blah blah.

where blah blah blah is some enormously complicated statement involving the variable x. After expanding G to get an explicit sentence of arithmetic, the use of “true” is no longer necessary. True is only necessary if we are using a *name* for the sentence instead of explicitly quoting the sentence.

How about the question “Does every Goodstein sequence terminate?” Perfectly arithmetical statement…

I’m not sure what you mean by your last two sentences. What reality are we ascribing to X, if any?