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I've been studying Gödel's incompleteness theorems and I am stuck with the Tarskian notion of truth. I 've searched all over the internet, in books, in relevant topics here , but I didn't find a satisfactory answer. Ok , let me state exactly what my problem is :

The semantic version of the first incompleteness theorem states that there are true sentences of arithmetic which cannot be proven by PA. It perfectly makes sense to me that there are sentences which cannot be proven or refuted by PA but what does it mean to say that there are true sentences that cannot be proven by PA. The immediate response is that these sentences are true in the standard interpretation of PA, namely in the standard model of the natural numbers. But what does this mean ? When is a sentence "true" in the standard model of the natural numbers ? I know that the second-order Peano arithmetic is categorical and thus a unique set ( up to isomorphism ) is a model of these axioms. Therefore it seems that every sentence has a definite truth value, either it is true or it is false, according to whether it holds for that unique model. But that seems to me completely wrong. Let's take Goldbach's conjecture for the sake of argument. Suppose that it is neither provable or disprovable. In what sense does this conjecture hold ? Someone could say : trivially because IN FACT every even number can be written as a sum of two primes. Maybe it's the Aristotelian logic that I reject. I simply cannot accept that there is an absolute truth besides what we can prove. My current view is that Tarski's definition of truth in a formal system simply presupposes that we magically have a metasystem where every question has simply a definite answer. Then we just translate every sentence into its relative meta-sentence and decide whether the first sentence is true or not. Maybe that's a naive view. An analytical exposition would be great.

Kostas Kartas
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    You can see Intuitionistic Logic and Intuitionism. – Mauro ALLEGRANZA Aug 24 '16 at 19:51
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    Though Wittgenstein didn't always go into great detail with his remarks/comments, it is worth pointing out that he strongly disagreed with our interpretation of the proof (as well as our interpretation of Cantor's Diagnolization proof). Point being, you can find reputable people who disagree with the widely held interpretations of various important proofs. And since you seem to be skeptical, you might find it worth it to read up on the works of these people, as it might be difficulty (as it would be in any community) to get sympathetic responses when questioning widely held interpretations. – Boolean_functions Aug 25 '16 at 00:18
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    Might be of interest: http://plato.stanford.edu/entries/wittgenstein-mathematics/ – Boolean_functions Aug 25 '16 at 00:28

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One contemporary reading of Tarski's truth definition is that it simply defines truth in a structure. So, from that point of view, you do need to start with a particular structure before you can talk about "truth". I don't believe this was Tarski's original aim in the 1930s, but it's a reasonable approach to "truth" from the current viewpoint of the field.

So, yes, when people talk about "true but unprovable statements", or about a "standard model", they are beginning with the viewpoint that there is a "standard model" to which we can refer. Once someone accepts that there is a standard model, the fact that it gives each sentence a unique truth value is just a matter of Tarski's definition of truth in a structure.

However, the reason that we know that Gödel sentence of a consistent theory $T$ is "true" but "unprovable in $T$" is that we can prove the Gödel sentence in some metatheory. If we couldn't prove the Gödel sentence, we wouldn't know it is true.

In general, if you don't like to talk about "truth", you can replace claims about "truth" with claims about provability in the metatheory. This is a very standard procedure, and authors often don't bother to even mention it. They may write as if truth is well determined, but this still allows people who worry about it to replace "true" with some other meaning. (Similarly, authors may use Platonistic terminology, but people with other tastes know how to re-interpret the language to suit their taste.)

In particular, the Gödel sentence of a theory $T$ is provable in a very weak metatheory - primitive recursive arithmetic (PRA) will suffice - under the additional assumption of Con($T$). This fact is part of a "formalized incompleteness theorem", where "formalized" means that we pay attention to the particular metatheory necessary to prove the theorem, including its claims about truth.

So what we have formally is that, when $T$ satisfies the hypotheses of the incompleteness theorems and $G_T$ is its Gödel sentence, then $$ \text{PRA} \vdash \text{Con}(T) \to G_T. $$ If we recognize the axioms of PRA as "true" in some sense, and that Con($T$) is "true" in that sense if and only if $T$ is consistent, then that formal result can be interpreted as: if $T$ is consistent then $G_T$ is "true" in the same sense.

Carl Mummert
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  • So we just assume , for example , that there is a standard structure of natural numbers ? What is that structure and why does it follow that every sentence concerning the natural numbers has a definite truth-value ? As for the "metatheory"...Well, the fact is that no book makes clear what exactly this "metatheory" is. Is it ZFC ? I've heard also that it generally involves everyday "informal" mathematics. That's utter nonsense in my opinion. I really want to make things as clear as possible. – Kostas Kartas Aug 27 '16 at 01:07
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    If you work with ZFC as your metatheory, you can prove there is a unique set of "standard natural numbers", which is to say a model of Peano Arithmetic in which every nonempty subset has a least element. So that would be your standard model of arithmetic. Also ZFC proves that every sentence in the language of arithmetic has a definite truth value in that model. So, if you want a formal metatheory, you could do this in ZFC. Many mathematicians work in more informal metatheory, though. The books don't make it clear because you can choose several options, and the claims make sense either way. – Carl Mummert Aug 27 '16 at 01:25
  • How does ZFC prove that every sentence in the language of arithmetic has a definite truth value ? – Kostas Kartas Aug 27 '16 at 14:57
  • By induction on the complexity of formulas. First, each formula is put into prenex normal form - all quantifiers in front. Let $F_n$ be the set of arithmetical formulas with $n$ leading coefficients. Then we can directly define the truth value of each formula in $F_0$, and we can define the truth values of $F_{i+1}$ using the truth values of formulas in $F_i$. This is a standard construction, it's the same way that ZFC would construct the truth set of any structure, we are just applying it to the standard model of arithmetic. – Carl Mummert Aug 27 '16 at 20:13
  • Okay ,so you are referring to the standard model-theoretic definition, I suppose. I am still confused though. It seems as though for every sentence ZFC can "decide" whether it holds in the standard model or not. If that's not the case can you elaborate your argument ? – Kostas Kartas Aug 28 '16 at 02:27
  • I'm not sure exactly what you're asking. As a metatheory, ZFC proves that every sentence of arithmetic has a truth value in the standard model. This means ZFC proves there's a function which takes a sentence in the language of arithmetic and gives its truth value in the standard model. ZFC can't prove everything about the values of that function, but it can prove the function exists. With ZFC as a metatheory, we can say "every sentence of arithmetic has a truth value in the standard model" even if we don't know what all the truth values are. This is the difference between proof and truth. – Carl Mummert Aug 28 '16 at 11:42
  • I understand the facts but I still cannot see how to derive them formally. Could you elaborate your previous argument on the existence of this function or provide some link/reference ? – Kostas Kartas Aug 28 '16 at 20:37
  • It is the same proof as for every other structure; it would be in any book on model theory and in most intro math logic books. The truth function is defined by first dividing the language of the structure into a sequence of sets based on the number of quantifiers in prenex form, and then defining a truth function for each of these sets, inductively. The usual construction of the truth function for any set-based structure - like the standard model of arithmetic - can be formalized without any trouble in ZFC. – Carl Mummert Aug 28 '16 at 23:33
  • I am following Enderton's textbook. It says in the beginning of Tarski's definition that ⊨ t1=t2[s] iff s(t1)=s(t2) , where s is a function from the set of terms into the universe. If our metatheory is ZFC I can only understand the equation s(t1)=s(t2) as saying that ZFC proves s(t1)=s(t2). What else could this mean ? If that's the case then in order to define the truth value of t1=t2[s] it must be the case that ZFC proves or disproves s(t1)=s(t2). – Kostas Kartas Aug 29 '16 at 01:07
  • The problem I am having is that it seems as though our meta-theory is somehow "complete" and we can use this completeness to assign values to our sentences. – Kostas Kartas Aug 29 '16 at 01:09
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I simply cannot accept that there is an absolute truth besides what we can prove.

This is a perfectly "sound" point of view, shared by Intuitionism and Constructivism.

But it is simply not the point of view of Tarski.

See e.g.:

The predicate "true" is sometimes used to refer to psychological phenomena such as judgments or beliefs, sometimes to certain physical objects, namely, linguistic expressions and specifically sentences, and sometimes to certain ideal entities called "propositions." [...] For several reasons it appears most convenient to apply the term "true" to sentences, and we shall follow this course.

The word "true," like other words from our everyday language, is certainly not unambiguous. And it does not seem to me that the philosophers who have discussed this concept have helped to diminish its ambiguity. In works and discussions of philosophers we meet many different conceptions of truth and falsity, and we must indicate which conception will be the basis of our discussion.

We should like our definition to do justice to the intuitions which adhere to the classical Aristotelian conception of truth - intuitions which find their expression in the well-known words of Aristotle's Metaphysics:

To say of what is that it is not, or of what is not that it is, is false, while to say of what is that it is, or of what is not that it is not, is true.

If we wished to adapt ourselves to modern philosophical terminology, we could perhaps express this conception by means of the familiar formula:

The truth of a sentence consists in its agreement with (or correspondence to) reality.

If, on the other hand, we should decide to extend the popular usage of the term "designate" by applying it not only to names, but also to sentences, and if we agreed to speak of the designata of sentences as "states of affairs," we could possibly use for the same purpose the following phrase:

A sentence is true if it designates an existing state of affairs.

Mauro ALLEGRANZA
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  • I totally agree with the correspondence theory of truth ,which is what I think Tarski had in mind. But to consider mathematical facts as real facts in the same sense that you consider the fact "snow is white" as real, seems to me untenable. Also that is a serious metaphysical assumption that at least should be mentioned in a honest way. Anyway , if that is the case then I 'll stick to the syntactic version of Godel's theorem. – Kostas Kartas Aug 27 '16 at 01:16