# Re: Tarski's Proof of the undefinability of truth.

```
On 25 Mar 2013, at 11:10, Alberto G. Corona wrote:```
```
```
I suspect that this impossibility is because math uses concept of a model, while truth refers to the match of facts of the model with facts of the reality . Or at least to facts of a metamodel outside of the model. That is AFAIK the Tarsky idea.
```
Tarski is with an "i". I change the title of the thread accordingly.

```
Tarski is talking about mathematical truth, and this is the match with some model (in the logician's sense).
```
```
Logician does not assume a notion of reality, per se, but in applied logic we can "model" some reality by models (mathematical structure with a notion of satisfying formal formula).
```
```
For example the meaning of "ExP(x)" is that it exists an object m in the model such that it is the case that P(m)".
```
```
Tarski theorem is a consequence of the diagonal lemma. That is: the fact that for all formula F in the language used by the theory, you can find a sentence S provably (by the theory) equivalent with F('S') where ' ' represents some fixed coding of S in the language of the theory. So if we could defined a predicate defining "truth of sentence" (and thus non-truth of a sentence) in the theory, we would find a sentence K provably equivalent with NOT-TRUE (K). But for a truth predicate we ask that the theory proves, for all sentence X, that X is equivalent with TRUE('X'), and so any reasonably rich theory would prove that the K above is both true and false. Like Stephen said, it is really a form of Epimenides "paradox". Gödel's theorem is also an easy consequence of the diagonal lemma, with "not-provable" in place of not-true. Yet "provable" and "not-provable" are definable in the theory, and this entails that if the machine is consistent (does not prove the false), the fixed point "K" will be true but not probable in the theory/by the machine.
```
```
All Löbian machines satisfy the diagonalization or diagonal lemma. They cannot define an all encompassing notion of truth for all the sentences that they can talk about, but they can still define very large notions of truth, restricted to very big subset of their language. Peano Arithmetic can already define a notion of truth for any sigma_i sentences, for example.
```
Bruno

```
```
```
For example, "All men are mortal is true" . Here suppose that "all men are mortal" is a fact of a model that admit inductive silogisms.
```
```
True in this case can express a match of the fact "All men are mortal" with reality. or a match with a metamodel in which the trueness of the model "all men are mortal" is assumed as an axiom.
```
```
In both cases the whole statement "all men are mortal is true" is outside of the model in which the statement "all men are mortal" is
```
```
However I can make an ordinaty mathematical model of matches between models and metamodels. That is the boolean logic.
```

2013/3/23 Stephen P. King <stephe...@charter.net>
```
"In 1936 Tarski proved a fundamental theorem of logic: the undefinability of truth. Roughly speaking, this says there's no consistent way to extend arithmetic so that it talks about 'truth' for statements about arithmetic. Why not? Because if we could, we could cook up a statement that says "I am not true." This would lead to a contradiction, the Liar Paradox: if this sentence is true then it's not, and if it's not then it is.
```
```
This is why the concept of 'truth' plays a limited role in most modern work on logic... surprising as that might seem to novices! ..."
```

--
Onward!

Stephen

--
```
You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com.
```To post to this group, send email to everything-list@googlegroups.com.
```
Visit this group at http://groups.google.com/group/everything-list?hl=en .
```For more options, visit https://groups.google.com/groups/opt_out.

--
Alberto.

--
```
You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com.
```To post to this group, send email to everything-list@googlegroups.com.
```
Visit this group at http://groups.google.com/group/everything-list?hl=en .
```For more options, visit https://groups.google.com/groups/opt_out.

```
```
http://iridia.ulb.ac.be/~marchal/

--
You received this message because you are subscribed to the Google Groups
"Everything List" group.
To unsubscribe from this group and stop receiving emails from it, send an email