Is every theorem of PA true in the standard model of number theory $N$?

Question

My understanding is that every theorem $\phi$ of $PA$ is true in $N$ because

1. $N$ is a model for $PA$, $N\models PA$.
2. By completeness of first order logic, "$PA\vdash\phi$" implies that "if $N\models PA$ then $N\models \phi$".

Hence $\phi$ is true in $N$, $N\models \phi$.

But I was confused upon reading the following from Godel's Theorem: An Incomplete Guide to Its Use and Abuse, p31:

We know that there are consistent theories extending PA that prove false mathematical statement ... So we have no mathematical basis for concluding that (say) the twin prime conjecture is true from the two premises "PA is consistent" and "PA proves the twin prime conjecture".

According to my understanding above, "PA is consistent" and "PA proves the twin prime conjecture" are enough for concluding truth of the conjecture (in the standard model). Theories extending PA may prove false theorem (relative to the standard model), but surely this is not the case for PA?

Answer 1

First a minor remark: I personally prefer to refer to the implication that you use in (2) as soundness and to reserve the term completeness for the other direction. It takes no Gödel to prove soundness ;)

You are correct in the mathematics before the quotation. I think the point that the author tries to make in that quotation is merely that the truth of theorems of PA is not a consequence of the consistency of PA. He does not question the truth of theorems of PA. Note that you also never invoke the consistency of PA in your argument, but the (stronger, because of soundness) statement that the natural numbers are a model. Indeed, as Franzén points out, if you have a statement $\phi$ that is true in $N$ but not provable in $PA$ then $PA+\neg\phi$ is a consistent theory that proves $\neg\phi$, but this does not make $\neg\phi$ true. Truth is not a consequence of consistency.

And by the way, I think that this is an excellent book.

Answer 2

I think it might be helpful to have both paragraphs:

Similarly, when we talk about arithmetical statements being true but undecidable in PA, there is no need to assume that we are introducing any problematic philosophical notions. That the twin prime conjecture may be true although undecidable in PA simply means that it may be the case that there are infinitely many primes $p$ such that $p + 2$ is also a prime, even though this is undecidable in PA. To say that there are true statements of the form “the Diophantine equation $D(x_1, \ldots, x_n) = 0$ has no solution” that are undecidable in PA is to make a purely mathematical statement, not to introduce any philosophically problematic ideas about mathematical truth.

Similar remarks apply to the observations made earlier regarding consistent systems and their solutions of problems. It was emphasized that the mere fact of a consistent system S proving, for example, that there are infinitely many twin primes by no means implies that the twin prime hypothesis is true. Here again it is often thought that such an observation involves dubious metaphysical ideas. But no metaphysics is involved, only ordinary mathematics. We know that there are consistent theories extending PA that prove false mathematical statements—we know this because this fact is itself a mathematical theorem—and so we have no mathematical basis for concluding that the twin prime conjecture is true, which is to say,that there are infinitely many twin primes, from the two premises “PA is consistent” and “PA proves the twin prime hypothesis.”

From the earlier paragraphs in the section, it seems that 'mathematical truth' is essentially truth in $\mathbb{N}$ (or $\mathbb{R}$, or whatever natural universe we're working in). I believe the point here is that PA is not complete, so there are (or, at least, there could be) consistent extensions S of PA which could prove both that PA is consistent and that the twin prime hypothesis is true. But that these extensions could also state, for instance, that some other theorem of $\mathbb{N}$ is false. So the statements 'PA is consistent' and 'PA proves the twin prime hypothesis' formally imply that the twin prime hypothesis is true, but care should be taken as to what consistent system this is being proved in (in particular, what system is proving that PA proves the twin prime hypothesis).

But there's nothing wrong with your reasoning. If $PA \vdash \phi$, then of course $\mathbb{N} \vDash \phi$.

Answer 3

I am now convinced from a rereading of the extended extract provided by Salman that this obviously wrong statement is just the result of two typos. If you change the last two occurrences of "PA" in the extract to "S" it all makes perfect sense (S being the name for a consistent extension of PA introduced earlier in the extract).

Answer 4

I believe the quotation is intended to mean: one cannot conclude

the twin prime conjecture is true

from

PA is consistent

and

PA proves the twin prime conjecture

without invoking

$\mathbb N\models\mathrm{PA}$.

By writing this, I am by no means doubting the truth of $\mathbb N\models\mathrm{PA}$ though.