[FOM] Forward: Provability of Consistency

Richard Heck richard_heck at brown.edu
Mon Mar 25 22:40:23 EDT 2019

Let me ask a few naive questions that (I hope) might help to clarify
what is going on here.

On 3/25/19 9:27 AM, Artemov, Sergei wrote:
> ...a formal string, e.g., Con(PA), has exact semantics, which we should know and take into account. There are at least three relevant levels of understanding of an arithmetical formula F: 
> 1. “Naive” mathematical reading F as a mathematical statement about natural numbers.
> 2. Semantics in the standard model of arithmetic. This is actually a glorified version of (1) for logicians. 
> 3. Provability in PA. This one is quite relevant in the context of Goedel’s Second Incompleteness Theorem, G2. 
> Con(PA) is an adequate formalization of consistency of PA in the standard model (2), and, perhaps, that is why “general” mathematicians have been willing to accept Con(PA) as in (1). No problems here. 
> However, our goal is to make a semantic sense out of (3), to understand what is really going on in G2. 

I do not understand this, and I strongly suspect that it is confused.
What does "understanding" Con(PA) mean if it does not mean understanding
it as standardly interpreted? (I take it that the issue of Gödel
numbering is agreed to be irrelevant. We can reformulate all of this in
terms of a theory of expressions, a la Grzegorczyk.) Is your view that
Con(PA) means different things depending upon what theory of the natural
numbers I accept? E.g., it means something different for Nelson or
Hilbert or Feferman or Shapiro? If not, what precisely is the project
you describe at (3)?

> By Goedel’s Completeness Theorem, a formula F is derivable in PA iff F holds in all PA-models, most of them nonstandard. ...This sketch of the semantic analysis of G2 shows that G2 is not about real PA-derivations, each of which is in fact not a proof of 0=1. 

Similarly, the statement of the Paris-Harrington theorem (PH) is
provable in PA iff PH holds in all models of PA, which it does not, even
though it does hold in the standard model. Does that also show that PH
is not really about the natural numbers, as its understood within the
context of PA?

Let me ask if you would accept the following reformulation of what you
are claiming. One of the difficulties that I (and perhaps some others)
have is that every instance of

(*)    If S is a PA-derivation, then S is not a proof of 0=1.

is already provable in R. (These are \Delta_1, and they are all true.)
But the proof that *all* the instances of (*) are provable in R will
involve resources that go beyond PA. What you are adding, I take it, is
a proof that PA-proofs of instances of (*) can be made to be uniform in
S. (It's a nice question whether that can be extended to weaker
systems.) But it's still obscure to me why uniformity should matter as
much as you seem to insist it does.

Another question: Is your view that PH could be shown to be provable "in
PA" by showing that its instances can be proven uniformly in n?Maybe
that is your view, and you think that PH could be shown to be provable
in PA by showing that its instances can be proven uniformly. If so, then
that'd be an interesting claim, but not one that has much to do with
Con(PA) specifically.


Richard Kimberly (Riki) Heck
Professor of Philosophy
Brown University

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