[FOM] First-order arithmetical truth

[Timothy Y. Chow]

mario <chiari.hm at flashnet.it> wrote:
> 1. Do you argue for [1]->[3] above along the following (one line longer)
> sketch:
> 
> if    (1) somebody lacks the ability to distinguish the intended model
> of PA from another model,
> then  (2) (s)he lacks the ability to distinguish the intended model of
> meta-PM from another model, 
> then  (3) (s)he lacks the ability to distinguish the intended meaning of
> the term "formal system" from some other meaning.

Not really.  The argument is just that any skeptical argument about the 
natural numbers translates easily into a skeptical argument about formal 
systems.

One particular skeptical argument about the natural numbers might go as 
follows:

"I lack the ability to understand precisely what some alleged mathematical 
entity is unless you can show me a first-order language for it and a set 
of first-order axioms with the property that the alleged mathematical 
entity is the unique structure satisfying the axioms.  There is no such 
set of first-order axioms for N in the language of arithmetic.  So I don't 
know what N is."

The translation is:

"I lack the ability to understand precisely what some alleged mathematical 
entity is unless you can show me a first-order language for it and a set 
of first-order axioms with the property that the alleged mathematical 
entity is the unique structure satisfying the axioms.  There is no such 
set of first-order axioms for PA in the language of syntax.  So I don't 
know what PA is."

Here "language of syntax" is something like your "meta-PM."

Maybe you don't think much of the above skeptical argument about the 
natural numbers, and think that there is some other skeptical argument 
that is more convincing.  Fine.  I'm sure that your favorite argument can 
also be translated, since it's very straightforward to pass between 
assertions about numbers and assertions about strings.

Tim