[FOM] First-order arithmetical truth
Timothy Y. Chow
tchow at alum.mit.edu
Sun Nov 5 17:57:15 EST 2006
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
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