[FOM] First-order arithmetical truth

Timothy Y. Chow tchow at alum.mit.edu
Sun Oct 22 16:15:25 EDT 2006

Vladimir Sazonov wrote:
>That is, you postpone answering my question. Or is this an attempt to 
>lead me to a contradiction?

It is an attempt to keep the discussion on the original track---which was 
my attempt to get Francis Davey to see Arnon Avron's point.  Notice that 
*you* are the one interrupting the discussion in order to inject your own 
agenda, so I do not feel obligated to answer *your* questions, as opposed 
to any questions Francis Davey might have about what I am trying to say.

But in any case, since you agree with the parallelism between (e.g.) 
"formal string of symbols" and "natural number," perhaps this will be 
helpful to Francis Davey too.  Any skepticism about the naive (resp. 
formal) concept of the integers carries over directly into skepticism 
about the naive (resp. formal) concept of a formal system.  Anyone who 
thinks that formal systems are crystal clear while integers are vague and 
suspect---and who tries to argue that the existence of nonstandard models 
lends support to that idea---is simply suffering from a blindspot that 
prevents him from seeing that his skeptical arguments apply equally to 
formal systems.

Here's another way to put it, taken from an email reply I just sent to 
another FOM subscriber (off-list).  The other subscriber wrote:

> It's easy to formulate a formal system as string rules, like one does 
> in Theory of Formal Languages courses in c.s.
> there seems no "standard" way to write a standard formal system on a 
> sheet of paper that isolates exactly the natural numbers w/o getting 
> non-standard numbers as well.

To this I replied:

> Let N be the standard natural numbers, and let PA be the standard 
> first-order system of Peano arithmetic.
> If there is no formal system in the language of arithmetic that isolates 
> exactly N, then there is no formal system in the language of syntax that 
> isolates exactly PA.  If the former state of affairs leads us to worry 
> about the determinacy of N, then the latter state of affairs should lead 
> us to worry about the determinacy of PA.  Given any text that 
> purportedly describes PA in terms of "string rules," as you suggest, I 
> can perversely interpret that text as referring to *nonstandard* string 
> rules, yielding something very different from the "intended" PA.


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