[FOM] First-order arithmetical truth

[Stephen Pollard]

Vladamir Sazonov asked whether anyone has the ability to distinguish  
"the intended model of arithmetic from another model." He continued:

> Is there any way to demonstrate this ability to see that this is  
> not just a fiction? Can we make any experiment to check who has and  
> who has not this ability?

This is a tricky and important question. I'm pretty sure of this  
much: there are forms of number theoretic discourse whose  
categoricity cannot be coherently challenged. For a fine discussion  
of this, see:

Charles Parsons (1990). The uniqueness of the natural numbers. Iyyun  
39, 13-44.

Hartry Field (in his book Truth and the Absence of Fact) has offered  
some acute criticisms of the Parsons argument. I think all these  
criticisms can be defused. (Parsons own responses appear in a paper  
he delivered in Iran in 2001.)

The recent philosophical literature on this topic is substantial and  
of high quality. Some other important contributors are Shaughan  
Lavine, Vann McGee, and Stewart Shapiro.

Stephen Pollard
Professor of Philosophy
Division of Social Science
Truman State University
spollard at truman.edu