[FOM] Is Goedel's Theorem Surprising?

William Tait williamtait at mac.com
Thu Dec 14 11:14:28 EST 2006

On Dec 11, 2006, at 10:00 PM, Vann McGee wrote:

> I've always supposed - although I hasten to admit I haven't any
> textual evidence for this - that, at the time the theorem was  
> published, it
> seemed highly probable that Dedekind's proof that any two "simply  
> infinite"
> systems are isomorphic could be deployed, if anyone took the time  
> to fill in
> the details, to show that Goedel's system P (which embeds second- 
> order Peano
> Arithmetic within the simple theory of types) is categorical, and  
> hence
> complete. The incompleteness of first-order PA came as no surprise,  
> but the
> first-order theory was widely regarded as an artificially  
> circumscribed
> fragment.

The paper

S. Awodey & E. Reck, "Completeness and Categoricity, Part
I: 19th Century Axiomatics to 20th Century Metalogic", History and
Philosophy of Logic 23:1, 2002, 1-30

cited in Reck's December 9 posting is relevant here. The paper  
mentions Veblen's distinction between semantical and syntactical  
completeness in 1906. Also, Fraenkel in the 1923 edition of his book  
on set theory distinguished between categoricity and semantical  
completeness (i.e. all models have the same true sentences) and in  
the 1928 edition between semantical and syntactical completeness.  
Nevertheless, there is also evidence that the latter distinction was  
not generally recognized.

Surprise over the incompleteness of PA could have rested on two  
misconceptions: One is the distinction between semantical and  
syntactical completeness for second-order systems, so that one might  
have been led by Dedekind's result to think that second-order number  
theory is syntactically complete. The second misconception would be  
to think that any first-order theorem of second-order number theory  
should only require mathematical induction applied to arithmetic  
properties and so should be deducible in PA. (I think that, if I had  
been around in 1930, I would have believed that. But then ...)

Bill Tait

More information about the FOM mailing list