[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
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