[FOM] Frege, Russell and type theory

[William Tait]

At 3:26 PM -0400 10/17/03, Alasdair Urquhart wrote:
>I am completely in agreement with Giovanni Sambin
>that prefigurations of type theory can be found
>much earlier than Frege and Russell.  Alonzo Church
>wrote a paper on Schroder's anticipation of the simple
>theory of types, and Russell himself went out
>of his way to emphasize the natural nature of type
>restrictions, and similarities to traditional notions.
>
>Nevertheless, I do think that the historical evidence
>shows that Russell's thinking on the theory of types,
>and especially type theory as a way out of the paradoxes,
>begins from Frege's functional hierarchy.  I did not
>mean in any way to make exaggerated claims for Frege
>(in spite of my earlier tongue in cheek comments about defending
>Frege from his detractors).  Putnam's article that I mentioned
>earlier is an excellent corrective to the tendency to think
>that Frege invented everything in modern logic (Whitehead
>and Russell in fact tended to conceal their considerable
>debt to the tradition of algebraic logic).

We might also mention Cantor's 1890 paper containing the diagonal 
argument to show that the totality of 2-valued functions on M is of 
power greater than M. He then remarks that this yields another 
sequence of powers cofinal with the ordinals (the other being his 
sequence of number classes in his 1883 paper).  This is essentially 
the cumulative hierarchy of types over N.

Bill