[FOM] Alternative foundations?

[Mitchell Spector]

Carl Hewitt wrote:
> Dana Scott [1967] pointed out that foundations need **both** types and **sets**:
>
> “there is only one satisfactory way of avoiding the paradoxes: namely, the use of some form of the
> theory of types... the best way to regard Zermelo's theory is as a simplification and extension of
> Russell's ...simple theory of types. Now Russell made his types explicit in his notation and Zermelo
> left them implicit. It is a mistake to leave something so important invisible...”
>
> “As long as an idealistic manner of speaking about abstract objects is popular in mathematics,
> people will speak about collections of objects, and then collections of collections of ... of
> collections. In other words **set theory is inevitable**.”


Yes, as Scott said, a theory of types appears to be needed to escape the paradoxes (while still 
having an interestingly large mathematical universe).

This shows a compelling foundational feature of ZF: a theory of types, in the form of well-ordered 
set-theoretic ranks, arises naturally and elegantly, emerging surprisingly from simply stated and 
straightforward properties of the membership relation.


Mitchell