[FOM] Alternative foundations?

[Carl Hewitt]

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*."
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