[FOM] FW: Alternative foundations?

Carl Hewitt hewitt at concurrency.biz
Fri Feb 21 16:47:09 EST 2014

Harvey raises a good point:

                 What is the status of categoricity results for Alternative Foundations à la Peano, Dedekind, and Zermelo?

From: Carl Hewitt
Sent: Friday, February 21, 2014 11:23
To: fom at cs.nyu.edu
Cc: 'Jay Sulzberger'; 'Kreinovich, Vladik'; 'Staffan Angere'; 'Michael Carroll'; 'David Roberts'
Subject: RE: [FOM] Alternative foundations?

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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