[FOM] Alternative foundations?
Kreinovich, Vladik
vladik at utep.edu
Fri Feb 21 14:26:04 EST 2014
True, in the HTT approach, sets are there - they are just NOT the fundamental object, they are defined in defined of what the authors call Homotopy Types.
From: Carl Hewitt [mailto:hewitt at concurrency.biz]
Sent: Friday, February 21, 2014 12:23 PM
To: Foundations of Mathematics
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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