Talk:Implementation of mathematics in set theory

It is (really!) not the aim of this article to support a polemic for ZFC or NFU. I do notice that NFU gets credit for allowing certain natural abstractions to be first-class objects which are often missed in ZFC (the universe, the Frege natural numbers). However, it must also be noted that constructions in NFU often get rather baroque (especially where large sets need to be taken into account) and it is pretty clear that the world of NFU contains nonstandard objects (in particular, nonstandard ordinals). Easy access to the "nonstandard" phenomena coded by the external endomorphism of the ordinals (the T operation) allows strong axioms of infinity to be stated in appealingly (perhaps deceptively) simple forms (see the New Foundations article for some of these). I actually think that ZFC on the whole is better (because easier) but not that much better (despite the fact that I study NF and related systems, I'm not a partisan of some wholesale revolution in set theoretical practice!) But I think it is good for those who study foundations to be aware of alternatives. Randall Holmes 03:19, 22 December 2005 (UTC)[reply]

More will be coming. Randall Holmes 03:19, 22 December 2005 (UTC)[reply]

The handling of indexed families is a little different from that in my book: I save a level of typing by allowing only index sets of singletons. It is not completely general but is a little easier -- but I still made some mistakes setting it up! Randall Holmes 06:26, 22 December 2005 (UTC)[reply]