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Article 5525 of comp.ai.philosophy:
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>From: holmes@opal.idbsu.edu (Randall Holmes)
Subject: New Foundations (was Re: penrose)
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References: <1992May1.025230.8835@news.media.mit.edu> <1992May6.220605.26774@unixg.ubc.ca> <1992May8.015202.10792@news.media.mit.edu>
Date: Sat, 9 May 1992 18:44:56 GMT
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In article <1992May8.015202.10792@news.media.mit.edu> minsky@media.mit.edu (Marvin Minsky) writes:
...lots of deletions...
>  Wow, gee.  Now "being careful" becomes part of the thesis!  Well,
>van Quine was awfully careful when he formulated "new foundations".
>And Barkley Rosser found the mistake.  The system was in fact
>inconsistent.
>

This is deceptive.  "New Foundations" is not known to be inconsistent;
Quine made a mistake in stating the stratification restrictions for
relations, as opposed to sets, which enabled Rosser to deduce the
Burali-Forti paradox.  (Quine has made a number of mistakes in stating
conqeuquences of NF and its variants, like everyone else who has
worked with this system). The set comprehension axiom of New
Foundations is consistent, and leads to an adequate theory of
relations as well (NFU, "New Foundations" with individuals, was proven
to be consistent by R. B. Jensen in 1969, and shown at the same time
to be extendible in strength as far as the usual set theory can be
extended).  It remains an _open_ question whether NF with full
extensionality is consistent.

Rosser himself wrote a book, after he published the paper in question,
in which he founded mathematical logic on NF (_Logic for
Mathematicians_).  Unfortunately, this book was published in the same
year that Specker showed that the Axiom of Choice can be proven to be
false in NF, a result which caused a sudden loss of interest in this
theory!  Jensen showed later that AC is consistent with NFU, but by
that time the damage was done.  Quine himself remarked that NFU may
have been the theory he really should have proposed; strong
extensionality in stratified set theory turns out to be more than the
technical convenience it is in the usual set theory.

There is a great deal of misinformation circulating about NF.  To
counter usual points: NF is not known to be inconsistent; it does
_not_ have great consistency strength, as far as is known (it is
probably consistent with the strength of the Theory of Types with the
Axiom of Infinity, thus much weaker than ZFC) -- this misconception
probably comes from the fact that NF has "large" sets like the
universe, but NFU has the same "large" sets and is weaker than Peano
arithmetic (when not extended with axioms of infinity).  The sense in
which measurable cardinals, for example, are "large" has nothing to do
with the "largeness" of the universe in NFU; in NFU + AC + measurable
cardinals (same strength as ZFC + measurables), the smallest
measurable cardinal would be one of the larger "small" cardinals in a
suitable sense.  The difficulties of the theory have nothing to do
with the presence of large sets like the universe; NFU + Axiom of
Choice + Infinity is a theory with the strength of the Theory of Types
with Infinity, shown to be consistent by Jensen, and a fairly friendly
mathematical environment (remarkably similar to Zermelo set theory (Z
is very slightly stronger) when one figures out how to deal with the
differences between the technical limitations).
-- 
The opinions expressed		|     --Sincerely,
above are not the "official"	|     M. Randall Holmes
opinions of any person		|     Math. Dept., Boise State Univ.
or institution.			|     holmes@opal.idbsu.edu


