Randall Holmes holmes at
Wed Feb 4 11:08:20 EST 1998

This posting is from M. Randall Holmes

re my own proposal in my last posting that the continuum might be a 
proper class (I do not actually think that this is the case!):

the kind of formal theory I had in mind is

Morse-Kelley - Power Set + all infinite sets are countable +
all proper classes are the same size

"Morse-Kelley" rather than "ZFC" because one would like to be able to
quantify over classes of reals to do analysis conveniently (without
technical problems with the least upper bound axiom).  This is
essentially third-order arithmetic.

The consequences of "all proper classes are the same size" (an
axiom due to von Neumann) include replacement, choice and (in this
context) CH, so some of the Morse-Kelley axioms are redundant.

One might adopt a further axiom to express the idea that this is a
proposal that the continuum is very (implausibly) large rather than a
proposal that the universe is small, such as

"the proper class ordinal (the real aleph-one) is inaccessible in L".

The alternative set theory of Vopenka has the same two cardinals.
Rucker proposes something similar as a motivation for CH in his book
Infinity and the Mind (he considers von Neumann's axiom in the context
of hereditarily countable sets).

It appears that such a system (even without the last axiom) is
adequate for almost all mathematics outside of set theory itself. One
hopes for successes from the program of reverse mathematics to make
set theorists more indispensible!

I don't believe that any observation in this posting is in the least
original; I just thought that I ought to be a little more concrete.

Mathematics is a religion!	|   --Sincerely, M. Randall Holmes
We are Reform Pythagoreans      |   Math. Dept., Boise State Univ.
(we eat beans).	No official BSU |   holmes at
endorsement of above opinions!  |

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