From: boshuck@triples.math.mcgill.ca (William Boshuck)
Subject: Re: Expansion of set theory
Organization: Dept Of Mathematics and Statistics
Date: Wed, 12 Aug 92 16:44:08 GMT

In article <1992Aug7.002122.24601@access.usask.ca> choy@skorpio.usask.ca writes:
>Suppose I take an oft used theory of sets and tack on axioms that are
>consistent but independent. The things that can be proven may alas
>be unrealistic, but any axiom used to prove an unrealistic result can be
>either negated or left to rot. So set theory grows. Is there a good
>algorithm for tacking on consistent and independent axioms to a
>set of axioms?
>
>Henry Choy
>choy@cs.usask.ca

I don't think that there could be an algorithm per say, to 
do this sort of thing, but there is a natural heuristic which
is probably how the axioms of set theory got there in the first
place. Namely, when things exploded after the Russell paradox, 
the idea to put things together was to begin with a more or
less concrete model of what was to become set theory and then
start writing down thigs which just had to be true of this model.

The model is, of course, V (the cumulative hierarchy) given by

         V_0 = the empty set
         
         V_1 = P(V_0)   (the power set of V_0)
          .       .   
          .       .
          .       .
         V_{omega} = the union of V_n for n<{omega}

and continue FOREVER taking power sets and then unions and power 
sets and then unions...

The union of all these things is V. Taking this informal 
description of V and then looking at the axioms of (say ZFC)
set theory we see that they are virtually "obvious" in V.
(There might be some quibbling over the axiom of choice, but
I don't want to get into this. If you want to drop the "C" from
ZFC, fine; but I still say that it's more or less obvious from the
description of V that I've given.)

One way to go beyond ZFC is to look at "axioms" which seem to say
that V contains very large sets, that is, we might be led to
believe that anything we do to "push up" the cumulative hierarchy
ought to be more or less harmless if it does not immediately
give rise to one of the classical paradoxes. The search for such
extended "axioms of infinity" is part of the study of "Large
Cardinal Axioms". Informally, a large cardinal is a cardinal
number which is so big that the assumption that it exists is
not even provably consistent with set theory (in ZFC). So the
existence of such a thing is a genuine addition to set theory.

The reason why these axioms are sometimes referred to a axioms 
infinity is that they behave with respect to a "modest" V
similarly to the way that the axiom of infinity behaves with
respect to the axioms of ZFC less the axiom of infinity. That
is, the hereditarily finite sets form a model of ZFC less the
axiom of infinity AND the axiom of infinity (the existence
of omega, the first infinite cardinal) allows the construction
of the hereditarily finite sets as a SET; an "inner model"
of ZFC less infinity, if you will. The way things usually go
with large cardinals is that the assumption that they exist
allows the construction of an inner model of ZFC (e.g., in any
model of ZFC plus "there exists a strongly inaccesible cardinal"
one can build the cumulative hierarchy up to such a
cardinal. This segment of V is a SET in the model AND it is a
model of ZFC. Of course its not always this simple.).

An intersting facet of all of this is that large cardinal axioms
actually have a bearing on relatively small things, like Borel
sets of real numbers, and sets of natural numbers in the definable
hierarchy. I hope to understand this phenomenon one day.

There is some discussion of large cardinals in Shonfield's (spelling?)
book on mathematical logic and is Jech's book on set theory. There is
also a book by Drake on the subject of large cardinals. My personal 
knowledge of the stuff (which is not very much yet) comes from seminars
given at McGill (by M. Makkai and S. Mann) on the topic.

I hope that I have been helpful.

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