From: columbus@strident.think.com (Michael Weiss)
Newsgroups: sci.math
Subject: Re: Beloved Books + Request
Date: 27 Aug 92 11:38:29
Organization: Thinking Machines Corporation, Cambridge MA, USA
NNTP-Posting-Host: strident.think.com
In-reply-to: mccolm@darwin.math.usf.edu.'s message of 24 Aug 92 15:31:59 GMT

In article <1992Aug24.153159.1240@ariel.ec.usf.edu>
mccolm@darwin.math.usf.edu. (Gregory McColm) writes: 

   Set theory nowadays is divided into 3 major fields:

   1.  Combinatorial set theory:  use of transfinite structures 
       to do transfinite arithmetic.  This is what is meant by 
       "classical set theory".  This is (distantly) related to 
       the combinatorics of finite sets, which is part of 
       (finite) combinatorics.

   2.  Forcing: [...]

   3.  Descriptive set theory: [...]

I gather you are including large cardinals under Combinatorial set theory?
Although as Drake's book shows, the boundary between 1&3 can get a little
blurred. 

   Some surveys:

   Drake's Set Theory
     A good book, out of date and out of print;  tilted towards 
     combinatorics

To expand on this a little, Drake explicitly omitted all discussion of
forcing, since there are many other treatments available.  I second the
recommendation.

   Halmos Naive Set Theory
     Some love it, some hate it.  No ZFC stuff.  I found it 
     incomprehensible

In the "set theory without axioms" category, Kaplansky's book should also
be mentioned. I thought it was better than Halmos.  For that matter,
Cantor's original papers were republished by Dover sometime ago, and are
very readable.

   [other refs omitted]

   Cohen's Set Theory and the Continuum Hypothesis
     The grand original on forcing:  opaque and not recommended
     (Shoenfeld wrote a decent paper on forcing--I forget where 
     it is.  Burgess's article in the Handbook of Math Logic is 
     error-riddled.  The accounts in Jech's books are okay;  Kunen
     is too fond of complex notation)

Shoenfield made significant technical simplifications to Cohen's original
forcing proofs (as did Scott and Solovay); instead of digging up the
article, I suggest looking at the last chapter of his book on mathematical
logic.

I'd recommend looking at Abraham Robinson's model-theoretic forcing, or
Fefferman's (sp?) forcing in Peano arithmetic, before tackling forcing in
set theory; the key ideas come across more simply here than in their
original mileau, ZF.

Finally, I consider Cohen's book to be one of the great classics of
mathematical exposition.  (Some I'm sure will strongly disagree with me.)
The first three chapters provide a lucid treatment of axiomatic set theory
(meaning ZF) from the ground up, and of Godel's proof of the relative
consistency of AC and GCH.  The concepts come across clearly, unencumbered
by excessive notation.  The "readability index" drops abruptly in last
chapter on forcing, simply because the technical simplification of Scott,
Solovay, and others lay still in the future; on the other hand, original
treatments possess a certain vitality that is often missing in the later
streamlined reworkings.

   ===================================================================
			     |
   Greg McColm               |    I see it, but I don't believe it.
   Dept of Mathematics       |
   Univ of S Florida         |        -----Georg Cantor
   mccolm@math.usf.edu       |
Xref: sparky sci.math:10901 comp.lang.misc:2900 comp.lang.fortran:3391