[FOM] Explanation/Continuum Hypothesis

Alasdair Urquhart urquhart at cs.toronto.edu
Mon May 8 10:09:10 EDT 2006

I managed to locate the passage of Goedel that I was
referring to in my earlier posting.  In his 1947 survey,
"What is Cantor's Continuum Problem?"  he conjectures
that CH is independent because the axioms of set theory
hold for two quite different conceptions, namely
the constructible sets, and the notion of the full cumulative
hierarchy of "arbitrary" sets.

	There are two quite differently defined classes of 
	objects which both satisfy all axioms of set theory
	written down so far.  One class consists of the sets
	definable in a certain manner by properties of their
	elements, the other of sets in the sense of arbitrary
	multitudes irrespective of if, or how, they can
	be defined.  (Collected Works, Vol. II, p. 183)

In Hao Wang's book, "A Logical Journey" some interesting
comments of Goedel about the notion of generic sets and
forcing are recorded.  On pages 251-252, he compares 
Cohen's generic sets to totalities generated by some physical process:

	The easiest way to understand Cohen's idea is to imagine
	sets to be physical sets ... If an arbitrary physical set is
	envisaged, empirical knowledge cannot define a definite
	limit: but Cohen nonetheless teaches us how generic
	statements could be made about it.

He also makes some remarks about his own unpublished independence
proof for AC:

	In fact, I had previously developed a part of a related method
	 -- not from constructible sets but from some idea stimulated
	by reading some work of Brouwer's -- and proved the independence
	of the axiom of choice.

I interpret this to mean that he was inspired by Brouwer's idea of free choice
sequence, but the nature of Goedel's purported independence proof remains
shrouded in obscurity.

Alasdair Urquhart

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