[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He also makes some remarks about his own unpublished independence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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