[FOM] Paul Cohen was wrong
Monroe Eskew
meskew at math.uci.edu
Thu Sep 8 22:49:04 EDT 2011
Consider the following quote from Paul Cohen's book, Set Theory and
the Continuum Hypothesis:
<<"A point of view which the author [Cohen] feels may eventually come
to be accepted is that CH is obviously false. The main reason one
accepts the axiom of infinity is probably that we feel it absurd to
think that the process of adding only one set at a time can exhaust
the entire universe. Similarly with the higher axioms of infinity. Now
\aleph_1 is the cardinality of the set of countable ordinals, and this
is merely a special and the simplest way of generating a higher
cardinal. The set C [the continuum] is, in contrast, generated by a
totally new and more powerful principle, namely the power set axiom.
It is unreasonable to expect that any description of a larger cardinal
which attempts to build up that cardinal from ideas deriving from the
replacement axiom can ever reach C.
Thus C is greater than \aleph_n, \aleph_\omega, \aleph_a where a =
\aleph_\omega, etc. This point of view regards C as an incredibly rich
set given to us by one bold new axiom, which can never be approached
by any piecemeal process of construction. Perhaps later generations
will see the problem more clearly and express themselves more
eloquently.">>
But his argument is self-defeating! Because in fact, \aleph_1 cannot
be built without the powerset axiom. The collection HC of
hereditarily countable sets satisfies ZFC minus powerset. The bold,
powerful principle which allows C to exist is no different from that
which allows \aleph_1 to exist.
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