[FOM] Paul Cohen was wrong
Brian White
white at math.stanford.edu
Sat Sep 10 00:53:46 EDT 2011
I think you've misunderstood what Cohen is asserting.
He doesn't say you can get aleph_1, aleph_2, etc using
ZFC minus the power set axiom.
Rather, he says you can get them with something weaker than
the power set axiom, in particular with higher axioms of
infinity.
He doesn't say what those axioms
are, but for example the following would do:
A1: there is an ordinal that contains all countable ordinals.
(Of course there is then a smallest such ordinal, namely aleph_1).
A2: there is an ordinal that contains all ordinals equipotent to aleph_1
etc.
On Sep 8, 2011, at 7:49 PM, Monroe Eskew wrote:
> 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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