[FOM] Paul Cohen was wrong

Monroe Eskew meskew at math.uci.edu
Sat Sep 10 15:43:22 EDT 2011

I know the existence of \aleph_1 can be logically separated from the
existence of P(\omega).  But in his quote, he says that cardinals are
built "from ideas deriving from the replacement axiom."  I disagree.
There is no "powerful new principle" that separates powersets from
cardinal successors.  The same principle underlies both, and
restricting our attention to fragments of ZFC, the same axiom
underlies both.  You can introduce new axioms weaker than full
powerset to tease them apart.  But what is the difference in character
between your A1 and A2 below and the following?

B1: There is a set that contains all subsets of omega.

B2: There is a set that contains all subsets of P(omega).

The A's and B's are both just special cases of powerset.

On Fri, Sep 9, 2011 at 9:53 PM, Brian White <white at math.stanford.edu> wrote:
> 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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