[FOM] Fwd: Paul Cohen was wrong

[Brian White]

On Sep 10, 2011, at 12:43 PM, Monroe Eskew wrote:

> 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.

Actually Cohen does not say that (in the passage you quote).
But he does certainly suggest it.  
The phrase  "ideas deriving from the replacement axiom"
is rather vague, and I have no idea what he meant by it.
Do you?

In your original message ("Paul Cohen was wrong"), you wrote:

>>> 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. 

Do you think Cohen wasn't very well aware of that?

> .  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 difference is that B1 is more powerful than A1:  B1 implies
A1 but not vice versa.   In a sense, it is much more powerful than
A1 because even A1 plus A2 plus A3 etc do not imply B1.

***********




> 
> 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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>> 
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