[FOM] Cohen was right
ali.enayat at gmail.com
Tue Sep 13 14:00:24 EDT 2011
The following two examples justify Cohen's position challenged by
Monore Eskew's recent postings.
In particular, the first ones addresses Eskew's comment that he sees
no philosophical difference between "completed R" (set of real
numbers) and "completed \omega_1." (set of countable ordinal), while
the second one shows the fundamental difference between "completed R"
and "completed alephs of all orders".
Let N be a model of ZFC in which the continuum is aleph_2; Cohen
showed us how to build N assuming Con(ZF).
Let M be H(aleph_2) as computed within M, i.e., M is the collection of
sets that are *hereditarily* of cardinality at most that aleph_1, as
viewed in N,
Then we have (1)-(3) below:
(1) All of the axioms of ZFC with the exception of the power set axiom
hold in M;
(2) The collection of real numbers DO NOT form a set in M;
(3) The collection of countable ordinals DO form a set in M (and they
are the last aleph in M).
So in M, "completed R" does not exist, but "completed omega_1" exists;
hence illustrating Cohen's claim.
Assuming Con(ZF + there exists an inacccessible cardinal), there is a
model N* of ZFC in which the continuum is a regular limit cardinal
(i.e., a weakly inaccessible cardinal). This is a consequence of
Solovay's classical modificaion of Cohen's argument in his "The
continuum can be anything it ought to be" paper, in which he
demonstarted that the continuum can be arranged to be any prescribed
aleph of uncountable cofinality in a cofinality-preserving generic
extension of the universe (Easton, in turn, generalized Solovay's
theorem, but that's a different story).
In such a model N*, if we define M* as H(continuum), i.e., then we have:
(1*) All of the axioms of ZFC with the exception of the power set
axiom hold in M*;
(2*) The collection of real numbers DO NOT form a set in M*';
(3') There is no last aleph in M*.
More information about the FOM