[FOM] set theory comments
Andreas Blass
ablass at umich.edu
Wed Jul 16 12:35:43 EDT 2003
I'd like to comment on some set-theoretic questions raised on fom
a while back. I apologize if someone else already made these comments and
I missed them.
Todd Eisworth asked "How much of the strength of the axiom of
infinity comes from the fact that it essentially posits the existence of
the set of natural numbers, as opposed to simply asserting the existence
of a set that is 'not finite'?" It may be interesting to note that, in
Zermelo set theory (i.e., without the axiom of replacement), various forms
of the axiom of infinity that, intuitively, posit the existence of the set
of natural numbers are still inequivalent. See Gabriel Uzquiano's paper
"Models of second-order Zermelo set theory" [Bulletin of Symbolic Logic 5
(1999) 289-302, http://www.math.ucla.edu/~asl/bsl/0503/0503-001.ps] for
details.
In connection with this question, Allen Hazen pointed out that the
double power set of any Dedekind infinite set (or indeed of any set that
isn't finite) includes a copy of the natural numbers. That observation
also answers negatively the next question in Eisworth's message, "Is it
consistent ... that ... the collection of D-finite sets is a model of ZF
minus infinity, but different from the hereditarily finite sets?"
In a message of 18 June, Matt Insall proposes that "for each
cardinal k, there is an ordinal a>k that is of cardinality k but is not
constructibly of cardinality k." Another way to say this is that the
successor cardinal k+ as computed in the constructible universe L is
strictly smaller than the real k+. Unless I'm mistaken, the following
three statements, of which Matt's proposal is the first, are provably
equivalent (in ZFC):
(1) For all infinite cardinals k, k+ of L is < k+.
(2) For at least one singular cardinal k, k+ of L is < k+.
(3) 0# exists.
[Partial explanation of (3): The existence of 0# is morally but not
strictly a large cardinal axiom. It is a statement about natural numbers
and sets thereof --- a Sigma^1_3 statement of second-order number theory
--- so if it is true then it remains true if we truncate the universe at
the first inaccessible cardinal. So it doesn't imply the existence of
inaccessibles. On the other hand, it implies that the constructible
universe satisfies a lot of large cardinal axioms. For example (to use
large cardinals that have appeared on fom before), it implies that there
are (really countable) ordinals that are, in the sense of L, subtle
cardinals. The existence of 0# follows from te existence of measurable
cardinals, and, as my preceding comments show, it very strongly
contradicts V=L.] In the equivalence of (1), (2), and (3), the implication
from (3) to (1) is quite easy once one has the basic theory of 0#, and the
implication from (1) to (2) is obvious. The implication from (2) to (3)
is a consequence of a deeper (than the basic theory of 0#) result,
Jensen's covering theorem.
Andreas Blass
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