[FOM] RE: FOM New Axioms
E. Todd Eisworth
eisworth at math.uni.edu
Sat Jun 28 16:39:28 EDT 2003
[Matt Insall (previously)]
(**) When possible, sets behave as do finite sets. Otherwise,
proofs or new axioms are required to explain why some sets
stray from behaving as if they are finite.
[Ralph Hartley]
I can't off hand think of a statement that is true/provable for all finite
sets which, if extended to all sets, implies CH or ~Choice, but can you
prove there isn't one?
How hard is it to find a pair of statements that are both true for all
finite sets, but which can't both be true for all sets (including infinite
ones)? ]
[Matt Insall]
Here is one: if X is a set of real numbers, then X is measurable. This
statement holds for all finite sets X, but when extended to all sets,
implies ~AC.
[Ralph Hartley]
How hard is it to find a pair of statements that are both true for all
finite sets, but which can't both be true for all sets (including infinite
ones)?
[Matt Insall]
I do not think it is very hard. But I expect it to be somewhat unnatural in
most cases. (Whatever that means.)
############################################################################
Recall that a set S is Dedekind finite (D-finite) if there is no one-to-one
mapping of S onto a proper subset of S.
One of Cohen's original models for the failure of AC presents us with an
infinite set that is nevertheless D-finite.
Thus, in a sense, it is possible to have an infinite (= non-finite) set that
behaves like the finite sets in a very strong sense.
How do you counter an argument that the principle "sets should behave like
finite sets whenever possible" implies that we must admit infinite D-finite
sets into our mathematical universe?
###############
While I'm thinking about it, maybe some of you can comment on the following
question: [Warning: What follows is a "random thought" (tm) and as such
there is no guarantee of it being of interest!]
How much of the strength of the Axiom of Infinity comes from the fact that
it essentially posits the existence of the set of all natural numbers, as
opposed to simply asserting the existence of a set that is "not finite"?
Is it consistent (relative to ZFC) that there is a transitive model of ZF
such that the collection of hereditarily D-finite sets is a model of ZF
minus Infinity, but different from the Hereditarily Finite sets?
Is there any "meaningful mathematics" that can be done in such a universe
that cannot already be done in HF?
Thanks!
Todd
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