[FOM] Hume's principle and the axiom of choice
Stewart Shapiro
shapiro+ at osu.edu
Wed May 14 20:01:19 EDT 2003
Apologies if this is not appropriate for this list. Feel free to ignore if so.
There was an extended discussion of Hume's principle on this list some time
ago (including, as I recall, a discussion of the extent to which the name
is appropriate). I have some questions concerning its relation to the
axiom of choice and set theory.
First, a reminder on what Hume's principle (HP) is. Let F and G be
monadic, second-order variables and let # be a name of a function from
items of type F,G to objects (i.e., items in the range of the first-order
variables). HP is the following sentence
(F)(G)[#F=#G iff EQU(F,G)],
where EQU is the second-order statement that there is a function mapping
the F's one to one onto the G's. It is abbreviated, "F is equinumerous
with G".
As some of you may recall, Frege's two great foundational works contain the
essentials of a derivation of the Peano postulates from HP. This theorem
is sometimes called "Frege's theorem".
George Boolos observed that HP is satisfiable on any infinite set. His
proof makes use of the axiom of choice, however. What George noted is that
HP is satisfiable on any infinite, well-ordered set. The number of sizes
of subsets of aleph-l is the maximum of aleph-0 and l, which is less than
or equal to aleph-l. So we can interpret the #-function symbol.
I wondered if this use of choice is necessary. For example, is it
consistent with ZF that HP is not satisfiable on the continuum.
Some time ago, I put this question in an email to Professor Shelah, and got
a negative answer. I could not reconstruct the model he suggested (my
forcing is rusty, especially when choice is falsified). But I (think I)
could see that the model he suggested did the trick. Here are its
properties (as I understand it):
In this model, there is a cardinal k such that every b<k is less then or
equal to the continuum. k is a fixed point in the alpehs (the number of
cardinals less then or equal to k is k). For each b<l, there is a one to
one map from b into the set of reals. And there is no one to one map from
k into the set of reals.
I got some helpful comments from Tom Jech and Tim Bays.
I'd be grateful if anyone could explain the model a bit more.
Second question. What happens to HP when it is interpreted on V (so that F
and G range over proper classes)? (n.b., anybody who finds talk of proper
classes incoherent or silly should stop reading.)
To apply the Boolos construction to V, we would need global choice (or
global well-ordering), so that all of the proper classes are
equinumerous. In that case, HP is easily satisfied. Just have # map each
set onto its cardinal and have it map each proper class onto a designated
set that is not a cardinal.
What if we don't assume global choice. Is it consistent with ZFC (i.e.,
with local choice only) that HP cannot be satisfied on the universe?
What I am asking about is the third-order sentence obtained from HP by
replacing # with a variable and binding it with an existential
quantifier. Is that sentence independent of ZFC?
I put that question to Saharon too, and he said that it is independent, but
the only further remark he made is that this is shown by a "standard
forcing argument". Can anybody shed more light on this? Or point to a
reference.
I have the wonderful Jech book handy, but could not find the relevant
results in it.
In case anybody is still reading, I am not sure if any of this has any
bearing on the philosophical discussion of HP in the neo-logicist
literature. They sometimes speak of abstraction principles (like HP) being
conservative, but the indicated notions are not sufficiently articulated to
be sure how relevant this stuff about choice may be.
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