[FOM] What makes a large cardinal axiom plausible?
Timothy Y. Chow
tchow at alum.mit.edu
Thu May 6 10:28:31 EDT 2004
I want to pick up on Roger Bishop Jones's suggestion that what makes a
large cardinal axiom plausible is that it "merely" sets a lower bound
on the size of the cumulative hierarchy.
This sort of makes sense for, say, inaccessible cardinals, because it
seems that all one is postulating is that all the sets that can be
generated using processes that have already been clearly defined can
be gathered together into a single set.
It doesn't seem, however, that this kind of thinking applies to many of
the larger cardinals, e.g., measurable cardinals. The definition of these
does not have any immediate relationship to the concept of collecting
together all smaller sets, and although it was known that they were
inaccessible, it took a long time before people realized how large they
really had to be. A lot of these cardinals seem to be motivated by
analogies from combinatorics and measure theory, not by the cumulative
hierarchy directly.
Now the existence of (at least some of) these cardinals can be motivated
by informal arguments that the set of all sets has the property in
question, and so "by reflection" there must be an actual cardinal with the
property (unless it's inconsistent). Though somehow this seems to me to
be a more general concept than that of "putting a lower bound on the
cumulative hierarchy," I guess I can see how someone might think that
it's all pretty much the same idea. So, would it be fair to say that the
plausibility of large cardinal axioms comes from these kinds of arguments?
Just in case there's any temptation to answer "yes" too hastily, let me
say that my impression is that there are several other criteria that drive
the acceptance of new axioms (not necessarily large cardinal axioms). For
example, set theorists seem to like axioms that lead to rich and beautiful
theories. Large cardinal axioms might be accepted for such reasons.
Conversely, an axiom like V=L might appeal to the typical working
mathematician who's getting a first exposure to set theory, because
it eliminates what he considers "pathologies" by settling lots of
"irrelevant" questions, but is disliked by many set theorists (perhaps
for the same reasons!).
In any case, to touch on another of RBJ's questions, the work of
Harvey Friedman and others has disillusioned us from the naive thought
that large cardinal axioms "merely" deal with how big things can get
"up there" and don't influence life "down here." Infinitary combinatorial
properties of large cardinals translate, using methods that are getting
increasingly direct, into finitary combinatorial properties of finite
sets. If the "irrelevance" of large cardinals to finitary mathematics
was ever a criterion for their acceptance, I don't think it should be
any longer.
Tim
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