[FOM] Re: What makes a large cardinal axiom plausible?
Timothy Y. Chow
tchow at alum.mit.edu
Sat May 8 14:29:03 EDT 2004
Roger Bishop Jones wrote:
> Tim in his message also observed that some large
> cardinals axioms seem not to connect in with this
> intuitive picture at all.
> This however, does not necessarily rule out the
> principle. Surely those large cardinals axioms
> which do not appear directly to address the
> question of height are only called large
> cardinal axioms because they have been shown
> to entail and be entailed by lower bounds on height?
I'm not sure this is the right intuitive picture. Something like a
measurable cardinal is not defined in terms of a lower bound on height, in
the sense that it is not phrased in terms of "if the cumulative hierarchy
goes up high enough then you have to have a measurable cardinal." Now,
it's true that it's called a large cardinal because it *entails* a lower
bound on height, but I don't think that it would need to "be entailed
by" a lower bound on height for people to call it a large cardinal. In
fact, as Joe Shipman suggested, it's something of a surprise that all
these large cardinal axioms line up nicely; this is not at all an obvious
or trivially "built-in" consequence of their definitions (if it were, then
the ordering according to consistency strength would presumably match with
the ordering according to size, and it doesn't), as one might expect if
the underlying thought behind them all were simply "height."
When I mentioned Harvey Friedman's work, I was not ignoring your comment
that large cardinals imply new theorems in arithmetic. Rather, I thought
that the "theorems in arithmetic" you were referring to were those that
were covered by your phrase "byproduct of a lower bound on height," e.g.,
the sorts of things that follow more or less immediately from Goedel's
work. What I was trying to point out was that some large cardinals enjoy,
and are even defined in terms of, *combinatorial* properties that, in
increasingly direct ways, imply *combinatorial* properties of finite sets.
As far as I can tell, "height" and its "byproducts" do not play a key
role in the intuition behind these facts. However, perhaps Harvey
Friedman can weigh in and educate both of us here.
To summarize my point: for the most part, all that it takes for a cardinal
to be called large is that it be strongly inaccessible, and "height" in a
restrictive sense is not the only reason for thinking them plausible; it's
usually enough if the cardinal enjoys some property (not only size) that
the set of all sets "ought" to have. Beyond that, I'm not sure what
reasons people have for believing in them, and I'd be interested in
others' responses to the issues you've raised.
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