FOM: Existence=Consistency for large cardinals?
Joe Shipman
shipman at savera.com
Fri Feb 25 10:59:26 EST 2000
Davis says:
>>where large cardinals are concerned, it is hard to see any basis for
>believing that the corresponding consistency statements (Pi-0-1 of
course)
>are true, except for some kind of ontological belief. So in that highly
>important case, the Hilbert principle is useless.<<
I disagree with this. Steel has argued similarly, and I quote from a
post of his last month:
>>It is sometimes argued that one can have
>all the benefits of the consistency strength of large cardinal
hypotheses
>while still maintaining that V=L, by adding to ZFC + V=L statements
like
>"there is a transitive model of "ZFC + there is a measurable cardinal"
"
>and the like.The new theory has all the Pi^0_1 consequences (in fact
all
>the Sigma^1_2 consequences) of ZFC + "there is a measurable cardinal",
but
>also contains "V=L", and therefore settles the GCH and many other
>questions of general set theory. It is strong in both the realm of the
>concrete and the realm of sets of arbitrary type.
> One might as well take the view above to its logical conclusion: Why
>not adopt Peano (or primitive recursive) Arithmetic plus Con(there is a
>measurable), Con(there is a supercompact), Con(...),... as our official
>theory? We get all the Pi^0_1 consequences we had before, and we decide
>every question about sets of high type by simply declaring there are no
>such sets. Isn't this progress?
>
> One can see what is wrong with this theory by noting that it is
>parallel to the theory that there is no past, that the world popped
into
>existence on, say, Jan.1,1998, complete with fossils, microwave
background
>radiation, and all the other evidence of a past. The theory is that
there
>is no past, but the world behaves exactly as if it had one. The
problem
>with this theory is that in using it, one immediately asks "what would
the
>world be like if it had a past", and then one is back to using the
>standard theory. The assertion that the world began Jan.1 cannot be
used
>in that context, and since that is the context we're always in, the
>assertion is just clutter. We've taken the standard theory and added
>meaningless window dressing.
>
> The parallel with PA + Con(ZFC + measurables) is pretty clear: one
is
>meant to use this theory by dropping into ZFC + measurables.( Of
course,
>one may not then use all of that theory in a given context; mostly,
>2nd or 3rd order arithmetic is enough. But then, big-bang theory isn't
>used too much either.) The assertion that there are no sets of infinite
>rank, but the sets of finite rank behave as if there were, is analogous
to
>the assertion that there is no past, but the world behaves as if there
>were. ( The fact that we could have chosen any "birthday" for the
universe
>corresponds to the fact that we could have taken 2nd order arithmetic,
>Zermelo, ZFC + V=L, or something else as the window dressing for our
>mathematical theory.)
> In sum, we can PRETEND to adopt V=L as part of our universal
framework
>theory, while reaching consistency strengths at the level of a
>measurable and beyond. But as it is imagined above, this is only a
>pretense: our mathematical behavior, which is what gives meaning to our
>theory, is that of one who believes that there are measurable
cardinals.
>The framework theory where the action takes place contains
>"there is a measurable cardinal", and "V=L" is just an incantation
which
>does no work in the important arena. Put another way: we have just
found a
>very strange way of saying that there are measurable cardinals.<<
I respond that if large cardinals do NOT exist , we have no reason for
believing that ZFC is powerful enough to prove that they don't exist.
The axioms of ZFC are "local" in a certain sense, they show how to build
up sets from earlier sets by replacement, separation, union,
set-of-subsets, etc., without presuming to assert anything limitative
about the universe of sets as a whole. If the universe contained no
inaccessible cardinals, we wouldn't expect ZFC to "know" that. Turning
this around, we get from this statement "nonexistence of large cardinals
shouldn't lead us to expect their inconsistency" the Bayesian
contrapositive "consistency of large cardinals shouldn't lead us to
expect their existence". When Steel says
" 'V=L' is just an incantation which does no work in the important
arena"
he forgets his point that 'V=L' does work in a *different* arena
(non-arithmetic statements). To Davis's
"it is hard to see any basis for believing that the corresponding
consistency statements (Pi-0-1 of course) are true, except for some kind
of ontological belief"
I respond that the work of set theorists over the last 40 years or so
has unveiled a rich and coherent structure and it is foreign to the
historical experience of mathematicians for such a complex field of
mathematical study to collapse into inconsistency. I know of no
comparable case of an inconsistent system of mathematics having been so
interesting before the inconsistency was discovered. This observation
does not depend upon ontological belief in large cardinals, because
without them the theory is not "about nothing", it is about models of ZF
(of accessible size) satisfying the large cardinal axioms, which are
quite interesting in and of themselves, as well as being usefully
applicable (for example, to show that there is no hope of proving
existence of a measure on all subsets of [0,1]).
To summarize, I am not ontologically rejecting large cardinals at all; I
am just criticizing two related arguments:
1) that their consistency is good evidence for their existence (Steel)
2) that the only basis for believing their consistency is an ontological
belief in their existence (Davis).
Contra Davis, I say that we have other reasons for believing in their
consistency (their intrinsic richness and coherence and the historical
experience that has led to strong intuitions of their consistency);
contra Steel, I say that believing in their consistency doesn't make me
much more likely to believe in their existence; but as he recognizes,
there are other reasons for believing in their existence (for example,
their extrinsic fruitfulness in other parts of math).
-- Joe Shipman
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