[FOM] Arithmetical compatibility of higher axioms
joeshipman@aol.com
joeshipman at aol.com
Sat Aug 15 02:17:58 EDT 2009
Dmytro, thanks for your thoughtful reply. I think the right answer to
my query is that there are plausible disagreements about the truth of
statements of second-order arithmetic, but that they all seem to
involve one statement that is a consequence of V=L and one statement
that implies a nonconstructible real. This isn't so bad, because one
can develop a pretty good intuition for V=L even if coding it up in
second-order arithmetic is horrible.
I think it is possible to take the argument for V=L further and claim
that the only sets which exist are the sets which must exist by the ZFC
axioms. This ends up giving you the axiom V=M, where M are Cohen's
"strongly constructible sets". V=M is equivalent to the conjunction of
V=L and ~SM (there is no standard model of ZF). ~SM is really not
anything like perverse axioms like ~Con(ZF+Inacc); it should be viewed
as saying "when you keep adding sets as needed to satisfy ZF, you
really end up adding ALL the sets there really are". This axiom ~SM can
be expressed in second-order arithmetic, and it is nearly as extreme as
you can go in the "small" direction. (You could go even further by
adding the axiom that all reals are constructible, but I don't think
you could get all the way to V=M within second-order arithmetic.)
But it is also possible to argue for plausible axioms in the "large"
direction, such as the existence of a real-valued measure on the
continuum, which entails weakly inaccessibly many real numbers.
My main point is that if we ever encounter an alien civilization which
has developed mathematics, I think there will probably be mathematical
statements we disagree with them about the truth of, but there will NOT
be arithmetical statements we disagree with them about the truth of.
They may regard an arithmetical statement A as proven while we think it
has not been proven or vice versa, but they will never think A has been
proven while we think A has been DISproven (except for the relatively
short period of time required to catch mistakes in mathematical papers
[which really is a short time compared with the age of most important
theorems]).
I am coming around to the attitude that Kronecker's dictum should be
flipped. It is not that God made the integers and the rest is the work
of man;
rather, the integers are the only mathematical objects man can know and
the rest should be left to God. That doesn't mean higher infinities
don't exist, just that, unlike God, we cannot know anything about them
except what we can prove from ZFC. Human attempts to settle questions
like CH are fruitless in this life; but the compensating miracle is
that we are free to assume anything non-perverse about infinite sets
without being led into error about the integers! (In other words, we
can obtain knowledge about the integers that goes beyond what we can
prove in ZFC, even though we cannot obtain knowledge about higher type
sets, because of the observed compatibility of non-perverse extensions
of ZFC with respect to arithmetical statements.)
The attitude that higher infinities are useful fictions that are
permissible in proofs of arithmetical statements goes back to the 19th
century, but Hilbert's hopes that they could be shown to be
conservative fictions (useful in shortening and simplifying proofs but
not in proving new things) were dashed by Godel. However, the reaction
to Godel should have been that infinite sets are not fictions and it's
a good thing they're not conservative because that makes them even more
useful. (Of course, such a reaction only looks correct after many
decades of work revealed that the set-theoretic landscape really does
hang together consistently with respect to arithmetical statements, an
observation that would have surprised logicians in the 1930's.) We can
therefore reconcile our differing views on which higher axioms should
be added to ZFC by taking all of them but replacing them with
arithmetical axiom schemes of the form "If Phi is an arithmetical
consequence of X then Phi is true". (This has to be a scheme because
you can't define arithmetical truth within arithmetic, but you can use
the arithmetical definitions of 1-quantifier truth, 2-quantifier truth,
etc.).
-- JS
-----Original Message-----
From: Dmytro Taranovsky <dmytro at mit.edu>
.... A mathematician who rejects general impredicative
definitions but accepts the notion of an arbitrary ordinal -- and the
ability
to iterate a construction any ordinal (even uncountable) number of
times -- may find ZFC + V=L to be a natural theory.
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