[FOM] Arithmetical compatibility of higher axioms

[joeshipman@aol.com]

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.