[FOM] How much of math is logic? ORACLE question

[Allen Patterson Hazen]

 >  whether there is any question in ordinary mathematics that would
 > not be "settled" by an oracle for "second-order validity" in the
 > standard semantics for second-order logic.

I think that, on one natural way of understanding it, THIS question admits
of a clear mathematical answer  (clear, that is, within classical
mathematics: so perhaps not satisfactory to those with strongly
anti-platonistic philosophical stances):

Answer: NO.

Relevant understanding: An ORACLE is, as the word is used in recursion
theory, something that gives yes/no answers about membership in some set
of (or codable as a set of) natural numbers: in this case, the set of
sentences of Second-Order Logic that are valid in the "standard" sense.
(So: if your philosophical scruples make you sceptical about the
meaningfulness of this sense of validity, the rest of the answer won't be
of much interest to you.  Go read footnote 535 of Church's "Introduction"
for a statement of the non-sceptical viewpoint.)  A question of ORDINARY
mathematics is one which doesn't ask about large cardinalities: one whose
possible answers all refer only to sets of size less than the first strong
inaccessible.

Argument for answer: Let AX be the conjunction of the SECOND-ORDER ZF
axioms with the statement "there is no strong inaccessible."  AX is
categorical: it describes the cumulative type structure, restricted to
ranks less than the first strong inaccessible, uniquely (up to
isomorphism).  So, to answer a question "Is it the case that P?" of
ordinary mathematics, all you have to do is ask the oracle whether or not
"Ax->P" is valid.

ONE version of "logicism" would be a kind of "if-then-ism": mathematical
truths are propositions that follow LOGICALLY from (perhaps arbitrarily
chosen) "axioms," or, more precisely put, the REAL content of a
mthematical statement is simply that the statement follows logically from
certain axioms.  So, if you are willing to settle for this (weak?)
understanding of "logicism" and willing to take "logic" to be "standard"
Second (or higher) Order... logicism is true for "ordinary" mathematics.

David Lewis's "framework" in his "Parts of Classes" (which he says he
would be happy to call "logic" if the word didn't already have a different
usage) is between Second and Third Order Logic in expressive power (at
least if "reality" contains mereological atoms).  In the appendix to that
book, and again in "Mathematics is megethology" (originally published in
"Philosophia Mathematica" in 1993; repr. in Lewis's "Papers in
Philosophical Logic"), he presents an argument for a related version of
logicism ("frameworkism"?): "Ramsified" versions of very strong systems of
axiomatic set theory (that is: statemets to the effect that there exist
objects and a relation such that the objects are organized under the
relation in the way the set theory says sets are under membership) can be
formulated in the framework, and follow "logically" ("frameworkically")
from hypotheses about the number of atomic objects there are.

Allen Hazen
Philosophy Department
University of Melbourne