FOM: CH and second-order

[Allen Hazen]

     Most of the famous independence results are proof theoretic: such and
such statement is neither provable nor refutable from such and such axioms
BY MEANS OF the rules of such and such formalized logic.  There is,
however, no reason not to consider also the semantic notion of
independence: such and such statement is neither implied by nor
contradictory with such and such axioms by the lights of such and such,
semantically described and possibly non-axiomatizable, logic. (Church
discusses the two kinds of question in section  55 of his "Introduction to
Mathematical Logic" (1956).)  So **the** question of the independence of CH
from Second Order logic is really two different questions.  Both have known
answers.
    (i) Proof-theoretic question: does (someone's favorite axiomatization
of) Second Order Logic allow for the deduction of CH or not-CH from some
bunch of standard set-theoretic axioms.  Answer: no.  Second Order ZF is,
after all, a well-known set theory: Morse-Kelly set theory (used as a
metalogic in Mostowski's textbook on set-theoretic independence results,
"Constructible Sets with Applications" (North-Holland "Studies in Logic"
series, 1969).  Stewart Shapiro's useful textbook/encyclopedia of Second
Order Logic, "Foundations without Foundationalism" (Oxford Logic Guides,
1991) attributes the proof of the  independence of CH from Second Order ZF
to Weston, article in NDJoFL 18 (1977), pp. 499-503.
    (ii) Semantic (or theological) question.  Answer: yes.  Second Order ZF
has models with differing lengths of the series of ordinals, so various
large cardinal axioms ARE independent of Second Order ZF, so there is no
general reason to think Second Order ZF decides interesting questions...
but CH is one of the "lucky" ones. <<<Well, I suppose I'm making
theological assumptions in saying that Second Order ZF "has" models-- to
put it in an acceptably agnostic way: in ZF + "There are 2 or more
inaccessibles" it is possible to prove that [2nd O. ZF + "there is an
inaccessible"] and [2nd O. ZF + "there isn't..."] both have models.>>>  By
a result that goes back, in essence, to Cantor, the real line is
CATEGORICALLY definable by Second Order axioms (just as, for example, the
Dedekind-Peano axioms CATEGORICALLY define the natural number sequence).
It follows from similar arguments that any two STANDARD models of Second
Order ZF will have isomorphic "bottom ends": the segment of the cumulative
hierarchy up to (say) rank omega+13 (13 or any other natural number: what
matters is HIGH ENOUGH to include real numbers, defined in any standard
way, sets of reals, functions from sets of reals to reals, etc) in any such
model will be isomorphic to the corresponding segment in any other such
model.  Thus Second Order ZF decides the CH in the sense that either CH or
not-CH is (semantically) entailed by the ZF axioms on the standard
(semantic) characterization of Second Order entailment.  (SOME theological
arguments are clear enough that even the agnostic can see that the
conclusion follows from the theological assumptions!)  Weston discussed
this and its philosophical implications in "Kreisel, the continuum
hypothesis, and second -order set theory," Journal of Philosophical Logic 5
(1976), pp. 281-298.
--
Allen Hazen
Philosophy Department
University of Melbourne