[FOM] CH in standard models

A.P. Hazen a.hazen at philosophy.unimelb.edu.au
Sun May 25 04:08:50 EDT 2003

   Roger Bishop Jones has tried to precisify the issue by restricting it to
the status of CH in "standard" models, in the hope of finding something
about which there is more "consensus."
   This doesn't help answer the question about truth value, but it is a
familiar way of arguing for* the MEANINGFULNESS of CH: cf. discussion by
Kreisel (1), in turn discussed by Weston (2).
   The bad news is that it only works for people who believe there IS a
standard model.  I recall a conversation a decade ago with a distinguished
logician (someone whose work in set theory has been discussed on the FoM
forum) who seemed almost unable to comprehend the question: CH is true in
some models and false in others, true of some cumulative hierarchies and
false of others, and it took several minutes before he seemed to recognize
the logical possibility that someone might think that one of those
cumulative hierachies was  "standard."  (Which one?  Well, the one in which
each rank contains ALL the sets of objects of lower rank... but of course
in every model, each rank contains all OF THAT MODEL'S sets of objects of
lower rank: you can imagine how the conversation went!)  So I fear that
even with this precisification, consensus will not be universal.
* Maybe "suggesting" or "trying to make plausible" would be better than
"arguing for" here.
(1) Kreisel, "Informal rigour and  completenessproofs," in I. Lakatos, ed.,
"Problems in the philosophy of mathematics," North-Holland, 1967.  (The
first half of this paper was reprinted in J. Hintikka, ed., "The Philosophy
of Mathematics," Oxford U.P. ("Oxford Readings in Philosophy" series),
1967, and so would  have been familiar to many philosophy students before
Oxford allowed the Hintikka collection to go out of print.)
(2) T. Weston, "Kreisel, the continuum hypothesis, and second-order set
theory," in "Journal of Philosophical Logic" 5 (1976), pp. 281-298.
(Weston showed that the  move to second-order ZF doesn't help solve the
problem of truth value axiomatically in "The continuum hypothesis is
independent of second-order ZF," in "Notre Dame Journal of Formal Logic" 18
(1977), pp. 499-503.)
Allen Hazen
Philosophy Department
University of Melbourne

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