FOM: Lowenheim numbers
John Mayberry
J.P.Mayberry at bristol.ac.uk
Thu Mar 18 06:49:31 EST 1999
Joe Shipman has asked about what is known about upper and lower
bounds on the Lowenheim number for second order logic. The same
questions can be raised about the *set-Lowenheim number*, which is the
smallest cardinal k such that for every set, S, of formulas of second
order logic, if S has a model, then S has a model of cardinality at
most k. There is a clear and illuminating discussion these and related
matters in section 6.4 of Stuart Shapiro's book (Foundations without
Foundationalism, p. 147ff).
In fact, this question is relevant to the general discussion we
have been conducting on 2nd order logic and its relation to set theory.
The essential problem consists in the fact that, prima facie, the
definitions of these Lowenheim numbers are not absolute, even for
models of 2nd order ZF. How you approach this problem depends on how
you see the relationship between the universe of sets - Cantor's
Absolute - and 2nd order logic. Does the universe of sets itself
constitute a model of 2nd order ZF? If so, it gives us an example of a
structure for interpreting the language of 2nd order ZF which lies
outside the universe of sets. What other such structures are there? Are
there models of 2nd order ZF among them? What is the general theory of
such structures? These questions force us to ask ourselves what we mean
when we say the universe of sets is "absolute".
John Mayberry
School of Mathematics
University of Bristol
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John Mayberry
J.P.Mayberry at bristol.ac.uk
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