FOM: Lowenheim numbers

[John Mayberry]

	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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