[FOM] An example of an axiomatizable second order theory that is complete but non-categorical?
Robert M. Solovay
solovay at Math.Berkeley.EDU
Tue May 16 22:26:43 EDT 2006
Some further thoughts on Aatu's question:
1) In my last posting on this topic I considered only theories
extending V= L + "my ordinals are a true" cardinal. I would now like to
consider more general signatures and theories. I first continue to assume
V=L. Because of the more general situation, it is necessary to have second
order variables for each n which range over n-ary relations.
The following results are easily proved by the methods of my prior
posting.
a) If A is an axiomitizable complete theory [in second order
logic] then all its models have the same cardinality.
b) If A is finitely axiomitizable and complete, then A is
categorical. {This uses ideas from the paper of Marek cited by Enayat.}
Even assuming V=L, I don't see how to prove every axiomitizable
complete second order theory is categorical.
Next I want to present an example of a model where there is a
finitely axiomitizable complete second order theory which *is not*
categorical.
Start with a countable transitve model, M, of ZFC + V=L. Add
aleph_1 generic reals a la Cohen to the model getting a model N. [If it
needs saying aleph_1 here is interpreted in the sense of the model M.]
Work inside N. The following theory is easily seen to be complete,
finitely axiomitizable, but not categorical.
The signature of A is that of ZFC.
A asserts: 1) ZF - power set {The two schemes are asserted inthe strong
second order sense as single second order formulas.}
2) There is a largest cardinal and it is aleph_omega.
3) A second order sentence saying that the order type of the
ordinals is really a cardinal.
4) A second order sentence saying the ordinals of the model are
well-ordered.
5) A first order sentence saying that the model is a Cohen
extension of its constructibles obtained by adding a single generic real.
--Bob Solovay
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