[FOM] An example of an axiomatizable second order theory that is complete but non-categorical?

Aatu Koskensilta aatu.koskensilta at xortec.fi
Mon May 15 23:56:34 EDT 2006

On May 16, 2006, at 3:19 AM, Marcin Mostowski wrote:

> As I understand the problem is the following:
> Give a second order theory T such that
> 1.	T is axiomatizable (so recursively enumerable);
> 2.	T is complete;
> 3.	T is not categorical.

This is not the question I was interested in, and as you note, the 
answer is rather obviously that there are no such theories. My question 
was rather:

Give a recursive set of sentences T such that

  1. for all A, either T |= A or T |= ~A
  2. T is not categorical

with the additional requirement that T be "nice" in some fashion, e.g. 
we don't need to bend backwards to produce it, and 1. and 2. don't 
depend on the existence of a proper class of Woodin cardinals.

I think the gist of Robert Solovay's idea is right, and there are no 
such T, but I'm not absolutely certain.

Aatu Koskensilta (aatu.koskensilta at xortec.fi)

"Wovon man nicht sprechen kann, darüber muss man schweigen"
  - Ludwig Wittgenstein, Tractatus Logico-Philosophicus

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