# Special Talk - Saharon Shelah

Simplicity and universality

Fixing a complete first order theory T, countable for transparency, we had known quite well for which cardinals T has a saturated model. This depends on T of course - mainly of
whether it is stable/super-stable. But the older, precursor notion of having
a universal notion lead us to more complicated answer, quite partial so far, e.g
the strict order property and even SOP_4 lead to having "few cardinals"
(a case of GCH almost holds near the cardinal). Note  that eg GCH gives a complete
uninteresting answer and so is the situation eg in the Easton model.
It seems that necessarily the answer involves sufficient
conditions for non-existence of a universal model
(in ZFC) and consistency for additional existence.
We conjecture that simplicity of T is crucial in answering this.
We shall speak on  recent advances: a new criterion covers the simplest non-
simple theory, so called $T_feq$. We may also speak complementary results
on having a universal model for simple theories.

## Date:

Sun, 23/06/2019 - 16:00 to 18:00

## Location:

Manchester Building, Room 110