This Week in Logic at CUNY:

——– Friday, Apr 15, 2011 ——–

Set Theory Seminar

Friday, April 15, 2011 10:00 am GC 6417

Professor Joel David Hamkins (The City University of New York)

What is the theory ZFC without Powerset?

Abstract. The theory ZFC- is usually described as having all the

axioms of ZFC except the Powerset axiom. If axiomatized by

Extensionality, Foundation, Pairing, Union, Infinity, Separation,

Replacement and the Axiom of Choice, however, then it is weaker than

commonly supposed, and suffices to prove neither that a countable

union of countable sets is countable, nor that omega_1 is regular, nor

that the Los theorem holds for ultrapowers, even for well-founded

ultrapowers on a measurable cardinal, nor that the Gaifman theorem

holds, that is, that every Sigma_1-elementary cofinal embedding

between models of the theory is fully elementary, nor that Sigma_n

sets are closed under bounded quantification. Nevertheless, these

deficits of ZFC- are completely repaired by strengthening it to the

theory obtained by using Collection rather than Replacement in the

axiomatization above. These results, proved in joint work with

Victoria Gitman and Thomas Johnstone, extend prior work of Zarach.

Model Theory Seminar

Friday, April 15, 2011 12:30 pm GC 6417

Mr. D. Dakota Blair (Ph.D. Program in Mathematics, Graduate Center of CUNY)

More interpretations

Logic Workshop

Friday, April 15, 2011 2:00 pm GC 6417

Professor Ali Enayat (American University)

Full Satisfaction Classes

Abstract. (joint work with Albert Visser)

A full satisfaction class on a model M (of say arithmetic or set

theory) is a binary relation S on M that `decides’ every first order

formula of M – including the nonstandard ones – while obeying the

usual Tarski conditions for a satisfaction predicate. By a fundamental

theorem of Kotlarski-Krajewski-Lachlan (1981), every countable

recursively saturated model of PA has a full satisfaction class. This

shows, in particular, that the theory PA plus “S is a full

satisfaction class” is conservative over PA.

The Kotlarski-Krajewski-Lachlan theorem was established using an

elaborate and rather mysterious infinitary logic known as `M-logic’.

In this talk we present a new versatile and perspicuous

model-theoretic technique for establishing the

Kotlarski-Krajewski-Lachlan theorem for theories that are even far

weaker than PA (more specifically, for sequential theories).

This new technique can be used to show that T plus “S is a full

satisfaction class” can be even interpreted in T for “reflexive”

theories such as T = PA and T = ZF. Moreover, by implementing the

construction in WKL_0, we also show that the conservativity of T plus

“S is a full satisfaction class” over a sequential theory T can be

verified within Primitive Recursive Arithmetic, thus obtaining a

model-theoretic proof of a result established by Halbach (1999) using

cut-elimination.