This Week in Logic at CUNY

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.

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