This Week in Logic at CUNY

Computational Logic Seminar
October 16, Time 2:00 – 4:00 PM, Room 3309.

Speaker: Giorgi Japaridze, Villanova
Title: Give Caesar what belongs to Caesar

Abstract: In this talk I will discuss the possibility and advantages
of basing applied theories (e.g. Peano Arithmetic) on Computability
Logic instead of the more traditional alternatives, such as Classical
or Intuitionistic Logics.

 

Set Theory Seminar
Friday, October 19, 2012, 10:00am GC 6417
Speaker: Thomas Johnstone
Title: Definability of the ground model in forcing extensions of ZF-models, I

Abstract: Richard Laver [2007] showed that if M satisfies ZFC and G is
any M-generic filter for forcing P of size less than delta, then M is
definable in M[G] from parameter P(delta)^M. I will discuss a
generalization of this result for models M that satisfy ZF but only a
small fragment of the axiom of choice. This is joint work with
Victoria Gitman.

Definition (ZF). P*Q has closure point delta if P is well-orderable of
size at most delta and Q is be well-orderable here.)

Theorem: If M models ZF+DC_delta and P is forcing with closure point
delta, then M is definable in M[G] from parameter P(delta)^M.

Model Theory Seminar
Friday, October 19, 2012, 12:30pm-1:45pm, GC 6417
Koushik Pal (University of Maryland)
Unstable Theories with an Automorphism

Abstract: Kikyo and Shelah showed that if T is a first-order theory in
some language L with the strict-order property, then the theory
T_\sigma, which is the old theory T together with an L-automorphism
\sigma, does not have a model companion in L_\sigma, which is the old
language L together with a new unary predicate symbol \sigma. However,
it turns out that if we add more restrictions on the automorphism,
then T_\sigma can have a model companion in L_\sigma. I will show some
examples of this phenomenon in two different context – the linear
orders and the ordered abelian groups. In the context of the linear
orders, we even have a complete characterization of all model complete
theories extending T_\sigma in L_\sigma. This is a joint work with
Chris Laskowski.

Logic Workshop
Friday, October 19, 2012 2:00 pm GC 6417
Prof. Andrej Bauer (University of Ljubljana)
Synthetic computability

Synthetic computability is a formulation of computability theory in
the style of synthetic differential geometry and synthetic domain
theory: we first “synthesise” a world of mathematics tailored for
computability, namely the effective topos, and then we work in it.
Computability theory becomes just ordinary mathematics in an
extraordinary world. Thus the c.e. sets are just the computable sets,
the computable functions are just functions, the
Kreisel-Lacombe-Shoenfield theorem is the Brouwerian continuity
principle, etc. Many classical theorems in computability theory can be
formulated and proved elegantly from simple, but unusual axioms, such
as “there are countably many countable subsets of natural numbers.”

After an excursion into synthetic computability we build another
synthetic world, based on infinite-time Turing machines. This one is
even stranger on the inside, as in it the real numbers form a subset
of the natural numbers. Continuity principles are invalid in the new
world, but we can look for their substitutes in the higher levels of
the hierarchies of sets.

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