**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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