**Computational Logic Seminar**

** Time 2:00 – 4:00 PM, September 11, Room 3309**

Can Baskent, IHPST, Université Paris 1 Panthéon-Sorbonne

Some Logical Approaches to Lakatos’s ‘Proofs and Refutations’

Abstract. Lakatos’s seminal work ‘Proofs and Refutations’ presents a rational analysis of theorem improvement. The case in question in Proofs and Refutations is a familar one – Euler’s Theorem for polyhedra. In my talk, I will present various approaches to Lakatos’s methodology. First, I will explicate the method of proofs and refutations. Then, I will present, first a dialethetic approach, second an interrogative approach, and time permitting, a computational approach. Finally, I will conclude with some remarks.

**Set Theory Seminar**

** Friday, September 14, 2012 10:00 am GC 6417**

Professor Joel David Hamkins (CUNY)

Title: The least weakly compact cardinal can be unfoldable, weakly

measurable and nearly theta-supercompact.

Abstract. Starting from suitable large cardinal hypothesis, I will explain

how to force the least weakly compact cardinal to be unfoldable, weakly

measurable and, indeed, nearly theta-supercompact. These results, proved in

joint work with Jason Schanker, Moti Gitik and Brent Cody, exhibit an

identity-crises phenomenon for weak compactness, similar to the phenomenon

identified by Magidor for the case of strongly compact cardinals.

**Logic Workshop**

** Friday, September 14, 2012 2:00 pm GC 6417**

Professor Russell Miller (Queens College & Graduate Center, CUNY)

Boolean subalgebras of the computable atomless Boolean algebra

Abstract. It is known that the spectrum of a Boolean algebra cannot contain a low4 degree unless it also contains the degree 0; it remains open whether the same holds for low5 degrees. We address the question differently, by considering Boolean subalgebras of the computable atomless Boolean algebra B. For such subalgebras A, we show that it is possible for the spectrum of the unary relation A on B to contain a low5 degree without containing 0. We also show how Rebecca Steiner improved this result to make the spectrum contain a low4 degree, but not 0. The proof uses techniques similar to those from recent joint work on linear orders in [1]. We will introduce the Boolean-algebra result by describing that work, which shows that there exists a unary relation on the computable dense linear order whose spectrum (as a relation) contains precisely the non-low Turing degrees. It remains open whether there is a linear order which has this same spectrum (as a structure).

[1] A. Frolov, V. Harizanov, I. Kalimullin, O. Kudinov, & R. Miller; Degree spectra of highn and non-lown degrees, Journal of Logic and Computation 22 (2012) 4, 755-777, see here.

[2] R. Miller, Low5 Boolean subalgebras and computable copies, Journal of Symbolic Logic 76 (2011) 3, 1061-1074, available here.