Schedule

Following up on Peter Koepke's CUNY Logic Workshop lecture of March 22, 2013, I will discuss Namba-like forcings which either exist or can be forced to exist at successors of singular cardinals.

We present some recent results on hyper-stationary subsets of regular cardinals. In particular, we shall look at (1) the connections between hyper-stationarity and second-order indescribability in the constructible universe, (2) the ideals associated to non-hyper-stationary sets, and (3) the consistency strength of hyper-stationarity.

We present an analogy between cardinal invariants of the continuum from set theory and highness properties of Turing degrees from computability theory. In particular, we develop a version of Cichon's diagram for highness properties and investigate to what extent results from set theory can be taken over to the computability context. This is joint work with Andrew Brooke-Taylor, Selwyn Ng, and Andre Nies.

The classical theorem of Silver states that GCH cannot break for the first time over a singular cardinal of uncountable cofinality. We would like to deal with  the situation once  blowing up power of singular cardinals is replaced by collapses of their successors.

I shall give a general introduction to the theory of infinite games, using infinite
chess---chess played on an infinite edgeless chessboard---as a central example. Since chess, when won, is won at a finite stage of play, infinite chess is an example of what is known technically as an open game, and such games admit the theory of transfinite ordinal game values. I shall exhibit several interesting positions in infinite chess with very high transfinite game values. The precise value of the omega one of chess is an open mathematical question.

It is known that the assumption that GCH first fails at aleph omega implies in ZFC the existence of inner models with large cardinals. Gitik and Koepke demonstrated that this is not so without the axiom of choice. Namely there is a cardinal-preserving symmetric-generic extension of L, in which GCH holds at every cardinal aleph n but there is a surjection from the power set of aleph omega onto any previously chosen cardinal in L, as large as one wants, and the axiom of choice by necessity fails. In other words, in such an extension GCH holds in the proper sense for all cardinals aleph n but fails at aleph omega in Hartogs' sense. The goal of this talk is to analyse the system of automorphisms involved in the Gitik-Koepke construction.

This talk will describe successful applications of descriptive set theory to symbolic dynamics, and list related open problems which might be similarly susceptible.

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In the talk I will sketch a notion of forcing that establishes that it is consistent relative to ZFC that there a simple P-aleph-1-point and a simple P-aleph-2-point.
The main technical tool are iterands that destroy ultrafilters in one direction and preserve all P-points outside this direction. We will also clarify whether destruction means diagonalisation.

We discuss whether Martin's Maximum is consistent with or even implies Woodin's axiom (*). This is joint work with D. Aspero and W. H. Woodin

The talk will address the question to what extent text linguistic features can be identified in mathematical proof texts and what they could tell us about the communicative content and the cognitive representation of proofs.