Welcome to New York Logic!

Mar 1 Sunday | Mar 2 Monday | Mar 3 Tuesday | Mar 4 Wednesday | Mar 5 Thursday | Mar 6 Friday | Mar 7 Saturday |
---|---|---|---|---|---|---|

**→** go to the Calendar

## Upcoming talks and events:

# Syntactic Epistemic Logic and Games III

We discuss the Muddy Children puzzle MC, identify and fix the gap in its solution: a completeness analysis is required if the problem is specified syntactically, but analyzed by reasoning on a specific model. We show that the syntactic description of MC is complete w.r.t. its model used in the standard solution. We will present two proofs of completeness: syntactic, by induction on a formulas, and semantic, which makes use of bounded morphisms, a special case of bisimulations. We then present a modification of MC, MClite, and show that it is not deductively complete, has no semantic characterization and cannot be handled by traditional semantic methods. We give a syntactic solution of MClite.

Time permitting, we will venture further into extensive games and their epistemic models

# Definability in linear fragments of Peano arithmetic III

In this series of talks, I will provide an analysis of definable sets in models of linear arithmetics. Here, for a cardinal k, the k-linear arithmetic (LA_k) is a full-induction arithmetical theory extending Presburger arithmetic by k non-standard scalars (= unary functions of multiplication by distinguished elements). Note that each model of a linear arithmetic naturally corresponds to a discretely ordered module over the ordered ring generated by the scalars.

I will prove a quantifier elimination result (QE up to disjunctions of bounded pp-formulas) for LA_1 and give a complete characterisation of definable sets in its models. On the other hand, I will construct an example of a model of LA_2 (or any LA_k with k at least 2) where multiplication is definable on a non-standard initial segment (and thus no pp-elimination is possible). Finally, I will show that all the theories LA_k satisfy quantifier elimination (at least) up to bounded formulas.

I will mention several applications and corollaries of these results including a construction of a non-NIP ordered module, or a result on automorphisms in saturated models of PA.

# Embeddings of the universe into the constructible universe, current state of knowledge

I shall describe the current state of knowledge concerning the question of whether there can be an embedding of the set-theoretic universe into the constructible universe. The main question is: can there be an embedding $j:Vto L$ of the set-theoretic universe $V$ into the constructible universe $L$, when $Vneq L$? The notion of *embedding* here is merely that $xin y$ if and only if $j(x)in j(y)$, and such a map need not be elementary nor even $Delta_0$-elementary. It is not difficult to see that there can generally be no $Delta_0$-elementary embedding $j:Vto L$, when $Vneq L$. Nevertheless, the question arises very naturally in the context of my previous work on the embeddability phenomenon, which shows that every countable model $M$ does admit an embedding $j:Mto L^M$ into its constructible universe. More generally, any two countable models of set theory are comparable; one of them embeds into the other. Indeed, one model $langle M,in^Mrangle$ embeds into another $langle N,in^Nrangle$ just in case the ordinals of the first $text{Ord}^M$ order-embed into the ordinals of the second $text{Ord}^N$. In these theorems, the embeddings $j:Mto L^M$ are defined completely externally to $M$, and so it was natural to wonder to what extent such an embedding might be accessible inside $M$. Currently, the question remains open, but we have some partial progress, settling it in a number of cases.

This is joint work of myself, W. Hugh Woodin, Menachem Magidor, with contributions also by David Aspero, Ralf Schindler and Yair Hayut. See more information at the links below:

Blog post for this talk | Related MathOverflow question | Article

# Lengths of roots of polynomials over *k((G))*

Mourgues and Ressayre showed that any real closed field $R$ can be mapped isomorphically onto a truncation-closed subfield of the Hahn field $k((G))$, where $G$ is the natural value group of $R$ and $k$ is the residue field. If we fix a section of the residue field and a well ordering < of $R$, then the procedure of Mourgues and Ressayre yields a canonical section of $G$ and a unique embedding $d: R$ → $k((G))$. Julia Knight and I believed we had shown that for a real closed field $R$ with a well ordering < of type ω, the series in $d(R)$ have length less than ω^{ωω}, but we found a mistake in our proof. We needed a better understanding of what happens to lengths under root-taking. In this talk, we give partial answer, which allows us to prove the original statement and generalize it. We make use of unpublished notes of Starchenko on the Newton-Puiseux method for taking roots of polynomials.

# The undecidability of lattice-ordered groups

In this talk I will discuss conjugacy in the automorphism group of the rationals as a linearly ordered set, and then show that the integers, together with addition and multiplication, can be interpreted in Aut(Q), and thus that the theory of Aut(Q) is undecidable.

# Title TBA

# Countable Arithmetically Saturated Models and the Small Index Property.

In 1994 Lascar proved that countable arithmetically saturated models of PA have the Small Index Property. In this talk we outline the proof and discuss related results and open problems.

# Generalized Baire spaces and closed Maximality Principles

Given an uncountable regular cardinal $\kappa$, the generalized Baire space of $\kappa$ is set ${}^\kappa\kappa$ of all functions from $\kappa$ to $\kappa$ equipped with the topology whose basic open sets consist of all extensions of partial functions of cardinality less than $\kappa$.

A subset of this space is $\mathbf{\Sigma}^1_1$ (i.e. a projection of a closed subset of ${}^\kappa\kappa\times{}^\kappa\kappa$) if and only it is definable over $\mathrm{H}(\kappa^+)$ by a $\Sigma_1$-formula with parameters. This shows that the class of $\mathbf{\Sigma}^1_1$-subsets contains a great variety of set-theoretically interesting objects. Moreover, it is known that many basic and interesting questions about sets in this class are not decided by the axioms of $\mathrm{ZFC}$ plus large cardinal axioms.

In my talk, I want to present examples of extensions of $\mathrm{ZFC}$ that settle many of these questions by providing a nice structure theory for the class of $\mathbf{\Sigma}^1_1$-subsets of ${}^\kappa\kappa$. These forcing axioms appear in the work of Fuchs, Leibman, Stavi and Väänänen. They are variations of the *maximality principle* introduced by Stavi and Väänänen and later rediscovered by Hamkins.

# TBA

# Harrington’s Principle and remarkable cardinals

It is open whether $\Pi^1_1$ determinacy implies the existence of $0^{\#}$ in 3rd order arithmetic, call it $Z_3$. We compute the large cardinal strength of $Z_3$ plus “there is a real $x$ such that every $x$-admissible is an $L$-cardinal.” This is joint work with Yong Cheng.

Friday, April 10, 2015

# Spring break

CUNY’s spring vacation is April 3-11, 2015. Therefore, no seminars will meet at the Graduate Center on April 3, nor on April 10.