**Computational Logic Seminar**

**Tuesday, May 7, 2013 2:00 pm Graduate Center, rm. 3209**

Speaker: Yoram Moses Israel Institute of Technology – Technion

Title: Knowledge and the Passage of Time

This talk will discuss how knowledge, nested knowledge, and common knowledge are gained in the presence of clocks and time bound information. It will complement the previous talk, in providing the causal structure underlying knowledge gain, from which the causal structure underlying basic coordination follows.

The talk will be based on joint work with Ido Ben Zvi.

**Models of PA**

**Wednesday, May 8, 2013 5:00 pm GC 4214.03**

Speaker: Ermek Nurkhaidarov Penn State Mont Alto

Title: The automorphism group of a model of arithmetic: recognizing standard system

Let M be countable recursively saturated model of Peano Arithmetic. In the talk I will discuss ongoing research on recognizing standard system of M in the automorphism group of M.

**Set theory seminar**

**Friday, May 10, 2013 8:00 am GC 5383**

Speaker: Joel David Hamkins The City University of New York

Title: Algebraicity and implicit definability in set theory

An element

*a*is

*definable*in a model M if it is the unique object in M satisfying some first-order property. It is

*algebraic*, in contrast, if it is amongst at most finitely many objects satisfying some first-order property φ, that is, if { b | M satisfies φ[b] } is a finite set containing

*a*. In this talk, I aim to consider the situation that arises when one replaces the use of definability in several parts of set theory with the weaker concept of algebraicity. For example, in place of the class HOD of all hereditarily ordinal-definable sets, I should like to consider the class HOA of all hereditarily ordinal algebraic sets. How do these two classes relate? In place of the study of pointwise definable models of set theory, I should like to consider the pointwise algebraic models of set theory. Are these the same? In place of the constructible universe L, I should like to consider the inner model arising from iterating the algebraic (or implicit) power set operation rather than the definable power set operation. The result is a highly interest new inner model of ZFC, denoted Imp, whose properties are only now coming to light. Is Imp the same as L? Is it absolute? I shall answer all these questions at the talk, but many others remain open.

This is joint work with Cole Leahy (MIT).

**Model theory seminar**

**Friday, May 10, 2013 12:30 pm GC 6417**

Speaker: Athar Abdul-Quader CUNY Grad Center

Title: Transplendent models of rich theories

Following up on a talk by Roman Kossak earlier this semester, I will discuss work by Engstrom and Kaye which address the question of existence of transplendent models (models with expansions omitting a type). If there is time, I will talk about transplendent models of PA.

**CUNY Logic Workshop**

**Friday, May 10, 2013 2:00 pm GC 6417**

Speaker: Thomas Johnstone The New York City College of Technology (CityTech), CUNY

Title: What is the theory ZFC without power set?

The theory ZFC-, consisting of the usual axioms of ZFC but with the power set axiom removed — specifically axiomatized by extensionality, foundation, pairing, union, infinity, separation, replacement and the assertion that every set can be well-ordered — is weaker than commonly supposed and is inadequate to establish several basic facts often desired in its context.

For example, there are models of ZFC- in which a countable union of countable sets is not countable. There are models of ZFC- for which the Los ultrapower theorem fails, even for wellfounded ultrapowers on a measurable cardinal. Moreover, the theory ZFC- is not sufficient to establish that the union of Σ_{n} and Π_{n} sets is closed under bounded quantification. Lastly, there are models of ZFC- for which the Gaifman theorem fails, in that there exists cofinal embeddings *j:M–>N* between ZFC- models that are Σ_{1}-elementary, but not fully elementary.

Nevertheless, these deficits of ZFC- are completely repaired by strengthening it to the theory obtained by using collection rather than replacement in the axiomatization above. This is joint work with Joel David Hamkins and Victoria Gitman, and it extends prior work of Andrzej Zarach.

arxiv preprint | post at jdh.hamkins.org | post on Victoria Gitman’s blog

**NY Philosophical Logic Group
Time: 4-6pm, Monday, May 13th. (last meeting of the semester)
**

Topic: The topology of gunk

Abstract: Space as we typically conceive of it in mathematics and physics is co posed of dimensionless points. Over the years, however, some have denied that points, or point-sized parts are genuine parts of space. Space, on an alternative view, is ‘gunky’: every part of space has a strictly smaller subpart. If this thesis is true, how should we model space mathematically? The traditional answer to this question is most famously associated with A.N. Whitehead, who developed a mathematics of pointless geometry that Tarski later modeled in regular open algebras. More recently, however, Whiteheadian space has come under attack, because it does not allow us to talk about the size or measure of regions in a nice way. A newer approach to the mathematics of gunk, advanced by F. Arntzenius, J. Hawthorne, and J.S. Russell, models space via the Lebesgue measure algebra, or algebra of (Lebesgue) measurable subsets of Euclidean space modulo sets of measure zero. But problems arise on this approach when it comes to doing topology. According to Arntzenius, the standard topological distinction between ‘open’ and ‘closed’ regions “is exactly the kind of distinction that we do not believe exists if reality is pointless.” I argue that the turn to non-standard topology in the measure-theoretic setting rests on a mistake. Once this is pointed out, the newer approach to gunk can claim two important advantages: it allows the gunk lover to talk about size and topology—both in perfectly standard ways.