# Blog Archives

# Topic Archive: definability

# Studying profinite monoids via logic

This talk is about my ongoing joint research project with Benjamin Steinberg (CCNY). We begin with the observation that the free profinite aperiodic monoid over a finite set *A* is isomorphic to the Stone dual space (spectrum) of the Boolean algebra of first-order definable sets of finite *A*-labelled linear orders (“*A*-words”). This means that elements of this monoid can be viewed as elementary equivalence classes of models of the first-order theory of finite *A*-words. We exploit this view of the free profinite aperiodic monoid to prove both old and new things about it using methods from model theory, in particular (weakly) saturated models.

The talk is aimed at anyone with a basic knowledge of model theory, not necessarily of profinite monoids; in particular I will take care to review some background on profinite monoids and on how they relate to logic and regular languages.

# The definability of radicals in noncommutative rings

We will survey some results in the definability of radicals in rings, focusing on some recent results about noncommutative rings. In particular, we will show that there is no simple definition of the prime radical in the noncommutative setting, thus differentiating prime radicals in commutative/noncommutative rings.

# NERDS on April 2, 2016

The Spring 2016 New England Recursion and Definability Seminar (“NERDS”) will take place on Saturday, April 2, 2016 at Springfield College, in Springfield, MA. Further details and abstracts of talks will be posted on nylogic.org as they become available.

# NERDS October 17, 2015 at Assumption College

The 2015 autumn meeting of the New England Recursion & Definability Seminar will be held on the campus of Assumption College, in Worcester Massachusetts, on Saturday, October 17, from 10:00 until 4:15. Titles and abstracts are now posted at the link for NERDS, under the Seminars tab. Directions and visitor information are available here.

# When does every definable set have a definable member?

Although the concept of `being definable’ is not generally expressible in the language of set theory, it turns out that the models of ${rm ZF}$ in which every definable nonempty set has a definable element are precisely the models of $V={rm HOD}$. Indeed, $V={rm HOD}$ is equivalent to the assertion merely that every $Pi_2$-definable set has an ordinal-definable element. Meanwhile, this is not true in the case of $Sigma_2$-definability, because every model of ZFC has a forcing extension satisfying $Vneq{rm HOD}$ in which every $Sigma_2$-definable set has an ordinal-definable element. This is joint work with François G. Dorais and Emil Jeřábek, growing out of some questions on MathOverflow, namely,

- Definable collections without definable members
- A question asked by Ashutosh five years ago, in which François and I gradually came upon the answer together.
- Is it consistent that every definable set has a definable member?
- A similar question asked last week by (anonymous) user38200
- Can $Vneq{rm HOD}$ if every $Sigma_2$-definable set has an ordinal-definable member?
- A question I had regarding the limits of an issue in my answer to the previous question.

In this talk, I shall present the answers to all these questions and place the results in the context of classical results on definability, including a review of basic concepts for graduate students.

# The implicitly constructible universe

The implicitly constructible universe, **IMP**, defined by Hamkins and Leahy, is produced by iterating implicit definability through the ordinals. **IMP** is an inner model intermediate between **L** and **HOD**. We look at some consistency questions about the nature of **IMP**.

# Model theory and exponentiation

Methods from mathematical logic have proved surprisingly useful in understanding the geometry and topology of sets definable in the real field with exponentiation. When looking at the complex exponential field, the definability of the integers is a seemingly insurmountable impediment, but a novel approach due to Zilber leads to a large number of interesting new questions.

# NERDS

The Autumn 2014 meeting of NERDS, the New England Recursion & Definability Seminar, will take place on Saturday, October 18 at Assumption College, in Worcester, MA, beginning at 10:30 a.m. The principal organizers are Brooke Andersen, Damir Dzhafarov, and Reed Solomon. All talks are posted on nylogic.org (find NERDS under the “Conferences” tab), and they will take place in the Carriage House building on the campus of Assumption College. Directions.

# Definability versus Algebraicity

In a recent paper, Hamkins and Leahy introduce the concept of algebraicity in the set theoretic context. Thus, a set is algebraic in a model of set theory if it belongs to a finite set definable in the model. Clearly, algebraicity is a weak form of definability, and it can be varied in similar ways as definability, for example by allowing parameters. While the authors showed that the class of hereditarily ordinal algebraic sets is equal to the class of hereditarily ordinal definable sets, many fundamental questions on the relationship between algebraicity and definability were open: in particular, the question whether these concepts can be different in a model of set theory. I will show how to produce models of set theory in which there are algebraic sets that are not ordinal definable, and construct a model in which there is a set which is internally algebraic (i.e., which belongs to a definable set the model believes to be finite), but not externally.

# 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).

# Preservation of DC delta by forcing with a closure point at delta

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 <=delta strategically closed. (Q need not 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.

# Definability of the ground model in forcing extensions of ZF-models

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 <=delta strategically closed. (Q need not 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.