# Blog Archives

# Topic Archive: HOD

# Singular in V, regular and non-measurable in HOD

Getting a model where $\kappa$ is singular in $V$ but measurable in ${\rm HOD}$ is somewhat straightforward however ensuring that $\kappa$ is regular but not measurable in ${\rm HOD}$ is a surprisingly more difficult problem. Magidor navigated around the issues and I will present his result starting with one measurable. His technique can be extended for set many cardinals.

# Does the HOD dichotomy preclude a model of a proper class of supercompact cardinals which are not HOD-supercompact?

I will be presenting my version of the notes of Woodin’s talk at the Appalachian Set Theory seminar on his paper, the HOD Dichotomy. In particular I will be discussing Woodin’s version and definition of a model of set theory being “far from HOD”. I will also be discussing how this relates to some possible future results (see title) as well as a recent result of Cheng,Friedman and Hamkins, which seem on their face to contradict Woodin’s premise.

# 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.

# 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.

# Covering, core model induction, and hod mice (Part II)

We will discuss recent covering principles and show how they can be used to derive strength from failure of square.

# Covering, core model induction and hod mice

We will discuss recent covering principles and show how they can be used to derive strength from failure of square.

# Large cardinals need not be large in HOD

I will demonstrate that a large cardinal need not exhibit its large cardinal nature in HOD. I shall begin with the example of a measurable cardinal that is not measurable in HOD, and then afterward describe how to force more extreme examples, such as a model with a supercompact cardinal, which is not weakly compact in HOD. This is very recent joint work with Cheng Yong.

# 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).

# Extending a property of HOD-supercompactness. Or Not.

I will discuss a result of Sargsyan and his method of proof in order to show why a theorem from my dissertation was incorrect, and some of the interesting results we discovered in an effort to save the theorem.