# Blog Archives

# Topic Archive: algebraicity

# 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).