Algebraicity and implicit definability in set theory

TL;DR

This paper explores the use of algebraicity as a weaker alternative to definability in set theory, revealing that algebraic and definable classes coincide in certain contexts and introducing the algebraic analogue of the constructible universe.

Contribution

It introduces the algebraic universe Imp, analyzes its properties, and compares it with the classical constructible universe L, highlighting new insights and open questions.

Findings

Hereditarily ordinal algebraic sets equal hereditarily ordinal definable sets (HOA=HOD)

Every pointwise algebraic model of ZF is pointwise definable

Imp can differ from L, with unique properties emerging

Abstract

We analyze the effect of replacing several natural uses of definability in set theory by the weaker model-theoretic notion of algebraicity. We find, for example, that the class of hereditarily ordinal algebraic sets is the same as the class of hereditarily ordinal definable sets, that is, HOA = HOD. Moreover, we show that every (pointwise) algebraic model of ZF is actually pointwise definable. Finally, we consider the implicitly constructible universe Imp---an algebraic analogue of the constructible universe---which is obtained by iteratively adding not only the sets that are definable over what has been built so far, but also those that are algebraic (or equivalently, implicitly definable) over the existing structure. While we know Imp can differ from L, the subtler properties of this new inner model are just now coming to light. Many questions remain open.