## Notre Dame Journal of Formal Logic

- Notre Dame J. Formal Logic
- Volume 57, Number 3 (2016), 431-439.

### Algebraicity and Implicit Definability in Set Theory

Joel David Hamkins and Cole Leahy

#### 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, $\mathrm{HOA}=\mathrm{HOD}$. Moreover, we show that every (pointwise) algebraic model of $\mathrm{ZF}$ 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 that $\mathrm{Imp}$ can differ from $L$, the subtler properties of this new inner model are just now coming to light. Many questions remain open.

#### Article information

**Source**

Notre Dame J. Formal Logic, Volume 57, Number 3 (2016), 431-439.

**Dates**

Received: 27 February 2012

Accepted: 30 December 2013

First available in Project Euclid: 20 April 2016

**Permanent link to this document**

https://projecteuclid.org/euclid.ndjfl/1461157794

**Digital Object Identifier**

doi:10.1215/00294527-3542326

**Mathematical Reviews number (MathSciNet)**

MR3521491

**Zentralblatt MATH identifier**

06621300

**Subjects**

Primary: 03E47: Other notions of set-theoretic definability

Secondary: 03C55: Set-theoretic model theory

**Keywords**

set-theoretic definability models of set theory

#### Citation

Hamkins, Joel David; Leahy, Cole. Algebraicity and Implicit Definability in Set Theory. Notre Dame J. Formal Logic 57 (2016), no. 3, 431--439. doi:10.1215/00294527-3542326. https://projecteuclid.org/euclid.ndjfl/1461157794