## Notre Dame Journal of Formal Logic

### Algebraicity and Implicit Definability in Set Theory

#### 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
Accepted: 30 December 2013
First available in Project Euclid: 20 April 2016

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

#### 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

#### References

• [1] Enayat, A., “Models of set theory with definable ordinals,” Archive for Mathematical Logic, vol. 44 (2005), pp. 363–85.
• [2] Fuchs, G., and J. D. Hamkins, “Degrees of rigidity for Souslin trees,” Journal of Symbolic Logic, vol. 74 (2009), pp. 423–54.
• [3] Fuchs, G., J. D. Hamkins, and J. Reitz, “Set-theoretic geology,” Annals of Pure and Applied Logic, vol. 166 (2015), pp. 464–501.
• [4] Hamkins, J. D., D. Linetsky, and J. Reitz, “Pointwise definable models of set theory,” Journal of Symbolic Logic, vol. 78 (2013), pp. 139–56.
• [5] Hamkins, J. D., and R. Yang, “Satisfaction is not absolute,” preprint, arXiv:1312.0670v1 [math.LO].
• [6] Jech, T., Set Theory, 3rd millennium ed., Springer, Berlin, 2003.
• [7] Krajewski, S., “Mutually inconsistent satisfaction classes,” Bulletin of the Polish Academy of Sciences, Series Science, Mathematics, Astronomy, Physics, vol. 22 (1974), pp. 983–87.
• [8] Leahy, C., “Pointwise algebraic models of set theory,” MathOverflow question, 2011, http://mathoverflow.net/questions/71537.
• [9] Myhill, J., and D. Scott, “Ordinal definability,” pp. 271–78 in Axiomatic Set Theory (Los Angeles, 1967), edited by D. Scott, vol. 13 of Proceedings of Symposia in Pure Mathematics, American Mathematical Society, Providence, 1971.