Talk:Lévy hierarchy

What about bounded quantifiers?

"The first level of the Levy hierarchy is defined as containing only formulas with no quantifiers, and is denoted by $\vartriangle _{0}$"; but in Bounded quantifier#Bounded quantifiers in set theory I read: "A ZF formula which contains only bounded quantifiers is called $\Sigma _{0}$, $\Delta _{0}$, and $\Pi _{0}$." Thus I am puzzled. Boris Tsirelson (talk) 18:55, 16 August 2014 (UTC)

I believe the second definition is correct: the three terms are equivalent, and that the first level of the hierarchy shoulk be defined as "no unbounded quantifiers". The second definition agrees with these reliable sources: Jech, Thomas (2003). Set Theory. Springer Monographs in Mathematics (Third Millennium ed.). Berlin, New York: Springer-Verlag. p. 183. ISBN 978-3-540-44085-7. Zbl 1007.03002. Buss, S.R., ed. (1998). Handbook of Proof Theory. Studies in Logic and the Foundations of Mathematics. Elsevier. p. 215. ISBN 0080533183. Deltahedron (talk) 19:14, 16 August 2014 (UTC)

On the Examples. (Old topic: the Continuum Hypothesis and its negation?)

"Σ₂-formulas: the Continuum Hypothesis (and its negation)" — really, both? That is, both are Δ₂? Boris Tsirelson (talk) 18:46, 17 August 2014 (UTC)

I find it somewhat puzzling that Continuum Hypothesis is listed as Σ₂-formula whereas General Continuum Hypothesis is listed as Π₂-formula. Should CH not be Δ₂-formula, then, or am I missing something? Lapasotka (talk) 16:39, 30 November 2023 (UTC)

Levy's bounding of $\Sigma _{1}$-definable operations for ordinals

Today I saw a result about operations on ordinals that are $\Sigma _{1}$-definable, but I can't decide if it should go on this page.

If $f:{\textrm {Ord}}^{n}\rightarrow {\textrm {Ord}}$ (for some $n\in \omega$) is $\Sigma _{1}$-definable, then for all $(\alpha _{0},\ldots ,\alpha _{n})\in {\textrm {Ord}}^{n}$, $f(\alpha _{0},\ldots ,\alpha _{n})<(\alpha _{0}^{+})^{L}+(\alpha _{1}^{+})^{L}+\ldots +(\alpha _{n}^{+})^{L}$. Source: MarekSrebrny73

Should this go on this page? C7XWiki (talk) 16:44, 19 October 2021 (UTC) : Appearing to contradict this, Rathjen said otherwise here, saying $x\mapsto x^{+}$ is $\Delta _{0}$-definable. If so then I think $\alpha \mapsto \{y\in {\textrm {minimal}}\,{\textrm {admissible}}\,{\textrm {set}}\,{\textrm {containing}}\,\alpha \,\mid \,y\,{\textrm {is}}\,{\textrm {an}}\,{\textrm {ordinal}}\}$, i.e. $\alpha \mapsto \alpha ^{+}$ should be $\Delta _{0}$-definable too, violating the bound. C7XWiki (talk) 00:28, 8 March 2022 (UTC). The message in strikethrough is incorrect, ⁺ means cardinal successor in Marek and Srebrny's paper. C7XWiki (talk) 05:02, 25 April 2022 (UTC)

Reference request

Levy's original book "A hierarchy of formulas in set theory" doesn't contain a definition of $\Delta _{n}$. Does anyone have a reference where $\Delta _{n}$ is defined? If not I guess Levy used "$\Sigma _{n}^{S}\cap \Pi _{n}^{S}$" in place of "$\Delta _{n}$" C7XWiki (talk) 21:52, 7 March 2022 (UTC)

Update: In "Set Theory: An Introduction to Large Cardinals" by Drake, it does indeed appear as $\Sigma _{n}^{S}\cap \Pi _{n}^{S}$, but it is called $\Delta _{n}^{S}$. C7XWiki (talk) 02:41, 2 July 2022 (UTC)