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 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:    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)