Jump to content

Talk:Lévy hierarchy

Page contents not supported in other languages.
From Wikipedia, the free encyclopedia

What about bounded quantifiers?

[edit]

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

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)[reply]

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

[edit]

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

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)[reply]

Levy's bounding of -definable operations for ordinals

[edit]

Today I saw a result about operations on ordinals that are -definable, but I can't decide if it should go on this page.

If (for some ) is -definable, then for all , . 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 is -definable. If so then I think , i.e. should be -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)[reply]

Reference request

[edit]

Levy's original book "A hierarchy of formulas in set theory" doesn't contain a definition of . Does anyone have a reference where is defined? If not I guess Levy used "" in place of "" C7XWiki (talk) 21:52, 7 March 2022 (UTC)[reply]

Update: In "Set Theory: An Introduction to Large Cardinals" by Drake, it does indeed appear as , but it is called . C7XWiki (talk) 02:41, 2 July 2022 (UTC)[reply]