In this article we are going to explore Lévy hierarchy, a very relevant topic today that has aroused great interest in different sectors. Lévy hierarchy is a concept that has been the subject of debate and analysis in recent years, and its impact on society has been significant. Since its emergence, Lévy hierarchy has generated conflicting opinions and has been a source of reflection for experts and scholars in the field. Throughout this article, we will examine in depth the different aspects related to Lévy hierarchy, from its origin to its evolution over time, also addressing its implications and its influence in the corresponding field.
The first level of the Lévy hierarchy is defined as containing only formulas with no unbounded quantifiers and is denoted by .[1] The next levels are given by finding a formula in prenex normal form which is provably equivalent over ZFC, and counting the number of changes of quantifiers:[2]p. 184
If a formula has both a form and a form, it is called .
As a formula might have several different equivalent formulas in prenex normal form, it might belong to several different levels of the hierarchy. In this case, the lowest possible level is the level of the formula.[citation needed]
Lévy's original notation was (resp. ) due to the provable logical equivalence,[4] strictly speaking the above levels should be referred to as (resp. ) to specify the theory in which the equivalence is carried out, however it is usually clear from context.[5]pp. 441–442 Pohlers has defined in particular semantically, in which a formula is " in a structure ".[6]
The Lévy hierarchy is sometimes defined for other theories S. In this case and by themselves refer only to formulas that start with a sequence of quantifiers with at most i−1 alternations,[citation needed] and and refer to formulas equivalent to and formulas in the language of the theory S. So strictly speaking the levels and of the Lévy hierarchy for ZFC defined above should be denoted by and .
^F. R. Drake, Set Theory: An Introduction to Large Cardinals (p.83). Accessed 1 July 2022.
^F. R. Drake, Set Theory: An Introduction to Large Cardinals (p.127). Accessed 4 October 2024.
^ abcAzriel Lévy, "On the logical complexity of several axioms of set theory" (1971). Appearing in Axiomatic Set Theory: Proceedings of Symposia in Pure Mathematics, vol. 13 part 1, pp.219--230