Editing Axiom of constructibility (section)

== In arithmetic ==
Especially from the 1950s to the 1970s, there have been some investigations into formulating an analogue of the axiom of constructibility for [[Second-order_arithmetic#Subsystems|subsystems of second-order arithmetic]]. A few results stand out in the study of such analogues:

* John Addison's <math>\Sigma_2^1</math> formula <math>\textrm{Constr}(X)</math> such that <math>\mathcal P(\omega)\vDash\textrm{Constr}(X)</math> iff <math>X\in\mathcal P(\omega)\cap L</math>, i.e. <math>X</math> is a constructible real.<ref>[[Victor W. Marek|W. Marek]], Observations Concerning Elementary Extensions of ω-models. II (1973, p.227). Accessed 2021 November 3.</ref><ref>W. Marek, [http://matwbn.icm.edu.pl/ksiazki/fm/fm98/fm9818.pdf ω-models of second-order arithmetic and admissible sets] (1975, p.105). Accessed 2021 November 3.</ref>

* There is a <math>\Pi_3^1</math> formula known as the "analytical form of the axiom of constructibility" that has some associations to the set-theoretic axiom V=L.<ref name="beta2models">W. Marek, [http://matwbn.icm.edu.pl/ksiazki/fm/fm82/fm82112.pdf Stable sets, a characterization of β₂-models of full second-order arithmetic and some related facts] (pp.176--177). Accessed 2021 November 3.</ref> For example, some cases where <math>M\vDash\textrm{V=L}</math> iff <math>M\cap\mathcal P(\omega)\vDash\textrm{Analytical}\;\textrm{form}\;\textrm{of}\;\textrm{V=L}</math> have been given.<ref name="beta2models" />