Editing Analytical hierarchy (section)

== Properties ==
For each <math>n</math> we have the following strict containments:

:<math>\Pi^1_n \subset \Sigma^1_{n+1}</math>,
:<math>\Pi^1_n \subset \Pi^1_{n+1}</math>,
:<math>\Sigma^1_n \subset \Pi^1_{n+1}</math>,
:<math>\Sigma^1_n \subset \Sigma^1_{n+1}</math>.

A set that is in <math>\Sigma^1_n</math> for some ''n'' is said to be '''analytical'''. Care is required to distinguish this usage from the term [[analytic set]], which has a different meaning, namely <math>\boldsymbol\Sigma_1^1</math>.<ref>[[Thomas Jech|T. Jech]], "[https://projecteuclid.org/journalArticle/Download?urlid=bams%2F1183548432 The Brave New World of Determinacy]" (PDF download). Book review, [[Bulletin of the American Mathematical Society]], vol. 5, number 3, November 1981 (pp.339--349).</ref>