Invariants, Boolean algebras and ACA₀⁺

Abstract

The sentences asserting the existence of invariants for mathematical structures are usually third order ones. We develop a general approach to analyzing the strength of such statements in second order arithmetic in the spirit of reverse mathematics. We discuss a number of simple examples that are equivalent to ACA 0 _{0} . Our major results are that the existence of elementary equivalence invariants for Boolean algebras and isomorphism invariants for dense Boolean algebras are both of the same strength as ACA 0 + _{0}^{+} . This system corresponds to the assertion that X ( ω ) X^{(\omega )} (the arithmetic jump of X X ) exists for every set X X . These are essentially the first theorems known to be of this proof theoretic strength. The proof begins with an analogous result about these invariants on recursive (dense) Boolean algebras coding 0 ( ω ) 0^{(\omega )} .