Invariants, Boolean algebras and ACA$_{0}^{+}$

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$_{0}$. 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$_{0}^{+}$. This system corresponds to the assertion that $X^{(\omega)}$(the arithmetic jump of $X$) exists for every set $X$. 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 $0^{(\omega)}$.