The set of nonnegative whole numbers is in some sense equivalent to the set of natural numbers, the set of Dedekind cuts is equivalent to the set of infinite decimal fractions (excluding 999... thing), etc.
I try to formulate such equivalence exactly.
I will call two objects $A$ and $B$ “predicate-equivalent” if and only if there is a bijection $f$ mapping all predicates $P$ of one variable true for the argument $A$ into all predicates $Q$ of one variable true for the argument $B$, such that $P(x)$ is true if and only if $(f(P))(x)$ is true for every variable $x$ and predicate $P$ of one variable.
Does predicate equivalence describe what I want to describe (that is equivalencies like the above examples)? Particularly, are every two objects predicate-equivalent? (If yes, this makes my construct useless.)
If this way to describe equivalence of mathematical objects does not work, please help me to correct this, to describe what I want to describe.