# Equivalent mathematical objects

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.

• You should look up "isomorphism". – Crostul Mar 12 '17 at 22:48
• @Crostul I know what "isomorphism" is. But isomorphism of logical systems requires a set algebraic structure (which in my case may be missing or unknown) or more generally some category structure (which may also be missing or unknown) – porton Mar 12 '17 at 22:50
• How about "equivalence relation"? There are many of these. – hardmath Mar 12 '17 at 22:58
• @hardmath Haven't you noticed that I am about a particular special case of equivalence relations? – porton Mar 12 '17 at 23:00
• I get the idea from your examples that you are concerned with a relation of equal cardinality, but your use of "mathematical objects" made me suspect you were interested in something more general. – hardmath Mar 12 '17 at 23:04

We have $f(P)=P$ because $(f(P))x\Leftrightarrow P(x)$. Thus $f$ is the identify.