Is it possible to write a sentence (formula) that expresses successor function being injective?
And in general, is it possible to state injectivity or surjectivity of a function in first-order?
Is it possible to write a sentence (formula) that expresses successor function being injective?
And in general, is it possible to state injectivity or surjectivity of a function in first-order?
Suppose that $f$ is an unary function symbol, write:
$\text{Injectivity: }\forall x\forall y(f(x)=f(y)\rightarrow x=y)\\ \text{Surjectivity: }\forall y\exists x(f(x)=y)$
For functions with several variables we simply have to quantify several variables. Note that we do not care for the interpretation of the symbol, only that it is a function.