I'm reading "classic set theory for guided independent study", and i'm studying the construction of natural numbers using peano's axioms.
The three axioms they give me are: A Peano system is a set $X$ with a special element $0∈X$ and a funtion $S:X→X$ such that the following also hold:
- The function S is one-one
- For all $x∈X$, $0≠S(x)$
- For all subset $A⊆X$, if A contains $0$ and contains $S(x)$ whenever $x∈A$, then $A$ is all of $X$.
I can't come up with any set and function that doesn't satisfy them, you guys have any idea?
Satisfying 1 and 2: $X=\Bbb N_0$, $S(x)=x+2$
Satisfying 1 and 3: $X=\{0,1\}$, $S(x)=1-x$
Satisfying 2 and 3: $X=\{0,1\}$, $S(x)=1$