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I've found many diferent definitions of isomorphisms depending on the theory you are working on, sadly my book doesn't give an expicit definition. I'm trying to prove that given two Peano's systems there exists an unique isomorphism that moves first element to first element of the corrrespondant Peano's system.

What's the definition I should work with ?

José Osorio
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  • What is your definition of a Peano system? – Eric Wofsey Jun 07 '16 at 06:28
  • What context is this in? My very naive guess would be isomorphism as defined in model theory. –  Jun 07 '16 at 06:28
  • Bijective homomorphism – wesssg Jun 07 '16 at 07:01
  • Peano's system is an algebraic structure (P,Sc,1)in which:
    1. $Sc(x)\neq 1 \forall x\in P$
    2. $Sc$ is injective
    3. If $A\subseteq P$, $1\in A$, $Sc(A)\subseteq A \Rightarrow A=P$
     – José Osorio Jun 07 '16 at 08:37
  • The theorem I'm trying to prove says this: If $(P, Sc,1)$ and $(P', S'c,1')$ Ate two diferent Peano's systems, $\exists !$ isomorphism $\varphi : P \rightarrow P'$ with $\varphi (1) = 1'$ & $S'c \circ \varphi = \varphi \circ Sc$. And my question is what is the bijective homomorphism suposed to preserve? – José Osorio Jun 07 '16 at 08:52
  • just like you have just written, it preserves succesor ($S'c \circ \varphi = \varphi \circ Sc$) and it preserves the first element ($\varphi(1) = 1'$) – mercio Jun 07 '16 at 17:15

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