MHB The Set of Positive Integers as a Copy of the Natural Numbers ....

[Math Amateur]

I am reading Ethan D. Bloch's book: The Real Numbers and Real Analysis ...

I am currently focused on Chapter 1: Construction of the Real Numbers ...

I need help/clarification with an aspect of Theorem 1.3.7 ...

Theorem 1.3.7 and the start of the proof reads as follows:https://www.physicsforums.com/attachments/6994In the above proof we read the following:" ... ... By Part (a) of the Peano Postulates we know that $$p \ne 1$$. ... ... " Can someone please explain exactly how the Peano Postulate (a) implies that $$p \ne 1$$ ... ?
Help will be much appreciated ...

Peter
The above post mentions the Peano Postulates so I am providing Bloch's statement of these postulates for the natural numbers ... as follows:
https://www.physicsforums.com/attachments/6995Readers of the above question may well be helped by access to Bloch's definition of the integers as well as Bloch's theorem on the algebraic properties of the integers ... so I am providing both as follows:

https://www.physicsforums.com/attachments/6996

View attachment 6997

 

[Ackbach]

I would agree that this claim $(p\not=1)$ is quite confusing, if not downright wrong. By definition of Bloch's function $i$, we have $i(1)=[(1+1,1)]=:\hat{1}$. Because $1\in\mathbb{N}$, $i(1)\in i(\mathbb{N})$. So it could be that $y=i(1)$. But then $y=[(1+1,1)]$, with $p=1$. There's no contradiction here that I can see. While Peano's axiom part (a) does assure us that $s(n)\not=1 \; \forall \, n\in\mathbb{N},$ it's not clear that Bloch is claiming $p$ to be the successor of something. If $p$ was the successor of something - that is, if $p=s(n)$ for some $n\in\mathbb{N}$ - then I would agree $p\not=1$.

Your image of Bloch's theorem on the algebraic properties of the integers is too small to be legible, I'm afraid. It would help if you could enlarge that.

 

[HallsofIvy]

I am inclined to suspect that what was intended was "By part (a) of the Peano Postulates we know that $$p+1\ne 1$$" since that is what the "part (a) of the Peano Postulates" says.

 

[Math Amateur]

I am inclined to suspect that what was intended was "By part (a) of the Peano Postulates we know that $$p+1\ne 1$$" since that is what the "part (a) of the Peano Postulates" says.

Thanks to Ackbach ana HallsofIvy for the help and support ...

Agree it should be p+1 is not equal to 1 ...

Thanks again,

Peter