The Integers as an Ordered Integral Domain .... Bloch Theorem 1.4.6 ....

[Math Amateur]

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

I am currently focused on Section 1.4: Entry 2: Axioms for the Integers ... In this section Bloch defines the integers as an ordered integral domain that satisfies the Well Ordering Principle ... rather than defining the integers via the natural numbers ...

I need help/clarification with an aspect of Theorem 1.4.6 ...

Theorem 1.4.6 and the start of the proof reads as follows:View attachment 7030

In the above proof ... near the start of the proof, we read the following:

" ... ... From the definition of \(\displaystyle \mathbb{N}\), we observe that \(\displaystyle S \subseteq \mathbb{N}\). ... ..."Question: What exactly is the reasoning that allows us to conclude that \(\displaystyle S \subseteq \mathbb{N}\) from the definition of \(\displaystyle \mathbb{N}\) ... "
The above theorem is in the section where Bloch defines the integers as an ordered integral domain that satisfies the Well ordering Principle... ... as follows:View attachment 7031The definition of the natural numbers is mentioned above ... Bloch's definition is as follows ...

View attachment 7032Hope someone can help,

Peter

 

[Math Amateur]

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

I am currently focused on Section 1.4: Entry 2: Axioms for the Integers ... In this section Bloch defines the integers as an ordered integral domain that satisfies the Well Ordering Principle ... rather than defining the integers via the natural numbers ...

I need help/clarification with an aspect of Theorem 1.4.6 ...

Theorem 1.4.6 and the start of the proof reads as follows:

In the above proof ... near the start of the proof, we read the following:

" ... ... From the definition of \(\displaystyle \mathbb{N}\), we observe that \(\displaystyle S \subseteq \mathbb{N}\). ... ..."Question: What exactly is the reasoning that allows us to conclude that \(\displaystyle S \subseteq \mathbb{N}\) from the definition of \(\displaystyle \mathbb{N}\) ... "
The above theorem is in the section where Bloch defines the integers as an ordered integral domain that satisfies the Well ordering Principle... ... as follows:The definition of the natural numbers is mentioned above ... Bloch's definition is as follows ...

Hope someone can help,

Peter

Maybe I shouldn't answer my own questions ... but maybe I am worrying too much over a trivial point ... maybe the reasoning is simply ... as follows ... since \(\displaystyle S\) is a set made up of positive integers then it is a subset of \(\displaystyle \mathbb{N}\) ... is it as simple as that ..?if it is as simple as that then I apologise for the post ...Peter

 

[Evgeny.Makarov]

What exactly is the reasoning that allows us to conclude that \(\displaystyle S \subseteq \mathbb{N}\) from the definition of \(\displaystyle \mathbb{N}\)

I don't think the claim $S\subseteq\mathbb{N}$ is necessary for the rest of the proof. We have $\emptyset\ne S\subseteq\{x\in\mathbb{Z}\mid x>0\}$, so $S$ has a least element by the well-ordering principle for $\mathbb{Z}$. Unless something is said about the identification of positive integers and natural numbers, as they are defined in the book, the fact that $S\subseteq\mathbb{N}$ does seem not immediately obvious.

Maybe I shouldn't answer my own questions

It's perfecty fine.

 

[Math Amateur]

I don't think the claim $S\subseteq\mathbb{N}$ is necessary for the rest of the proof. We have $\emptyset\ne S\subseteq\{x\in\mathbb{Z}\mid x>0\}$, so $S$ has a least element by the well-ordering principle for $\mathbb{Z}$. Unless something is said about the identification of positive integers and natural numbers, as they are defined in the book, the fact that $S\subseteq\mathbb{N}$ does seem not immediately obvious.

It's perfecty fine.

Thanks Evgeny ...

Just to explain my confusion ...

... indeed my problem was how to derive \(\displaystyle S \subseteq \mathbb{N}\) ... but got confused (with Bloch's help ... :(

...)

To explain ...

Bloch investigates two approaches to defining/constructing the integers as he describes here ... https://www.physicsforums.com/attachments/7034In Section 1.4, where Theorem 1.4.6 occurs, Bloch is expounding the ordered integral domain approach to the integers ... so we should not go back to the Peano Postulates as I did - that is Bloch's approach number 1 ... under the second approach, the ordered integral domain approach, the Peano Postulates/Axioms become a theorem and are proved ...

Actually when we meet \(\displaystyle \mathbb{N}\) in the proof of Theorem 1.4.6 Bloch has not defined \(\displaystyle \mathbb{N}\) yet in this approach ... he does so after presenting Theorem 1.4.6 as follows:View attachment 7035
By the way, Evgeny, thanks for pointing out that the claim $S\subseteq\mathbb{N}$ is not necessary for the rest of the proof ... great point!

Thanks again,

Peter

 

[Evgeny.Makarov]

If natural numbers are defined in this way in the second approach, then it is clear that $S=\{z\in\mathbb{Z}\mid 0<z<1\}\subseteq\{x\in\mathbb{Z}\mid x>0\}=\mathbb{N}$. Maybe the author wrote $S\subseteq\mathbb{N}$ in Theorem 1.4.6 in anticipation of this definition, but yes, this is strange.