Integers as an Ordered Integral Domain ... Bloch Th. 1.4.6

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Main Question or Discussion Point

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:


?temp_hash=432383d02291abe058b7ea84c87f1574.png



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

" ... ... From the definition of ##\mathbb{N}##, we observe that ##S \subseteq \mathbb{N}##. ... ..."


Question: What exactly is the reasoning that allows us to conclude that ##S \subseteq \mathbb{N}## from the definition on ##\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:


?temp_hash=432383d02291abe058b7ea84c87f1574.png




The definition of the natural numbers is mentioned above ... Bloch's definition is as follows ...


?temp_hash=432383d02291abe058b7ea84c87f1574.png



Hope someone can help,

Peter
 

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  • #2
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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 ##S## is a set made up of positive integers then it is a subset of ##\mathbb{N}## ... is it as simple as that ..?

My apologies if it is that simple ...

Peter
 
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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 ##S## is a set made up of positive integers then it is a subset of ##\mathbb{N}## ... is it as simple as that ..?

My apologies if it is that simple ...

Peter
As I read it, yes. However, I do not see immediately from the Peano axiom how the natural numbers are all positive, but this depends on the definition of the order as well. So somewhere between 1.2.1. and 1.4.6. the author should have shown (or defined), that natural numbers are positive.
 
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As I read it, yes. However, I do not see immediately from the Peano axiom how the natural numbers are all positive, but this depends on the definition of the order as well. So somewhere between 1.2.1. and 1.4.6. the author should have shown (or defined), that natural numbers are positive.

Thanks fresh_42 ... indeed my problem was how to derive ##S \subseteq \mathbb{N}## ... but got confused (with Bloch's help ... :frown: ...)

To explain ...

Bloch investigates two approaches to defining/constructing the integers as he describes here ...

?temp_hash=1574f2bad2421b2b59fd08932580c001.png



In 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 his 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 ##\mathbb{N}## in the proof of Theorem 1.4.6 Bloch has not defined ##\mathbb{N}## yet in this approach ... he does so after presenting Theorem 1.4.6 as follows:

?temp_hash=1574f2bad2421b2b59fd08932580c001.png


Interestingly as Evgeny Makarov has pointed out in the Math Help Forum, the claim ##S \subseteq \mathbb{N}## seems unnecessary for the rest of the proof ...

Peter
 

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Interestingly as Evgeny Makarov has pointed out in the Math Help Forum, the claim S⊆NS⊆NS \subseteq \mathbb{N} seems unnecessary for the rest of the proof ...
It is not, because he needs ##p \in \mathbb{N}## provided by the well-ordering principle of ##S \subseteq \mathbb{N}.## Otherwise he wouldn't have the laws available, which he applies on ##p,## and which I assume were provided by the axioms (or derived statements) of the natural numbers.
 
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Hi fresh_42,

Well ... I do not think that that is the case ...

Maybe I did not make clear the approach to the integers that was relevant to Theorem 1.4.6 ...

Bloch is, in Section 1.4, dealing with the approach to defining/constructing the integers via defining the integers as an ordered integral domain that satisfies the Well-Ordering Principle ... not the approach through the natural numbers (see my previous post above). In the ordered integral domain approach the natural numbers the natural numbers are 'found'/defined as an embedded set within the integers. The relevant definition for the natural numbers is given in Definition 1.4.7 - unfortunately presented after Theorem 1.4.6. The definition reads as follows:

?temp_hash=13b178a1469c8fd5933105233e9a00d8.png


Once we have defined the integers as an ordered integral domain we have the following properties to call on in Theorem 1.4.6 ... and these properties include those named in the proof of the Theorem ...

?temp_hash=13b178a1469c8fd5933105233e9a00d8.png


Peter
 

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One needs the well-ordering to get ##p \in S## (definition 1.4.3). Together with definition 1.4.7. this abbreviates to ##S \subseteq \mathbb{N}##. It is just another language as to say all elements of ##S## are positive. So whether you say all elements of ##S## are natural numbers or all elements of ##S## are positive, doesn't make a difference, it is the same. And which ever way you look at it, one of them is needed. So the language might be redundant, the fact is not.
 
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Hi fresh_42 ... thanks for the post ...

Yes ... see the point you are making ...

Peter
 

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