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

In summary, The theorem states that the integers are an ordered integral domain and that it satisfies the well-ordering principle.
  • #1
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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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  • Bloch - Peano Postulates and Defn of the Natural Numbers ... ....png
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  • #2
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
 
  • #3
Math Amateur said:
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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  • #4
fresh_42 said:
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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  • Bloch - Section 1.4 Introduction ....png
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  • Bloch - Defn 1.4.7 ... ....png
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  • #5
Math Amateur said:
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.
 
  • #6
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
 

Attachments

  • Bloch - Defn 1.4.7 ... ....png
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  • Bloch - Defn 1.4 - Ordered Integral Domain ....png
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  • #7
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.
 
  • #8
Hi fresh_42 ... thanks for the post ...

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

Peter
 

1. What are integers as an ordered integral domain?

Integers as an ordered integral domain refer to the set of whole numbers, including positive, negative, and zero values, that can be arranged in a specific order based on their magnitude. This set forms a mathematical structure known as an integral domain, which has specific properties and operations defined for its elements.

2. What is the significance of the Bloch Theorem 1.4.6?

The Bloch Theorem 1.4.6 states that every nonzero integer in an ordered integral domain has a unique inverse. This means that for every integer, there exists another integer that, when multiplied together, result in the identity element of the integral domain, which is typically represented as the number 1.

3. How do integers behave under the operation of addition?

Integers follow the commutative, associative, and distributive properties under addition. This means that the order of the numbers does not affect the result, they can be grouped in different ways, and multiplication can be distributed over addition.

4. Can integers be negative in an ordered integral domain?

Yes, integers in an ordered integral domain can be negative. The negative integers are represented by a minus sign (-) before the number and are located to the left of the origin on a number line.

5. What is the role of the order property in an ordered integral domain?

The order property in an ordered integral domain allows for the arrangement of elements based on their magnitude. This order is preserved under the operations of addition and multiplication, making it a fundamental property in understanding the behavior of integers in this mathematical structure.

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