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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:
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:
The definition of the natural numbers is mentioned above ... Bloch's definition is as follows ...
Hope someone can help,
Peter
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 ##\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:
The definition of the natural numbers is mentioned above ... Bloch's definition is as follows ...
Hope someone can help,
Peter
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