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Principle of Induction and Well-Ordering Principle
My Analysis lecturer defined $\mathbb{N}$ as the smallest set such that $0 \in \mathbb{N}$ and if $n \in \mathbb{N}$ then $n+1 \in \mathbb{N}$. He then proceeded to "prove" the Principle of...
View ArticleAnswer by Arthur for Principle of Induction and Well-Ordering Principle
The Well-Ordering Principle states that any non-empty set can be well-ordered. This does not follow from the standard ZF axioms, and it is required to be assumed as its own axiom (conventionally, the...
View ArticleAnswer by user694818 for Principle of Induction and Well-Ordering Principle
Your lecturer is essentially defining $\mathbb N$ as a particular subset of $\mathbb R$ that you can do induction on. Since $\mathbb N$ is defined as the smallest set that contains $0$ and the...
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