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I'm trying to tie some loose ends here. My lecturer didn't bother to go into details, so I have to work it out myself. I usually hate to be pedantic, but these questions have been bugging me for a while.

First, what's the proper definition of the natural numbers? The one given to me is: "the smallest set such that $\emptyset\in N$ and $x\in N\implies x\cup\lbrace x\rbrace\in N$". This definition does not seem rigorous to me. What does "smallest" mean here? Does it mean that no proper subset have the same property? And how to prove the existence of such set (under ZFC)? Lastly, how to prove this set satisfies Peano's axioms, especially the last axiom (about induction)?

Next, how to prove that the relation $<$ is a total order? The definition is $x. I was thinking of changing the definition into "the transitive closure of $\lbrace(n,n')|n\in N\rbrace$" (to avoid the set theory stuffs), but it's still difficult to prove that it's a total order.

Lastly, how to prove the well ordering principle? I suspect this would be easy after proving the previous statement though.

Edit: Why didn't the curly bracket appear? I'll use the square bracket instead.

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    You prove the well ordering principle with the principle of mathematical induction (and vice versa).2011-09-30
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    \left \{ abc \right \} (in dollar signs) gives { *abc* }2011-09-30
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    I got out of bed to write an answer, and by the time I get to the computer two good answers have appeared. Instead I fixed the braces (which did not work due to the quotation marks, I suspect.)2011-09-30
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    I also can't figure out why you are trying "to avoid the set theory stuff" in a set theoretic construction.2011-09-30
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    Is the definition of order *really* $x\lt y\Leftrightarrow x\subset y$? It is more usual to define it as $x\lt y\Leftrightarrow x\in y$.2011-09-30
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    @Asaf: The coast is clear - you can head back to bed now!2011-09-30
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    @TheChaz: If only things were that simple... :-)2011-09-30

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