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First ever class of Modern Algebra today. I have a question about the well-ordering principle, introduced in class. This seems awfully foreign to me, as I haven't had a single theoretical math class. I'm worried that this question is super stupid so I elected to post it anonymously to the forum rather than ask the professor.

Here are my notes (they may not be what he wrote, maybe leading to my confusion. It was a stressful class):

well-ordering princ:

Every non empty subset of [itex]Z[/itex] that is bounded below contains a minimal element.

(I interpreted this as every set of numbers in z that is bounded below has a smallest number).

What does it mean to be bounded below????

he then said {n [itex]\in[/itex] [itex]Z[/itex] I (what does this vertical bar mean?)n^{3}>7}

Ok, this seemed pretty straightforward. But then he said it's bounded below by 0! In other words, 0 is the smallest number that satisfies n^3>7??? Or did he just arbitrarily choose 0 as the bound?

So then I have written 0[itex]\in[/itex]S (I'm assuming S is the set n^3>0). And then he said 2 is the minimal element...I suppose that makes sense. I just don't understand where he pulled 0 as the bounded below value. Wouldn't 2 be the bounded below value, as 0^3 is obviously not larger than 7?

He then wrote {r[itex]\in[/itex]R I r^3 >7}. the first thing he throws out is that 7^(1/3) would satisfy the inequality, but it's not in the set because it's not rational. Perhaps I accidentally wrote '>' in place of "=", because it makes no sense that 7>7.

He then started the division algorithm in Z, which you'll probably hear about tomorrow.

So, the summary of my questions is:

Why 0 as the bounded below value?

What's the vertical bar?

Why 7^(1/3) as something in the second set if, in fact, I correctly wrote '<'?

Also, can anybody give me a brief preview of how this principle will be important in Modern Algebra, just so I can get a jump?

Thanks a million,

Blue

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# Help! Well-ordering principle

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