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My textbook claims to prove the statement that for $2$ well-ordered sets, $X$ and $Y$, either $X$ and $Y$ are isomorphic or one of the sets is isomorphic to an initial segment of the other set (i.e. an upper bounded subset of the other set). The proof starts off:

"Let $X$ be well-ordered, and let $f:X \Rightarrow X$ be a monotonic map, i.e., $$Z_1<Z_2 \Rightarrow f(Z_1)<f(Z_2).$$ Then for all $Z \in X$ we have $f(Z)\geq Z$..."

1) Isn't this the definition of strictly increasing, not monotonic?

2) Take $X = [0,5]$ and $f(x) = 0.5x$. This is a strictly increasing map of $X$ to itself--note that the least element is preserved--for which $f(Z) \leq Z$.

Who's confused: me or the author?

Juniven Acapulco
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LumpyGrads
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    $[0,5]$ is not a well-ordered set. Take $(0,5]$, for instance. What is the least element of this set? Now let $x$ be the least element of $X$. Since $f$ is monotonic, you can deduce that $f(x)\geq x$. Now remove $x$, i.e. consider $X-{x}$ and pick a least element. Continuing in this fashion you get the result. – Levent Dec 26 '16 at 05:50
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    Well-ordered means every subset has a least element. The set $(0,5]={x\in\mathbb{R}\mid 0<x\leq 5}$ is a subset of $[0,5]$ and does not have a least element. – Levent Dec 26 '16 at 06:06
  • For $[0,5]$, wouldn't it be zero? Also, for $f(x)=0.5x$ $f(x) \leq x$ – LumpyGrads Dec 26 '16 at 06:07
  • You don't have to delete your comment to add something. There is a little 'edit' next to your comment, you can use it. – Levent Dec 26 '16 at 06:08
  • I'm not asking about (0,5], I'm proposing [0,5] as the counter-example... – LumpyGrads Dec 26 '16 at 06:08
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    As I said, $[0,5]$ is not well-ordered because it has a subset, i.e. $(0,5]$ which does not have a least element. Therefore $[0,5]$ is not a counter-example. – Levent Dec 26 '16 at 06:09
  • Well, now I feel stupid. – LumpyGrads Dec 26 '16 at 06:10

1 Answers1

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Let's start with (2). Your example is irrelevant because the set $[0,5]$ (I presume you mean the closed interval containing all reals in between) is not a well-ordered set.

As for (1), in set theory monotonic means what's stated in the definition (which does look like the definition of increasing functions in Calculus, but this is a different branch of mathematics with its own naming conventions). The reason for calling it monotonic is that it preserves the ordering.

zipirovich
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    See @Levent's comment above. I suspect you may be confusing the concepts of well-ordering and total ordering (of course, my suspicion may be wrong). Well-ordering is more than being totally ordered; it also requires that every nonempty subset has the least element. But for example $(0,5]$ (from Levent's comment) doesn't have the smallest element. – zipirovich Dec 26 '16 at 06:09