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The are many orderings that naturally occur in interesting but seemingly unrelated circumstances. Here are some examples:

  1. The volume spectrum of orientable hyperbolic 3-manifolds has order type $\omega^\omega$.

  2. Ordinals that play important roles in Conway's $\mathbf {On_2}$, most notably $\omega^{\omega^\omega}$, the algebraic closure of $2$. See Lenstra's papers 1 2, Conway's ONAG, and Lieven's blog posts.

  3. The set of fusible numbers has order type $\epsilon_0$ (quite likely but not proven, see my note).

  4. The Sharkovsky ordering of natural numbers, which does not have order type of an ordinal.

  5. There are proof theoretical ordinals, which I know little about.

Do you know any other examples or see any connection among aforementioned examples? Most of the examples above are ordinals, but other interesting examples are welcome.

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That's a good question! – Todd Trimble Jun 12 '11 at 14:41
Not certain if this is an appropriate for an answer so I'll leave it as a comment for now. Here's another nice-but-surprising way to get $\omega^\omega$: Let f(n) denote the smallest number of 1's needed to write n using any combination of addition and multiplication, e.g., f(7)=6 as shortest way for 7 is 7=(1+1+1)(1+1)+1. For any n, f(n)>=3log_3(n). So subtract off this lower bound and consider d(n)=f(n)-3log_3(n). Then the set of all values of d is a well-ordered set of real numbers, with order type $\omega^\omega$. Afraid I can't give a proper reference as we haven't published this yet! :) – Harry Altman Jun 13 '11 at 0:18
@Harry: your results are definitely interesting and also arise from a natural number theory question, so I'd like to vote it up if you post it as an answer:) By the way, there are two more examples became known to me recently: the order type of the set of Pisot numbers, and the $\sigma$-ordering on braid groups. – Junyan Xu Jun 15 '11 at 5:38

Alright, I'll put my comment as an answer and hopefully get this off the no-upvoted-answers queue. :)

Here's another nice-but-surprising way to get $\omega^\omega$: Let $\|n\|$ denote the smallest number of 1's needed to write n using any combination of addition and multiplication, e.g., $\|7\|=6$ as shortest way for 7 is $7=(1+1+1)(1+1)+1$. (This is known as the "integer complexity" of n; it's sequence A005245.)

Now, for any n, we have the lower bound $\|n\|\ge 3log_3 n$. So subtract this off and consider $\delta(n):=\|n\|-3log_3 n$. Then the set of all values of $\delta$ is a well-ordered subset of $\mathbb{R}$, with order type $\omega^\omega$.

For a proof, I refer you to my preprint:

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Hrbacek, following Ballard, has recently put a certain partial ordering called $\sqsubseteq$ on absolutely everything in order to do nonstandard analysis a la Nelson (i.e. internal set theory):

Let $\mathscr{L}$ be the language of ZFC (and Tarski-Grothendieck if you insist). We say a well-formed formula of $\mathscr{L}$ is an $\in$-formula. We assert that ZFC holds for all well-formed $\in$-formulae.

Now we throw in $\sqsubseteq$ to $\mathscr{L}$ to get a bigger language $\mathscr{HB}.$ First, we assume that ZFC holds for all $\in$-formulae. Let us write $x \sqsubseteq_\alpha y$ as an abbreviation of $x \sqsubseteq \alpha \vee x \sqsubseteq y.$ Suppose we have a well-formed formula $P$ of $\mathscr{HB}.$ We write $P^\alpha$ for the replacement of every instance of $\sqsubseteq$ with $\sqsubseteq_\alpha.$ Let us also write $x \sqsubset y$ for $x \sqsubseteq y \wedge y \not\sqsubseteq x.$ We also write $2^A_{\mathrm{fin}}$ for the set of all finite subsets of $A.$ We also write $(\forall u \sqsubseteq v) P(u,v)$ for $(\forall u)(u \sqsubseteq v \implies P(u,v)),$ and so on for $\exists,$ and for $\in$ as well.

Then Hrbacek's GRIST is the following condition on $\sqsubseteq,$ with four axiom schemata:

R elativization condition on $\sqsubseteq$: $\sqsubseteq$ is a total dense preordering with minimal element $\emptyset$ and no maximal element; i.e. the conjunction of

  1. Partial ordering: $(\forall u,v,w)((v\sqsubseteq u \wedge w \sqsubseteq v) \implies w \sqsubseteq u) \wedge u \sqsubseteq u;$

  2. Preordering: $(\forall u,v)(u \sqsubseteq v \wedge v \sqsubseteq u);$

  3. Minimality of $\emptyset$: $(\forall u)(\emptyset \sqsubseteq u);$

  4. Illimitability: $(\forall u) (\exists v) (u \sqsubset v);$

  5. Density: $(\forall u,v) (u \sqsubset v \implies (\exists w)(u \sqsubset w \sqsubset v)).$

Axiom schemata, in which we use words so as not to have our eyes completely glaze over:

For any well-formed formula $P$ of $\mathscr{HB}$ depending on finitely many variables,

  1. T ransfer: for all $u \sqsubseteq v$ and $x_1,\ldots, x_n \sqsubseteq u,$

    $P^u(x_1,\ldots,x_n) \iff P^v(x_1,\ldots,x_n)$

  2. S tandardization: for all $u \sqsupset \emptyset$ and for all $A, x_1, \ldots, x_n,$ there are $v \sqsubset u$ and $B \sqsubset v$ such that, for every $w$ with $v \sqsupseteq w \sqsupset u,$

    $(\forall y \sqsubseteq w)(y \in B \iff y \in A \wedge P^w(y,x_1,\ldots,x_n)).$

  3. I dealization: For all $A \sqsubset v$ and all $x_1,\ldots,x_n,$

    $(\forall a \in 2^A_{\mathrm{fin}})\big([a\sqsubset v \implies (\exists y)(\forall x \in a) P^v(x,y,x_1,\ldots,x_n)] \iff[(\exists y)(\forall x \in A)[x \sqsubset u \implies P^v(x,y,x_1,\ldots,x_n)]\big).$

  4. G ranularity: For all $x_1,\ldots,x_k,$ if $(\exists u) P^u(x_1,\ldots,x_k),$ then

    $(\exists u)[P^u(x_1,\ldots,x_k) \wedge (\forall v)(v \sqsubset u \implies \neg P^v(x_1,\ldots,x_n))]$

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