How many ways can we totally order a set $X$ (but not necessarily the whole set $X$, perhaps just a subset) up to isomorphism?

Question

Here's Attempt #2 at asking this question.

Suppose $X$ is a set (not necessarily finite), and let $T$ denote the collection of all totally ordered sets $(A,\leq)$ such that $A \subseteq X.$ Now let $\cong$ denote the unique relation on $T$ such that $(A,\leq_A) \cong (B,\leq_B)$ holds iff they're isomorphic as totally ordered sets.

Question. What is the cardinality of $T /\!\cong$ in terms of the cardinality of $X$?

If $X$ is finite, this is easy: the answer is $|T/\!\cong\!| = |X|+1.$ So its really the infinite case that I'm interested in. For the infinite case, all I know is that $|X|^+ \!\leq |T/\!\cong\!|,$ where $\kappa^+$ denotes the cardinal successor of $\kappa$. This follows more or less from the definition of cardinal successor.

@AndresCaicedo, hey any idea what the answer is for scattered linear orderings? — goblin GONE, Apr 18 '14 at 17:47
I haven't thought much about it... The counting in that case should probably be based on Hausdorff's result, see here. — Andrés E. Caicedo, Apr 18 '14 at 18:43
@AndresCaicedo, okay I'll have a think about it. Probably just end up asking another question at some point. — goblin GONE, Apr 18 '14 at 19:03

score 2 · Accepted Answer · edited Apr 13 '17 at 12:58

OK, we get $2^\kappa$ orders, as explained here.

The answer does not change if we restrict our attention to scattered orders: Given any element of $(\kappa^+)^{<\kappa^+}$, say $(\alpha_\iota\mid \iota<\tau)$, simply considered the ordered (transfinite) sum $$\alpha_0+(\omega^*+\omega)+\alpha_1+(\omega^*+\omega)+\dots$$ Different sequences give rise to non-isomorphic scattered orders, so we get at least $|(\kappa^+)^{<\kappa^+}|=2^\kappa$ orders this way.

(Related.)

How many ways can we totally order a set $X$ (but not necessarily the whole set $X$, perhaps just a subset) up to isomorphism?

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