It seems that one can construct ordinals from bottom up by successively introducing a new symbol each time a limit is taken: $$1,\ 2,\ \ldots,\ \omega,\ \omega +1,\ \omega +2,\ \ldots,\ \omega\cdot 2,\ \omega\cdot 2 +1,\ \ldots,\ \omega^{2},\ \ldots,\ \omega^{3},\ \ldots\ \omega^{\omega},\ \ldots,\ \omega^{\omega^{\omega}},\ \ldots, \epsilon_{0},\ \ldots$$ Can this be taken as a (mechanical) definition of ordinals? More abstract definitions like "an ordinal is a transitive well-ordered set satisfying certain properties" are much more appealing to me. Is this mechanical definition sufficient to prove things like "each well-ordered set is order isomorphic to exactly one ordinal?"

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    $\begingroup$ What do you do when you reach an uncountable ordinal - do you have an uncountable number of symbols to choose from? What if we just use each ordinal as a symbol for itself? $\endgroup$ – Carl Mummert Oct 11 '11 at 1:37
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    $\begingroup$ Worse yet, the first uncountable ordinal $\omega_1$ cannot be reached as the limit of a countable sequence of smaller ordinals. So your process will give you at most the countable ordninals. $\endgroup$ – Henning Makholm Oct 11 '11 at 1:44
  • $\begingroup$ @Henning: this is an argument in favor of taking each ordinal as a symbol for itself. $\endgroup$ – Carl Mummert Oct 11 '11 at 1:50
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    $\begingroup$ @Carl, they are awfully hard to write down on paper (except by use of other symbols, and even then we don't get most of them), which strikes me as a rather basic requirement for symbols. $\endgroup$ – Henning Makholm Oct 11 '11 at 1:53
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    $\begingroup$ In fact, the intent of the OP's method won't even get you to a nonrecursive ordinal, but it will get you to things that make $\varepsilon_{0}$ pale into insignificance (e.g. $\Gamma_{0}$, $\Gamma_{\varepsilon_{0}}$, the Bachmann-Howard ordinal, etc.). See en.wikipedia.org/wiki/Recursive_ordinal and en.wikipedia.org/wiki/Large_countable_ordinal $\endgroup$ – Dave L. Renfro Oct 11 '11 at 14:52

As remarked in the comments, this is far from sufficient to cover even the countable ordinals. Personally, I see the problem with the three dots at the end which imply both an undefined idea of continuing this sequence, as well something that will terminate after at most $\omega_1$ many steps.

I imagine you might get to some large countable ordinals, perhaps $\epsilon_{\epsilon_0}$ or even higher. However this will terminate long before $\omega_1$.

Why is that a problem? Well, of course that we know about well-ordered sets whose order type is uncountable. But think of this reason: $\mu_0=\{\text{all those ordinals you wrote above}\}$, ordered by $\in$ this would be a transitive and well-ordered set. However it is not isomorphic to any of its members.

More generally, if you have a process which can only be used to generate set many ordinals then this set itself would be a well-ordered set not isomorphic to any of them. This would essentially break the theorem that every well-ordered set is isomorphic to exactly one ordinal.


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