defining countable ordinals, how far can we go?
- From: David Bernier <david250@xxxxxxxxxxxx>
- Date: Sat, 10 Nov 2007 19:17:21 -0500
It's easy to define or describe ordinals below omega^omega using polynomials in Z[x] which
have no negative coefficients. For example, omega^2 + omega*3 + 7 corresponds to:
x^2 + 3*x + 7 . Obviously, we can have exponential towers omega^(omega^omega) and
so on. Beyond that, there are the recursive ordinals, those alpha for which there exists a computable
function f: NxN -> {0, 1} where f(m,n) = 1 iff m =n or m precedes n, in some ordering
which happens to be a well-ordering isomorphic to the ordering by "subset_or_equal"
on elements of alpha, whether we can prove it or not.
Using sentences in first order set theory notation, one can no doubt go further.
Suppose all countable ordinals could be defined finitely in some way (informal notion
of finite definability). Then we would have aleph_1 definitions, but only aleph_0 finite
strings, which is a contradiction.
On the other hand, if we can define ordinals below a coutable ordinal alpha finitely,
can we define alpha finitely? It may depend on what is meant by "define finitely".
Probably the question of how far one can define finitely the countable
ordinals shouldn't be taken too seriously.
David Bernier
.
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