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infinite countable total orderings embeddable in the reals
Posted:
Jun 22, 2013 5:25 PM


I'm looking at infinite countable total orderings (S, <_{S} ) with  S  = aleph_0
 embeddable in the reals.
For example, any countably infinite ordinal beta < omega_1 is such.
In ordinal logic, some "large" countable ordinals have been defined:
 Veblen hierarchy,
< http://en.wikipedia.org/wiki/Veblen_function > .
The Feferman?Schütte ordinal is denoted Gamma_0: < http://en.wikipedia.org/wiki/Feferman%E2%80%93Sch%C3%BCtte_ordinal > .
The BachmannHoward ordinal,
http://en.wikipedia.org/wiki/BachmannHoward_ordinal
Then, one with a rather forbidding name is known by it's name, namely:
Psi_{0}(Omega_{omega}) ,
< http://en.wikipedia.org/wiki/%CE%A8%E2%82%80%28%CE%A9%CF%89%29 > .
Maybe some set theories without the full axioms can define a countable total ordering embeddable in the reals, but are incapable of proving that that ordering is not complete, like the real numbers are. I don't know...
David Bernier  On Hypnos, http://messagenetcommresearch.com/myths/bios/hypnos.html



