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Topic: infinite countable total orderings embeddable in the reals
Replies: 1   Last Post: Jun 22, 2013 9:44 PM

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David Bernier

Posts: 3,401
Registered: 12/13/04
infinite countable total orderings embeddable in the reals
Posted: Jun 22, 2013 5:25 PM
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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 Bachmann-Howard ordinal,

http://en.wikipedia.org/wiki/Bachmann-Howard_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



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