[FOM] Extension of well-founded relation well-ordering
alberto.marcone at uniud.it
Tue Feb 10 05:00:50 EST 2015
Il 10/02/2015 07:56, Gert Smolka ha scritto:
> Consider the following statement:
> Let X be a set with a choice function. Then every well-founded relationon X can be extended to a well-ordering of X.
> Is there a proof of the statement in the literature?
> I would expect that a proof of Zermelo's theorem can be extended to a proof of the statement.
> Gert Smolka
> FOM mailing list
> FOM at cs.nyu.edu
This statement is just saying that omega^* is extendible (also
enforceable). In general say that a linear order type tau is extendible
if a poset containing no copy of tau always has a linear extension
containing no copy of tau. this kind of statements have been stated from
the viewpoint of reverse mathematics.
I quote a bit of history from the introduction to
"Computability-theoretic and proof-theoretic aspects of partial and
linear orderings" by Downey, Hirschfeldt, Lempp, and Solomon (Israel J.
Math. 138 (2003), 271-289):
"The extendibility of various linear order types was studied extensively
by Bonnet, Corominas, Fraissé, Jullien, and Pouzet in France, as well as
independently by Galvin, Kostinsky, and McKenzie in the United States. A
complete characterization of the countable extendible linear order types
was obtained by Bonnet [Bo69]."
The final reference is to R. Bonnet, Stratifications et extension des
genres de chaînes dénombrables, C. R. Acad. Sci. Paris Sér. A-B 269
Alberto Marcone alberto.marcone at uniud.it
Dip. di Matematica e Informatica
Universita' di Udine tel: +39-0432-558482
via delle Scienze 206 fax: +39-0432-558499
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