Approximate Euler characteristic, dimension, and weak pigeonhole principles
Krajíček, Jan
J. Symbolic Logic, Tome 69 (2004) no. 1, p. 201-214 / Harvested from Project Euclid
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We define the notion of approximate Euler characteristic of definable sets of a first order structure. We show that a structure admits a non-trivial approximate Euler characteristic if it satisfies weak pigeonhole principle WPHP2nn: two disjoint copies of a non-empty definable set A cannot be definably embedded into A, and principle CC of comparing cardinalities: for any two definable sets A, B either A definably embeds in B or vice versa. Also, a structure admitting a non-trivial approximate Euler characteristic must satisfy WPHP2nn. ¶ Further we show that a structure admits a non-trivial dimension function on definable sets if and only if it satisfies weak pigeonhole principle WPHPn2n: for no definable set A with more than one element can A2 definably embed into A.
Publié le : 2004-03-14
Classification:  
@article{1080938837,
     author = {Kraj\'\i \v cek, Jan},
     title = {Approximate Euler characteristic, dimension, and weak
pigeonhole principles},
     journal = {J. Symbolic Logic},
     volume = {69},
     number = {1},
     year = {2004},
     pages = { 201-214},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1080938837}
}

Krajíček, Jan. Approximate Euler characteristic, dimension, and weak
pigeonhole principles. J. Symbolic Logic, Tome 69 (2004) no. 1, pp.  201-214. http://gdmltest.u-ga.fr/item/1080938837/