Journal of Symbolic Logic

Approximate Euler characteristic, dimension, and weak pigeonhole principles

Jan Krajíček

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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.

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J. Symbolic Logic, Volume 69, Issue 1 (2004), 201-214.

First available in Project Euclid: 2 April 2004

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Krajíček, Jan. Approximate Euler characteristic, dimension, and weak pigeonhole principles. J. Symbolic Logic 69 (2004), no. 1, 201--214. doi:10.2178/jsl/1080938837.

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