*PM : subsets of countable sets are countable, id = 7721 new !-- WP guess : subsets of countable sets are countable-- Status:
32.
*PM : lemma on projection of countable sets, id = 7700 new !-- WP guess : lemma on projection of countable sets-- Status:
33.
*PM : lemma on projection of countable sets, id = 7700 new !-- WP guess : lemma on projection of countable sets-- Status:
34.
The "'Cantor Bendixson theorem "'states that any Polish space can be written as the union of a countable set and a perfect set.
35.
Indeed, there are uncountably many nonisomorphic countable Boolean algebras, which Jussi Ketonen [ 1978 ] classified completely in terms of invariants representable by certain hereditarily countable sets.
36.
Moreover, since a countable set can be made into two copies of itself, one might expect that somehow, using countably many pieces could do the trick.
37.
At each stage you've removed only countably many points, and the countable union of countable sets is countable, so you're done, proof complete.
38.
As the comments above say, you either have lots of zeros in the set and just a countable infinity of non-zero numbers or the set is a countable set.
39.
The hereditarily countable sets form a model of Kripke Platek set theory with the axiom of infinity ( KPI ), if the axiom of countable choice is assumed in the metatheory.
40.
The set of equivalence classes of the relation \ sim ( defined by : x \ sim y iff x \ preceq y and x \ succeq y ) are a countable set.
