Assertion: set's element | /r/HasProperty | finite
Facets:
Source sentences:
text: It is also needed to prove the existence of such simple sets as the set of hereditarily finite sets, i.e., those finite sets whose elements are finite, the elements of which are also finite, and so on; or to prove basic set-theoretic facts such as that every set is contained in a transitive set, i.e., a set that contains all elements of its elements (see Mathias 2001 for the weaknesses of Zermelo set theory).
text: The theory of the hereditarily-finite sets, namely those finite sets whose elements are also finite sets, the elements of which are also finite, and so on, is formally equivalent to arithmetic.
text: The set S is said to be a semigroup generating set of G if each element of G is a finite sum of elements of S. Similarly, a set S is said to be a monoid generating set of G if each non-zero element of G is a finite sum of elements of S.
text: The theory of the hereditarily-finite sets, namely those finite sets whose elements are also finite sets, the elements of which are also finite, and so on, is formally equivalent to arithmetic.