Foundation

[foun-dey-shuh n] /faʊnˈdeɪ ʃən/

noun
1.
the basis or groundwork of anything:
the moral foundation of both society and religion.
2.
the natural or prepared ground or base on which some structure rests.
3.
the lowest division of a building, wall, or the like, usually of masonry and partly or wholly below the surface of the ground.
4.
the act of , setting up, establishing, etc.:
a policy in effect since the foundation.
5.
the state of being .
6.
an institution financed by a donation or legacy to aid research, education, the arts, etc.:
the Ford Foundation.
7.
an endowment for such an institution.
8.
a cosmetic, as a cream or liquid, used as a base for facial makeup.
9.
.
10.
Solitaire. a card of given denomination on which other cards are to be added according to denomination or suit.
/faʊnˈdeɪʃən/
noun
1.
that on which something is founded; basis
2.
(often pl) a construction below the ground that distributes the load of a building, wall, etc
3.
the base on which something stands
4.
the act of founding or establishing or the state of being founded or established
5.

6.
an institution supported by an endowment, often one that provides funds for charities, research, etc
7.
the charter incorporating or establishing a society or institution and the statutes or rules governing its affairs
8.
a cosmetic in cream or cake form used as a base for make-up
9.
See foundation garment
10.
(cards) a card on which a sequence may be built
n.

late 14c., “action of founding,” from Old French fondacion (14c.) or directly from Latin fundationem (nominative fundatio) “a founding,” noun of action from past participle stem of fundare (see found (v.1)). The Latin word is glossed in Old English by staþol. Meaning “that which is founded” (a college, hospital, etc.) is from 1510s; meaning “funds endowed” is early 15c. Sense of “solid base of a structure” is from late 15c.

foundation foun·da·tion (foun-dā’shən)
n.
The basis on which something stands or is supported; a base.

The axiom of foundation states that the membership relation is well founded, i.e. that any non-empty collection Y of sets has a member y which is disjoint from Y. This rules out sets which contain themselves (directly or indirectly).