downward closed การใช้

• Thus " I " contains the bottom element, is downward closed, and is closed under countable suprema ( which must exist ).
• A subset L of a partially ordered set P is said to be a lower set of P if it is downward closed : if y \ in L and x \ le y then x \ in L.
• It can be a simplification, in terms of notation if nothing more, to concentrate on one " half "  say, the lower one  and call any downward closed set " A " without greatest element a " Dedekind cut ".
• This definition has the advantages that one can recursively enumerate the predecessors of a given ordinal ( though not in the ordering ) and that the notations are downward closed, i . e ., if there is a notation for \ gamma and \ alpha then there is a notation for \ alpha.
• In 2009, Kontinen and V滗n鋘en, showed, by means of a partial inverse translation procedure, that the families of relations that are definable by IF logic are exactly those that are nonempty, downward closed and definable in relational \ Sigma _ 1 ^ 1 with an extra predicate R ( or, equivalently, nonempty and definable by a \ Sigma _ 1 ^ 1 sentence in which R occurs only negatively ).
• There are infinite ordinals as well : the smallest infinite ordinal is "'? "', which is the order type of the natural numbers ( finite ordinals ) and that can even be identified with the " set " of natural numbers ( indeed, the set of natural numbers is well-ordered as is any set of ordinals and since it is downward closed it can be identified with the ordinal associated with it, which is exactly how ? is defined ).