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In ZF, a set is infinite if and only if the power set of its power set is a Dedekind-infinite set, having a proper subset equinumerous to itself. [4] If the axiom of choice is also true, then infinite sets are precisely the Dedekind-infinite sets. If an infinite set is a well-orderable set, then it has many well-orderings which are non-isomorphic.
is a finite set with five elements. The number of elements of a finite set is a natural number (possibly zero) and is called the cardinality (or the cardinal number) of the set. A set that is not a finite set is called an infinite set. For example, the set of all positive integers is infinite:
Let be a set and a nonempty family of subsets of ; that is, is a nonempty subset of the power set of . Then is said to have the finite intersection property if every nonempty finite subfamily has nonempty intersection; it is said to have the strong finite intersection property if that intersection is always infinite.
Using first-order logic primitive symbols, the axiom can be expressed as follows: [2] ( ( ()) ( ( (( =))))). In English, this sentence means: "there exists a set 𝐈 such that the empty set is an element of it, and for every element of 𝐈, there exists an element of 𝐈 such that is an element of , the elements of are also elements of , and nothing else is an element of ."
However, some countably infinite sets of non-empty sets can be proven to have a choice function in ZF without any form of the axiom of choice. For example, V ω ∖ { ∅ } {\displaystyle V_{\omega }\setminus \{\emptyset \}} has a choice function, where V ω {\displaystyle V_{\omega }} is the set of hereditarily finite sets , i.e. the first set ...
Any family of sets without repetitions is a subclass of the proper class of all sets (the universe). Hall's marriage theorem, due to Philip Hall, gives necessary and sufficient conditions for a finite family of non-empty sets (repetitions allowed) to have a system of distinct representatives.
In any topological space X, the empty set is open by definition, as is X. Since the complement of an open set is closed and the empty set and X are complements of each other, the empty set is also closed, making it a clopen set. Moreover, the empty set is compact by the fact that every finite set is compact. The closure of the empty set is empty.
8 Ways of defining sets/Relation to descriptive set theory. ... Empty set; Finite set, Infinite set; Countable set, Uncountable set; Power set; Relative to a topology.