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Equivalently, a set is finite if its Cardinality , i.e., the number of its elements, is a natural number. More specifically, a set whose cardinality is n is also called an n-set . For instance, the set of Integer s between −15 and 3 (excluding the end points) has 17 elements, so it is finite; in fact, it is a 17-set. In contrast, the set of all Prime Number s has cardinality ℵ0 , so it is infinite. A set is called Dedekind-finite if there exists no bijection between the set and any of its Proper Subset s. If the axiom of Dependent Choice (a weak form of the Axiom Of Choice ) holds, then a set is finite If And Only If it is Dedekind-finite. Otherwise, paradoxically, there may be infinite Dedekind-finite sets (see Foundational Issues below). All finite sets are Countable , but not all countable sets are finite. (However, some authors use "countable" to mean "countably infinite", and thus do not consider finite sets to be countable.) CLOSURE PROPERTIES For any elements ''x'', ''y'', the sets {}, {''x''}, and {''x'', ''y''} are finite. The union of a finite set of finite sets is finite. The powerset of a finite set is finite. Any subset of a finite set is finite. The set of values of a function when applied to elements of a finite set is finite. The Cartesian product of a finite set of finite sets is finite. However, the set of natural numbers (whose existence is assured by the Axiom Of Infinity ) is not finite. NECESSARY AND SUFFICIENT CONDITIONS FOR FINITENESS In Zermelo–Fraenkel Set Theory (ZF), the following conditions are all equivalent: # ''S'' is a finite set. That is, ''S'' can be placed into a one-to-one correspondence with the set of those natural numbers less than some specific natural number. # ( Kazimierz Kuratowski ) ''S'' has all properties which can be proved by mathematical induction beginning with the empty set and adding one new element at a time. (See the section on foundational issues for the set-theoretical formulation of Kuratowski finiteness.) # ( Paul Stäckel ) ''S'' can be given a total ordering which is both Well-order ed forwards and backwards. That is, every non-empty subset of ''S'' has both a least and a greatest element in the subset. # Every function from P(P(''S'')) one-to-one into itself is onto. That is, the Powerset of the powerset of ''S'' is Dedekind-finite (see below). # Every function from P(P(''S'')) onto itself is one-to-one. # ( Alfred Tarski ) Every non-empty family of subsets of ''S'' has a minimal element with respect to inclusion. # ''S'' can be well-ordered and any two well-orderings on it are Order Isomorphic . In other words, the well-orderings on ''S'' have exactly one Order Type . If the Axiom Of Choice also holds, then the following conditions are all equivalent: # ''S'' is a finite set. # ( Richard Dedekind ) Every function from ''S'' one-to-one into itself is onto. # Every function from ''S'' onto itself is one-to-one. # Every partial ordering of ''S'' contains a Maximal Element . FOUNDATIONAL ISSUES s constitutes a model of Zermelo-Fraenkel Set Theory with the Axiom Of Infinity replaced by its negation. Even for those mathematicians who embrace infinite sets, in certain important contexts, the formal distinction between the finite and the infinite can remain a delicate matter. The difficulty stems from Gödel's Incompleteness Theorems . One can interpret the theory of hereditarily finite sets within Peano Arithmetic (and certainly also vice-versa), so the incompleteness of the theory of Peano arithmetic implies that of the theory of hereditarily finite sets. In particular, there exists a plethora of so-called Non-standard Models of both theories. A seeming paradox, non-standard models of the theory of hereditarily finite sets contain infinite sets --- but these infinite sets look finite from within the model. (This can happen when the model lacks the sets or functions necessary to witness the infinitude of these sets.) On account of the incompleteness theorems, no first-order predicate, nor even any recursive scheme of first-order predicates, can characterize the standard part of all such models. So, at least from the point of view of first-order logic, one can only hope to characterize finiteness approximately. More generally, informal notions like Set , and particularly finite set, may receive interpretations across a range of formal systems varying in their axiomatics and logical apparatus. The best known axiomatic set theories include Zermelo-Fraenkel Set Theory (ZF), Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC), Von Neumann–Bernays–Gödel Set Theory (NBG), Non-well-founded Set Theory , Bertrand Russell 's Type Theory and all the theories of their various models. One may also choose among classical first-order logic, various higher-order logics and intuitionistic logic. A Formalist might see the meaning of ''set'' varying from system to system. A Platonist might view particular formal systems as approximating an underlying reality. |
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