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Elements of Set Theory by Sidney Felder in the Words of Georg Cantor

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Elements of Set Theory by SidneyFelder In the words of GeorgCantor (1845-1918) ,the creator of Set Theory and the modern theory of the Infinite, a set (or class) is “a collection into a whole, of definite, well-distinguished objects of our perception or of our thought”. In thinking about anything, we mentally aggregate and separate things in all kinds of combinations. In any such case, we are forming sets. On various occasions, it will be convenient to speak of sets such as the fol- lowing: the set containing the twophysicists Michael Faraday and Marie Curie; the set containing the numbers 1, 2, 3, and 4; the set of all evennumbers; the set of all copies of this Handout; the set of all possible legal sequences of movesinChess; the set of all legalsequences of movesinChess that terminate in a draw; the set of all persons who have just read the words “the words”; the set of all persons who have not read this paragraph; the set of all fundamental particles in the universe; the set of all sets enumerated in this list; the set of all finite or infinite sets of natural numbers that contain the number 1,000; the set of all ways to make change from a 1 dollar bill; the set of all Kings of the United States. We express in writing the fact that selected entities form a set by exhibiting the symbolic representations of these entities enclosed in a pair of oppositely facing curly brackets (or braces), { and }. Thus {9,18,27} rep- resents the set composed of the three numbers 9, 18, and 27. These three numbers are called the members or elements that belong to the set {9,18,27}. We symbolize the fact that the number 9 belongs to the set {9,18,27} by the expression 9∈{9,18,27}. This same pattern of representation holds for sets whose members are sets. Thus {{1,2,3},{Mars,Neptune}} represents a two-element set that contains both the set {1,2,3} (a set of numbers) and the set {Mars,Neptune} (a set of planets). One may have noticed that both the set {9,18,27} and the set {{1,2,3},{Mars,Neptune}} contain sets that are intuitively of a “vertically homogeneous” kind. All the members of the set {9,18,27} (which are 9, 18 and 27) contain no members; and all the members of {{1,2,3},{Mars,Neptune}} (which are {1,2,3} and {Mars,Neptune}) consists solely of members (in one case numbers and in the other case planets) that themselves contain no members). Our choice of examples of this simple kind does not reflect anylimitation of the present rule “Place a single pair of curly brackets around any aggregate of objects and sets you already have”. Consequently,this same device works for “vertically hetero- geneous” sets also, sets that contain sets of all different levels or orders.Thus, for example, the dizzying expression {1,{1},{{{{2}}}},{{5,6},{{7,8}}}} is a completely unambiguous representation of a well-defined set.1 We speak of extensional and intensional definitions of sets. An intensional definition of a set is one that determines its membership by specifying some property or combination of properties that all its elements have in common.2 Thus we speak of all green objects; all objects that are both green and round; all objects that are either green or round; all students in a room; all evennumbers; all evennumbers that are also greater than 1 The expression {1,{1},{{{{2}}}},{{5,6},{{7,8}}}} represents a four-element set whose members are: a num- ber (the number 1) that contains no members; a set ({1}) that contains a single number (1) that contains no mem- bers; a set ({{{{2}}}}) that contains a single set ({{{2}}}) that contains a single set ({{2}}) that contains a single set {2} that contains a single number (2) that contains no members; and a set {{5,6},{{7,8}}} that contains only a)a single set of numbers ({5,6}) that contains twonumbers (1 and 2) both of which contain no members and b)a single set ({{7,8}}) that contains a single set ({7,8}) that contains twonumbers (6 and 8) both of which contain no members. 2 By the way,Ishould remark that I think it is doubtful that there is anymetaphysically absolute classification of properties into “simple” and “complex”—I would argue that anycombination of properties may be conceivedasa single property from some imaginable point of view, and vice-versa.This is of course not to say that certain points of vieware not more powerful or convenient in relation to a givencontext, or that (in any“normal” context) we cannot stand fast with a particular choice of properties that are to be considered “simple”. Course Notes Page 1 Elements of Set Theory eleven; all oxygen molecules in the atmosphere; etc. Although this is not sufficiently well appreciated, it is difficult to give a general characterization of extensional definitions. It is perhaps best to say that an extensional definition of a set is one that is givenbyanenumera- tion (listing) of all its elements. (It is amusing that the character of the class of extensional definitions will be best conveyed by fragments of an extensional definition—by an enumeration of a small number of extensional definitions of particular sets). The set that contains Abraham Lincoln, the number 328, and the gravitational field of Mars is a good example. If hard pressed, we can come up with some property that all three of these objects have incommon, but the only property that theyshare by design is that theyand only theybelong to the set {Abraham Lincoln,328,the gravitational field of Mars}. Obviously,asamethod or either constructing or exhibiting sets, enumeration is feasible only when very small sets are involved. Larger sets can only be defined intensionally,and hence require another mode of representation. According to the most common, the definition of the set of evennumbers ({0,2,4,6,8,10,...}) is written {x|x is even}, to be read “the set of all enti- ties x, such that x is even”. It is essential to understand that although we inevitably mark down the elements of a set in some particular order when giving an extensional definition of it, this is merely an artifact of the process of definition—some- thing that is unavoidably introduced in consequence of the fact that everything we do and say happens in tem- poral sequence and everything written exhibits spatial order.One is already accustomed in elementary geome- try to the fact that although a line is assumed to have nothickness, our representations of it on paper,black- board, or screen inevitably do. (This is by no means the same thing as saying that objects likepoints and lines don’thav e an exact expression in the physical world!). In particular,the elements of the set itself are not to be conceivedasbeing arrayed either in the order listed or in anyother order: A set has no structure beyond that which suffices to “contain” its elements in the most abstract sense imaginable. The creation of a set is the choice of a classification.This means that although a givenobject can be a member of anymulti- plicity of sets, it can only showuponce in a particular set because the only structures at this levelthat sup- port multiplicities are those that are implicit in the distinctness among objects and the distinctness among sets. The identity of a set is considered to be determined by its extension,that is, by its membership,the identity of the elements that belong to it. In all familiar variants of Axiomatic Set Theory,this assertion is called the ‘Axiom of Extension’, although it is more likeastipulation about the meaning of the term ‘set’ than a hypoth- esis of substance. With certain critical exceptions, every intension—every concept, property,and condition— determines a unique empty or non-empty set. (The postscript describing Russell’sParadox should give a sense of the character of these exceptions). In general, arbitrarily manydifferent intensions will determine the same set. All intensions that determine the same set are said to be co-extensive.For example, the set of all Presi- dents of the United States through the year 2005 contains the same 43 elements as does the set of all male Presidents of the United States through the year 2005—thus these twosets are identical, which means that the twodefining conditions that pick out these 43 persons are co-extensive.This is true despite the fact that the twodefining conditions are conceptually quite distinct, and that in another possible world differing from ours in only comparatively modest ways, these twoconditions pick out twosets with distinct memberships. (A possible world is a self-consistent way things could have been. Thus there is a possible world in which life neverevolved on Earth, but there is no possible world in which 2+2 =5). Further examples of mutually co-extensive sets: the set of all odd numbers between the numbers 2 and 8 and the set of all prime numbers between 2 and 8 (both intensions correspond to the set containing the elements 3, 5, and 7) ;the set of objects not identical to themselves, the set of round squares, the set of inhabitants of Atlantis, the inhabitants of Mars in 2005 (all four intensions correspond to the (necessarily unique) null set, the set containing no elements); the set containing the number 2, the set containing the number that is equal to the number 4 divided by the number 2, the set containing the number that is equal to the number 8 divided by the number 4, the set containing the number that is equal to the number 16 divided by the number 8,..
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