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Basic Concepts Page 1 of 39 39 Hardegree, Set Theory, Chapter 1: Basic Concepts page 1 of 39 39 Basic 1 Concepts 1. Membership .....................................................................................................................................2 2. Pure Set Theory versus Impure Set Theory.....................................................................................2 3. The Axiom of Extensionality...........................................................................................................4 4. Inclusion and Exclusion...................................................................................................................5 5. Set-Abstracts ....................................................................................................................................7 6. Whether Set-Abstracts Denote .........................................................................................................8 7. Classical Set Theory.......................................................................................................................11 8. Russell’s Paradox...........................................................................................................................12 9. Modern Set Theory........................................................................................................................13 10. The Axiom of Separation...............................................................................................................14 11. Russell’s Paradox Revisited...........................................................................................................15 12. The Empty Set................................................................................................................................16 13. Intersection and Set Difference......................................................................................................17 14. Union and Boolean Sum................................................................................................................18 15. Absolute versus Relative Complementation..................................................................................19 16. Singletons, Doubletons, etc............................................................................................................20 17. General Union and Intersection .....................................................................................................21 18. Power Sets......................................................................................................................................23 19. Fields of Sets..................................................................................................................................24 20. The Principle of Set-Abstraction and Contextual Definitions .......................................................25 1. Principle of Set-Abstraction...............................................................................................25 2. The Move from Explicit Definitions to Implicit Definitions .............................................26 3. Contextual Definitions .......................................................................................................26 4. An Aside on Contextuality.................................................................................................27 21. Axioms for Chapter 1.....................................................................................................................28 22. Definitions for Chapter 1 ...............................................................................................................29 1. Official (Explicit) Definitions ............................................................................................29 2. Grammatical Categories.....................................................................................................30 3. Contextual Definitions .......................................................................................................30 23. Theorems for Chapter 1 .................................................................................................................31 24. Exercises for Chapter 1..................................................................................................................33 1. Part 1:.................................................................................................................................33 2. Part 2:.................................................................................................................................33 3. Part 3:.................................................................................................................................33 4. Part 4:.................................................................................................................................33 5. Part 5. .................................................................................................................................33 25. Answers to Exercises for Chapter 1...............................................................................................34 26. Examples of Derivations of Theorems...........................................................................................35 Hardegree, Set Theory, Chapter 1: Basic Concepts page 2 of 39 39 1. Membership Basic to set theory is the notion of membership or elementhood. Indeed, set theory can be formulated in its entirety as a formal theory in which the only non-logical notion (symbol) is membership. The remaining notions can all be defined in terms of membership using the standard machinery of first-order logic (including identity and definite descriptions). Although set theory can be formulated exclusively in terms of membership, in actual practice, free use is made of defined expressions. Accordingly, very few set-theoretic theorems are actually displayed in primitive notation. Membership is a two-place relation, and its syntactic counterpart is a two-place predicate – Î (epsilon). Following the usual mathematical custom, we write this predicate in infix notation, rather than in prefix notation (as is customary in abstract logic and syntax). Specifically, in order to say that one thing is a member (element) of another thing, we write their respective names, and we infix the membership predicate, Î. Thus, the formula,1 x Î y may be read in either of the following ways. x is a member of y x is an element of y Infixing epsilon is analogous to the way the identity relation is expressed by the formula ‘x=y’. Also, in analogy with the identity predicate, we simplify negated formulas in the natural way, according to the following definitions. (d1) a ¹ b ü ;[a = b] (d2) a Ï b ü ;[a Î b] Here, ‘a’ and ‘b’ stand for arbitrary singular terms. The expression ‘ü’ is a metalinguistic expression used to indicate that expressions of one form abbreviate expressions of another form. 2. Pure Set Theory versus Impure Set Theory In the previous section, in saying that membership is the only non-logical notion of set theory, I mean pure set theory. Pure set theory may be formulated as a formal theory in a first-order language in which the only non-logical symbol is ‘Î’. This is because pure set theory talks about sets, and nothing else. Thus, in particular, in the formula ‘x Î y’, both x and y are sets. By contrast, impure set theory (for lack of a better name!) talks not only about sets, but also about other things, including first elements and classes (not necessarily both). Briefly, the notion of class is a generalization of the notion of set; sets are special kinds of classes (see below). Anything that isn’t a class is a first element, sometimes called a point. First elements, or points, which are very important in informal set theory, have no internal structure from the viewpoint of set theory. Of course, 1 It depends upon the author whether ‘Î’ and ‘=’ are treated like other two-place functors written in infix format, specifically whether the official formula has outer parentheses. We adopt the view that parentheses (or brackets) officially flank identity statements and membership statements, and that these are dropped when convenient. Hardegree, Set Theory, Chapter 1: Basic Concepts page 3 of 39 39 first elements may have very interesting internal structure from the viewpoint of, say, biology! The point is that they have no internal set theoretic structure. In informal set theory, as used throughout modern mathematics, points include numbers, geometrical points, vectors, group elements, etc. Points also include all concrete objects, as well as all abstract objects (other than classes!) In one particular formulation of impure set theory, there are two non-logical predicates; besides ‘Î’ there is the one-place predicate ‘C’ (‘...is a class’). Three other predicates may then be defined in terms of ‘C’ and ‘Î’, as follows. Fa ü ;Ca Sa ü Ca & $y[a Î y] Pa ü Ca & ;$y[a Î y] [Here, a is any singular-term, and y is any variable not free in a.] The predicate letters have the following intended readings. Cx : x is a class; Fx : x is a first element; Sx : x is a set; Px : x is a proper class. A proper class is a class that is not a set, by the following theorem, which follows immediately from the definitions. "x(Px « [Cx & ;Sx]) Alternatively, we could introduce three primitive one-place predicates, ‘C’, ‘S’, and ‘P’, and add the following as axioms. "x(Sx « [Cx & $y[x Î y]]); "x(Px « [Cx & ;$y[x Î y]]). Other than
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