04 Set Theory.pdf - Sets(a logical approach 1 Sets and Logic Early we introduced the symbol ∈ a binary predicate where a ∈ B means an object a(which

04 Set Theory.pdf - Sets(a logical approach 1 Sets and...

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Sets (a logical approach) 1
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Sets and Logic Early, we introduced the symbol , a binary predicate, where means an object (which may be a set) is in the set B. We then define as follows: e.g., proper subset (how would we define this?) a B a A B ↔ ∀ x ( x A x B ) {1,8,4} {8,4,1} {1,8,4} {8,9,4,2,1} {1,8,4} {8,9,4,2,1} 2
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We then define using logic as follows. A B ~ ࠵? ( x A x B) ࠵? ~(x A x B) negate ࠵? ~( x A x B) remove ࠵? ~(x A x B) defined a new symbol ࠵? (x A x B) de Morgan We are getting pretty “loose here”. We aren’t numbering statements, we are using logical equivalences to justify a proof step. We could even get rid of a few of those symbols. A B iff ~ ࠵? ( if x A then x B) iff ࠵? ~( if x A then x B) negate iff ࠵? ~(if x A x B) remove iff ࠵? ~(if x A x B) defined a new symbol iff ࠵? such that x A x B de Morgan ∼ ∈ ∼ ∈ 3
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Sets remind us why we need a universe . (Very over simplified history) When set theory was axiomatized, there was an idea that there were objects, and collections , or sets. Objects could be elements of sets, but so could other sets. The question arose – can a set contain itself? That is, is it possible there exists a set S such that ࠵? ࠵? ࠵? ? Then, some sets are “self-swallowing”, and some are “non- self-swallowing”. 4
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That is, the set of all walruses contains only walruses, and the set containing only Joan of Arc does not contain the set containing Joan of Arc. Call these run-of-the-mill sets. Then we should have S1 , the set of all self-swallowing sets, and S2 , the set of all run-of-the-mill sets. Clearly a set is either self-swallowing or run-of-the-mill. What about S2 ? Is it in S1 or S2 ? If S2 is run-of-the-mill, it can’t be in S1. But then it should be in S2, but then S2 is would not be the set of run-of-the- mill sets because now it contains itself. 5
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We could rewrite this using set builder notation and logic. Let be the set of run-of-the-mill sets. Let where S can be any set. (note that there is a implied in set builder notation). Is If so, - by the definition of X - every element of can’t contain itself? Suppose . Then . Why? Because X contains all the sets such that don’t contain themselves and . Symbolically, Either way, we end up with a contradiction, and our logical system collapses. This is Russell’s paradox, which you can read about. X X = { S | S S } X X ? X X X X X X X X X X X X X . 6
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Barber paradox: The barber shaves everyone who does not shave himself. Then who shaves the barber? This was used as a popular way of explaining the problem in the set theory of Frege. There is a simple way to solve the barber paradox, just rephrase it as “The barber shaves everyone who does not shave himself, with the exception that he shaves himself.” To avoid the contradiction in set theory, we need to define a universe U of objects, our universe of discourse, from which other sets are constructed.
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