Combining Texts
Ideas for
'works', 'Elements of Set Theory' and 'Set Theory and related topics (2nd ed)'
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17 ideas
4. Formal Logic / F. Set Theory ST / 1. Set Theory
15901
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Trying to represent curves, we study arbitrary functions, leading to the ordinals, which produces set theory [Cantor, by Lavine]
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4. Formal Logic / F. Set Theory ST / 2. Mechanics of Set Theory / b. Terminology of ST
13206
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A 'linear or total ordering' must be transitive and satisfy trichotomy [Enderton]
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13201
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∈ says the whole set is in the other; ⊆ says the members of the subset are in the other [Enderton]
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13204
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The 'ordered pair' <x,y> is defined to be {{x}, {x,y}} [Enderton]
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4. Formal Logic / F. Set Theory ST / 2. Mechanics of Set Theory / c. Basic theorems of ST
18098
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Cantor proved that all sets have more subsets than they have members [Cantor, by Bostock]
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13444
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Cantor's Theorem: for any set x, its power set P(x) has more members than x [Cantor, by Hart,WD]
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4. Formal Logic / F. Set Theory ST / 3. Types of Set / b. Empty (Null) Set
13200
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Note that {Φ} =/= Φ, because Φ ∈ {Φ} but Φ ∉ Φ [Enderton]
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13199
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The empty set may look pointless, but many sets can be constructed from it [Enderton]
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4. Formal Logic / F. Set Theory ST / 3. Types of Set / c. Unit (Singleton) Sets
13203
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The singleton is defined using the pairing axiom (as {x,x}) [Enderton]
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15505
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If a set is 'a many thought of as one', beginners should protest against singleton sets [Cantor, by Lewis]
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4. Formal Logic / F. Set Theory ST / 3. Types of Set / d. Infinite Sets
10701
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Cantor showed that supposed contradictions in infinity were just a lack of clarity [Cantor, by Potter]
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10865
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The continuum is the powerset of the integers, which moves up a level [Cantor, by Clegg]
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4. Formal Logic / F. Set Theory ST / 3. Types of Set / e. Equivalence classes
8920
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Equivalence relations are reflexive, symmetric and transitive, and classify similar objects [Lipschutz]
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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / d. Axiom of Unions III
13016
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The Axiom of Union dates from 1899, and seems fairly obvious [Cantor, by Maddy]
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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / h. Axiom of Replacement VII
13202
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Fraenkel added Replacement, to give a theory of ordinal numbers [Enderton]
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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX
13205
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We can only define functions if Choice tells us which items are involved [Enderton]
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4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / b. Combinatorial sets
14199
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Cantor's sets were just collections, but Dedekind's were containers [Cantor, by Oliver/Smiley]
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