Combining Texts
Ideas for
'The Sayings of Confucius', 'Understanding the Infinite' and 'Goodbye Descartes'
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18 ideas
4. Formal Logic / A. Syllogistic Logic / 2. Syllogistic Logic
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8081
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'No councillors are bankers' and 'All bankers are athletes' implies 'Some athletes are not councillors' [Devlin]
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4. Formal Logic / B. Propositional Logic PL / 1. Propositional Logic
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8085
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Modern propositional inference replaces Aristotle's 19 syllogisms with modus ponens [Devlin]
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4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / e. Axioms of PL
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8086
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Predicate logic retains the axioms of propositional logic [Devlin]
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4. Formal Logic / F. Set Theory ST / 1. Set Theory
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15945
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Second-order set theory just adds a version of Replacement that quantifies over functions [Lavine]
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4. Formal Logic / F. Set Theory ST / 2. Mechanics of Set Theory / b. Terminology of ST
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15914
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An 'upper bound' is the greatest member of a subset; there may be several of these, so there is a 'least' one [Lavine]
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4. Formal Logic / F. Set Theory ST / 3. Types of Set / a. Types of set
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15921
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Collections of things can't be too big, but collections by a rule seem unlimited in size [Lavine]
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4. Formal Logic / F. Set Theory ST / 3. Types of Set / d. Infinite Sets
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15937
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Those who reject infinite collections also want to reject the Axiom of Choice [Lavine]
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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / g. Axiom of Powers VI
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15936
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The Power Set is just the collection of functions from one collection to another [Lavine]
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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / h. Axiom of Replacement VII
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15899
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Replacement was immediately accepted, despite having very few implications [Lavine]
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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / i. Axiom of Foundation VIII
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15930
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Foundation says descending chains are of finite length, blocking circularity, or ungrounded sets [Lavine]
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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX
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15920
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Pure collections of things obey Choice, but collections defined by a rule may not [Lavine]
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15898
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The controversy was not about the Axiom of Choice, but about functions as arbitrary, or given by rules [Lavine]
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4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / c. Logical sets
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15919
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The 'logical' notion of class has some kind of definition or rule to characterise the class [Lavine]
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4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / e. Iterative sets
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15900
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The iterative conception of set wasn't suggested until 1947 [Lavine]
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15931
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The iterative conception needs the Axiom of Infinity, to show how far we can iterate [Lavine]
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15932
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The iterative conception doesn't unify the axioms, and has had little impact on mathematical proofs [Lavine]
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4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / f. Limitation of Size
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15933
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Limitation of Size: if it's the same size as a set, it's a set; it uses Replacement [Lavine]
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4. Formal Logic / F. Set Theory ST / 6. Ordering in Sets
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15913
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A collection is 'well-ordered' if there is a least element, and all of its successors can be identified [Lavine]
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