Combining Philosophers
Ideas for Archimedes, Michal Walicki and Feferman / Feferman
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11 ideas
4. Formal Logic / B. Propositional Logic PL / 1. Propositional Logic
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Post proved the consistency of propositional logic in 1921 [Walicki]
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Propositional language can only relate statements as the same or as different [Walicki]
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4. Formal Logic / B. Propositional Logic PL / 3. Truth Tables
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Boolean connectives are interpreted as functions on the set {1,0} [Walicki]
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4. Formal Logic / F. Set Theory ST / 3. Types of Set / b. Empty (Null) Set
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The empty set is useful for defining sets by properties, when the members are not yet known [Walicki]
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The empty set avoids having to take special precautions in case members vanish [Walicki]
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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX
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Cantor's theories needed the Axiom of Choice, but it has led to great controversy [Feferman/Feferman]
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The Axiom of Choice is consistent with the other axioms of set theory [Feferman/Feferman]
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Axiom of Choice: a set exists which chooses just one element each of any set of sets [Feferman/Feferman]
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Platonist will accept the Axiom of Choice, but others want criteria of selection or definition [Feferman/Feferman]
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The Trichotomy Principle is equivalent to the Axiom of Choice [Feferman/Feferman]
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4. Formal Logic / F. Set Theory ST / 6. Ordering in Sets
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Ordinals play the central role in set theory, providing the model of well-ordering [Walicki]
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