Combining Philosophers

Ideas for Hermarchus, Michal Walicki and Bernard Linsky

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4 ideas

4. Formal Logic / F. Set Theory ST / 3. Types of Set / b. Empty (Null) Set
17752 The empty set is useful for defining sets by properties, when the members are not yet known [Walicki]
     Full Idea: The empty set is mainly a mathematical convenience - defining a set by describing the properties of its members in an involved way, we may not know from the very beginning what its members are.
     From: Michal Walicki (Introduction to Mathematical Logic [2012], 1.1)
17753 The empty set avoids having to take special precautions in case members vanish [Walicki]
     Full Idea: Without the assumption of the empty set, one would often have to take special precautions for the case where a set happened to contain no elements.
     From: Michal Walicki (Introduction to Mathematical Logic [2012], 1.1)
     A reaction: Compare the introduction of the concept 'zero', where special precautions are therefore required. ...But other special precautions are needed without zero. Either he pays us, or we pay him, or ...er. Intersecting sets need the empty set.
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / p. Axiom of Reducibility
21705 Reducibility says any impredicative function has an appropriate predicative replacement [Linsky,B]
     Full Idea: The Axiom of Reducibility avoids impredicativity, by asserting that for any predicate of given arguments defined by quantifying over higher-order functions or classes, there is another co-extensive but predicative function of the same type of arguments.
     From: Bernard Linsky (Russell's Metaphysical Logic [1999], 1)
     A reaction: Eventually the axiom seemed too arbitrary, and was dropped. Linsky's book explores it.
4. Formal Logic / F. Set Theory ST / 6. Ordering in Sets
17759 Ordinals play the central role in set theory, providing the model of well-ordering [Walicki]
     Full Idea: Ordinals play the central role in set theory, providing the paradigmatic well-orderings.
     From: Michal Walicki (Introduction to Mathematical Logic [2012], 2.3)
     A reaction: When you draw the big V of the iterative hierarchy of sets (built from successive power sets), the ordinals are marked as a single line up the middle, one ordinal for each level.