Combining Texts

Ideas for 'fragments/reports', 'Introduction to Mathematical Logic' and 'Models and Reality'

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5 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 / o. Axiom of Constructibility V = L
13655 The Löwenheim-Skolem theorems show that whether all sets are constructible is indeterminate [Putnam, by Shapiro]
     Full Idea: Putnam claims that the Löwenheim-Skolem theorems indicate that there is no 'fact of the matter' whether all sets are constructible.
     From: report of Hilary Putnam (Models and Reality [1977]) by Stewart Shapiro - Foundations without Foundationalism
     A reaction: [He refers to the 4th and 5th pages of Putnam's article] Shapiro offers (p.109) a critique of Putnam's proposal.
9915 V = L just says all sets are constructible [Putnam]
     Full Idea: V = L just says all sets are constructible. L is the class of all constructible sets, and V is the universe of all sets.
     From: Hilary Putnam (Models and Reality [1977], p.425)
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.