Combining Philosophers

Ideas for Lynch,MP/Glasgow,JM, Arend Heyting and Jos L. Zalabardo

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5 ideas

4. Formal Logic / F. Set Theory ST / 1. Set Theory
10888 Sets can be defined by 'enumeration', or by 'abstraction' (based on a property) [Zalabardo]
     Full Idea: We can define a set by 'enumeration' (by listing the items, within curly brackets), or by 'abstraction' (by specifying the elements as instances of a property), pretending that they form a determinate totality. The latter is written {x | x is P}.
     From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §1.3)
4. Formal Logic / F. Set Theory ST / 2. Mechanics of Set Theory / b. Terminology of ST
10889 The 'Cartesian Product' of two sets relates them by pairing every element with every element [Zalabardo]
     Full Idea: The 'Cartesian Product' of two sets, written A x B, is the relation which pairs every element of A with every element of B. So A x B = { | x ∈ A and y ∈ B}.
     From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §1.6)
10890 A 'partial ordering' is reflexive, antisymmetric and transitive [Zalabardo]
     Full Idea: A binary relation in a set is a 'partial ordering' just in case it is reflexive, antisymmetric and transitive.
     From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §1.6)
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
10886 Determinacy: an object is either in a set, or it isn't [Zalabardo]
     Full Idea: Principle of Determinacy: For every object a and every set S, either a is an element of S or a is not an element of S.
     From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §1.2)
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / l. Axiom of Specification
10887 Specification: Determinate totals of objects always make a set [Zalabardo]
     Full Idea: Principle of Specification: Whenever we can specify a determinate totality of objects, we shall say that there is a set whose elements are precisely the objects that we have specified.
     From: José L. Zalabardo (Introduction to the Theory of Logic [2000], §1.3)
     A reaction: Compare the Axiom of Specification. Zalabardo says we may wish to consider sets of which we cannot specify the members.