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Single Idea 10887

[filed under theme 4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / l. Axiom of Specification ]

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.

Gist of Idea

Specification: Determinate totals of objects always make a set

Source

José L. Zalabardo (Introduction to the Theory of Logic [2000], §1.3)

Book Ref

Zalabardo,José L.: 'Introduction to the Theory of Logic' [Westview 2000], p.5

A Reaction

Compare the Axiom of Specification. Zalabardo says we may wish to consider sets of which we cannot specify the members.

The 18 ideas from 'Introduction to the Theory of Logic'

10886 Determinacy: an object is either in a set, or it isn't [Zalabardo]
10887 Specification: Determinate totals of objects always make a set [Zalabardo]
10888 Sets can be defined by 'enumeration', or by 'abstraction' (based on a property) [Zalabardo]
10889 The 'Cartesian Product' of two sets relates them by pairing every element with every element [Zalabardo]
10890 A 'partial ordering' is reflexive, antisymmetric and transitive [Zalabardo]
10891 If a set is defined by induction, then proof by induction can be applied to it [Zalabardo]
10893 Γ |= φ for sentences if φ is true when all of Γ is true [Zalabardo]
10892 We make a truth assignment to T and F, which may be true and false, but merely differ from one another [Zalabardo]
10894 A sentence-set is 'satisfiable' if at least one truth-assignment makes them all true [Zalabardo]
10895 'Logically true' (|= φ) is true for every truth-assignment [Zalabardo]
10896 Propositional logic just needs ¬, and one of ∧, ∨ and → [Zalabardo]
10897 A first-order 'sentence' is a formula with no free variables [Zalabardo]
10898 The semantics shows how truth values depend on instantiations of properties and relations [Zalabardo]
10901 Some formulas are 'satisfiable' if there is a structure and interpretation that makes them true [Zalabardo]
10900 Logically true sentences are true in all structures [Zalabardo]
10899 Γ |= φ if φ is true when all of Γ is true, for all structures and interpretations [Zalabardo]
10903 A structure models a sentence if it is true in the model, and a set of sentences if they are all true in the model [Zalabardo]
10902 We can do semantics by looking at given propositions, or by building new ones [Zalabardo]