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Single Idea 17747

[filed under theme 5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models ]

Full Idea

A specification of a domain of objects, and of the rules for interpreting the symbols of a logical language in this domain such that all the theorems of the logical theory are true is said to be a 'model' of the theory.

Gist of Idea

A 'model' of a theory specifies interpreting a language in a domain to make all theorems true

Source

Michal Walicki (Introduction to Mathematical Logic [2012], History E.1.3)

Book Ref

Walicki,Michal: 'Introduction to Mathematical Logic' [World Scientific 2012], p.33

A Reaction

The basic ideas of this emerged 1915-30, but it needed Tarski's account of truth to really get it going.

The 19 ideas from 'Introduction to Mathematical Logic'

17742 Scotus based modality on semantic consistency, instead of on what the future could allow [Walicki]
17747 A 'model' of a theory specifies interpreting a language in a domain to make all theorems true [Walicki]
17748 The L-S Theorem says no theory (even of reals) says more than a natural number theory [Walicki]
17749 Post proved the consistency of propositional logic in 1921 [Walicki]
17741 To determine the patterns in logic, one must identify its 'building blocks' [Walicki]
17752 The empty set is useful for defining sets by properties, when the members are not yet known [Walicki]
17753 The empty set avoids having to take special precautions in case members vanish [Walicki]
17754 Inductive proof depends on the choice of the ordering [Walicki]
17759 Ordinals play the central role in set theory, providing the model of well-ordering [Walicki]
17758 Ordinals are transitive sets of transitive sets; or transitive sets totally ordered by inclusion [Walicki]
17755 Ordinals are the empty set, union with the singleton, and any arbitrary union of ordinals [Walicki]
17756 The union of finite ordinals is the first 'limit ordinal'; 2ω is the second... [Walicki]
17760 Two infinite ordinals can represent a single infinite cardinal [Walicki]
17757 Members of ordinals are ordinals, and also subsets of ordinals [Walicki]
17762 In non-Euclidean geometry, all Euclidean theorems are valid that avoid the fifth postulate [Walicki]
17761 A compact axiomatisation makes it possible to understand a field as a whole [Walicki]
17763 Axiomatic systems are purely syntactic, and do not presuppose any interpretation [Walicki]
17764 Boolean connectives are interpreted as functions on the set {1,0} [Walicki]
17765 Propositional language can only relate statements as the same or as different [Walicki]