2012 | Introduction to Mathematical Logic |
History B.4 | p.13 | 17742 | Scotus based modality on semantic consistency, instead of on what the future could allow |
History E.1.3 | p.33 | 17747 | A 'model' of a theory specifies interpreting a language in a domain to make all theorems true |
History E.2 | p.34 | 17748 | The L-S Theorem says no theory (even of reals) says more than a natural number theory |
History E.2.1 | p.35 | 17749 | Post proved the consistency of propositional logic in 1921 |
History E.2.2 | p.36 | 17751 | Gödel proved the completeness of first order predicate logic in 1930 |
History Intro | p.2 | 17741 | To determine the patterns in logic, one must identify its 'building blocks' |
1.1 | p.45 | 17752 | The empty set is useful for defining sets by properties, when the members are not yet known |
1.1 | p.45 | 17753 | The empty set avoids having to take special precautions in case members vanish |
2.1.1 | p.69 | 17754 | Inductive proof depends on the choice of the ordering |
2.3 | p.88 | 17756 | The union of finite ordinals is the first 'limit ordinal'; 2ω is the second... |
2.3 | p.88 | 17755 | Ordinals are the empty set, union with the singleton, and any arbitrary union of ordinals |
2.3 | p.89 | 17758 | Ordinals are transitive sets of transitive sets; or transitive sets totally ordered by inclusion |
2.3 | p.89 | 17760 | Two infinite ordinals can represent a single infinite cardinal |
2.3 | p.89 | 17757 | Members of ordinals are ordinals, and also subsets of ordinals |
2.3 | p.89 | 17759 | Ordinals play the central role in set theory, providing the model of well-ordering |
4.1 | p.118 | 17761 | A compact axiomatisation makes it possible to understand a field as a whole |
4.1 | p.122 | 17763 | Axiomatic systems are purely syntactic, and do not presuppose any interpretation |
4.1 | p.122 | 17762 | In non-Euclidean geometry, all Euclidean theorems are valid that avoid the fifth postulate |
5.1 | p.143 | 17764 | Boolean connectives are interpreted as functions on the set {1,0} |
7 Intro | p.183 | 17765 | Propositional language can only relate statements as the same or as different |