Ideas of Michal Walicki, by Theme
[Norwegian, fl. 2012, At the University of Bergen, Norway.]
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4. Formal Logic / B. Propositional Logic PL / 1. Propositional Logic
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17749
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Post proved the consistency of propositional logic in 1921
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17765
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Propositional language can only relate statements as the same or as different
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4. Formal Logic / B. Propositional Logic PL / 3. Truth Tables
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17764
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Boolean connectives are interpreted as functions on the set {1,0}
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4. Formal Logic / F. Set Theory ST / 3. Types of Set / b. Empty (Null) Set
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17752
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The empty set is useful for defining sets by properties, when the members are not yet known
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17753
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The empty set avoids having to take special precautions in case members vanish
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4. Formal Logic / F. Set Theory ST / 6. Ordering in Sets
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17759
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Ordinals play the central role in set theory, providing the model of well-ordering
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5. Theory of Logic / A. Overview of Logic / 1. Overview of Logic
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17741
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To determine the patterns in logic, one must identify its 'building blocks'
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5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models
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17747
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A 'model' of a theory specifies interpreting a language in a domain to make all theorems true
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5. Theory of Logic / J. Model Theory in Logic / 3. Löwenheim-Skolem Theorems
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17748
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The L-S Theorem says no theory (even of reals) says more than a natural number theory
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5. Theory of Logic / K. Features of Logics / 1. Axiomatisation
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17763
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Axiomatic systems are purely syntactic, and do not presuppose any interpretation
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17761
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A compact axiomatisation makes it possible to understand a field as a whole
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6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
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17758
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Ordinals are transitive sets of transitive sets; or transitive sets totally ordered by inclusion
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17757
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Members of ordinals are ordinals, and also subsets of ordinals
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17755
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Ordinals are the empty set, union with the singleton, and any arbitrary union of ordinals
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17756
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The union of finite ordinals is the first 'limit ordinal'; 2ω is the second...
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17760
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Two infinite ordinals can represent a single infinite cardinal
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6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Geometry
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17762
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In non-Euclidean geometry, all Euclidean theorems are valid that avoid the fifth postulate
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6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / f. Mathematical induction
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17754
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Inductive proof depends on the choice of the ordering
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10. Modality / A. Necessity / 2. Nature of Necessity
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17742
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Scotus based modality on semantic consistency, instead of on what the future could allow
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