Combining Texts

All the ideas for 'Intuitionism and Formalism', 'First-Order Logic' and 'Last Fragments'

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18 ideas

4. Formal Logic / E. Nonclassical Logics / 7. Paraconsistency
12452 Our dislike of contradiction in logic is a matter of psychology, not mathematics [Brouwer]
5. Theory of Logic / A. Overview of Logic / 1. Overview of Logic
10282 Logic is the study of sound argument, or of certain artificial languages (or applying the latter to the former) [Hodges,W]
5. Theory of Logic / I. Semantics of Logic / 1. Semantics of Logic
10283 A formula needs an 'interpretation' of its constants, and a 'valuation' of its variables [Hodges,W]
10284 There are three different standard presentations of semantics [Hodges,W]
10285 I |= φ means that the formula φ is true in the interpretation I [Hodges,W]
5. Theory of Logic / J. Model Theory in Logic / 3. Löwenheim-Skolem Theorems
10288 Down Löwenheim-Skolem: if a countable language has a consistent theory, that has a countable model [Hodges,W]
10289 Up Löwenheim-Skolem: if infinite models, then arbitrarily large models [Hodges,W]
5. Theory of Logic / K. Features of Logics / 6. Compactness
10287 If a first-order theory entails a sentence, there is a finite subset of the theory which entails it [Hodges,W]
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / g. Applying mathematics
12451 Scientific laws largely rest on the results of counting and measuring [Brouwer]
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
10286 A 'set' is a mathematically well-behaved class [Hodges,W]
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / b. Intuitionism
12454 Intuitionists only accept denumerable sets [Brouwer]
12453 Neo-intuitionism abstracts from the reuniting of moments, to intuit bare two-oneness [Brouwer]
14. Science / B. Scientific Theories / 1. Scientific Theory
19594 General statements about nature are not valid [Novalis]
17. Mind and Body / A. Mind-Body Dualism / 8. Dualism of Mind Critique
19596 The whole body is involved in the formation of thoughts [Novalis]
19. Language / A. Nature of Meaning / 5. Meaning as Verification
10117 Intuitonists in mathematics worried about unjustified assertion, as well as contradiction [Brouwer, by George/Velleman]
25. Social Practice / E. Policies / 5. Education / d. Study of history
19593 Persons are shaped by a life history; splendid persons are shaped by world history [Novalis]
26. Natural Theory / A. Speculations on Nature / 1. Nature
19595 Nature is a whole, and its individual parts cannot be wholly understood [Novalis]
26. Natural Theory / A. Speculations on Nature / 4. Mathematical Nature
19592 The basic relations of nature are musical [Novalis]