Combining Texts

All the ideas for 'fragments/reports', 'works' and 'Grundlagen (Foundations of Theory of Manifolds)'

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5 ideas

4. Formal Logic / F. Set Theory ST / 1. Set Theory
15946 Cantor developed sets from a progression into infinity by addition, multiplication and exponentiation [Cantor, by Lavine]
     Full Idea: Cantor's development of set theory began with his discovery of the progression 0, 1, ....∞, ∞+1, ∞+2, ..∞x2, ∞x3, ...∞^2, ..∞^3, ...∞^∞, ...∞^∞^∞.....
     From: report of George Cantor (Grundlagen (Foundations of Theory of Manifolds) [1883]) by Shaughan Lavine - Understanding the Infinite VIII.2
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
15911 Ordinals are generated by endless succession, followed by a limit ordinal [Cantor, by Lavine]
     Full Idea: Ordinal numbers are generated by two principles: each ordinal has an immediate successor, and each unending sequence has an ordinal number as its limit (that is, an ordinal that is next after such a sequence).
     From: report of George Cantor (Grundlagen (Foundations of Theory of Manifolds) [1883]) by Shaughan Lavine - Understanding the Infinite III.4
10. Modality / A. Necessity / 8. Transcendental Necessity
490 Everything happens by reason and necessity [Leucippus]
     Full Idea: Nothing happens at random; everything happens out of reason and by necessity.
     From: Leucippus (fragments/reports [c.435 BCE], B002), quoted by (who?) - where?
27. Natural Reality / A. Classical Physics / 2. Thermodynamics / b. Heat
20645 Heat is a state of vibration, not a substance [Joule]
     Full Idea: We consider heat not as a substance but as a state of vibration.
     From: James Joule (works [1870]), quoted by Peter Watson - Convergence 01 'Nature's'
     A reaction: The puzzle is that giving accurate accounts of vibrations, heat and movement require a quantitative substance, energy. But all we have here is movement, and the denial of a substance. Energy is 'nature's currency system'.
20972 Joule showed that energy converts to heat, and heat to energy [Joule, by Papineau]
     Full Idea: James Joule established the equivalence of heat and mechanical energy, in the sense of showing that a specific amount of heat will always be produced by the expenditure of a given amount of energy, and vice versa.
     From: report of James Joule (works [1870]) by David Papineau - Thinking about Consciousness App 4.2
     A reaction: This was a major step towards the law of conservation of energy.