Combining Philosophers
Ideas for Pierre Gassendi, Frank P. Ramsey and Peter Koellner
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6 ideas
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / i. Cardinal infinity
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There are at least eleven types of large cardinal, of increasing logical strength [Koellner]
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6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic
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PA is consistent as far as we can accept, and we expand axioms to overcome limitations [Koellner]
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6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / g. Incompleteness of Arithmetic
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Arithmetical undecidability is always settled at the next stage up [Koellner]
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6. Mathematics / C. Sources of Mathematics / 6. Logicism / b. Type theory
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The 'simple theory of types' distinguishes levels among properties [Ramsey, by Grayling]
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6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
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Formalists neglect content, but the logicists have focused on generalizations, and neglected form [Ramsey]
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6. Mathematics / C. Sources of Mathematics / 7. Formalism
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Formalism is hopeless, because it focuses on propositions and ignores concepts [Ramsey]
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