Combining Philosophers

All the ideas for Charles Parsons, Hermann Weyl and Feferman / Feferman

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22 ideas

2. Reason / D. Definition / 1. Definitions
15927 Definition just needs negation, known variables, conjunction, disjunction, substitution and quantification [Weyl, by Lavine]
4. Formal Logic / D. Modal Logic ML / 4. Alethic Modal Logic
9470 Modal logic is not an extensional language [Parsons,C]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX
10147 The Axiom of Choice is consistent with the other axioms of set theory [Feferman/Feferman]
10148 Axiom of Choice: a set exists which chooses just one element each of any set of sets [Feferman/Feferman]
10149 Platonist will accept the Axiom of Choice, but others want criteria of selection or definition [Feferman/Feferman]
10150 The Trichotomy Principle is equivalent to the Axiom of Choice [Feferman/Feferman]
13418 The old problems with the axiom of choice are probably better ascribed to the law of excluded middle [Parsons,C]
10146 Cantor's theories needed the Axiom of Choice, but it has led to great controversy [Feferman/Feferman]
5. Theory of Logic / G. Quantification / 4. Substitutional Quantification
9469 Substitutional existential quantifier may explain the existence of linguistic entities [Parsons,C]
9468 On the substitutional interpretation, '(∃x) Fx' is true iff a closed term 't' makes Ft true [Parsons,C]
5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models
10158 A structure is a 'model' when the axioms are true. So which of the structures are models? [Feferman/Feferman]
10162 Tarski and Vaught established the equivalence relations between first-order structures [Feferman/Feferman]
5. Theory of Logic / J. Model Theory in Logic / 3. Löwenheim-Skolem Theorems
10159 Löwenheim-Skolem Theorem, and Gödel's completeness of first-order logic, the earliest model theory [Feferman/Feferman]
10160 Löwenheim-Skolem says if the sentences are countable, so is the model [Feferman/Feferman]
5. Theory of Logic / K. Features of Logics / 4. Completeness
10161 If a sentence holds in every model of a theory, then it is logically derivable from the theory [Feferman/Feferman]
5. Theory of Logic / K. Features of Logics / 7. Decidability
10156 'Recursion theory' concerns what can be solved by computing machines [Feferman/Feferman]
10155 Both Principia Mathematica and Peano Arithmetic are undecidable [Feferman/Feferman]
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / c. Counting procedure
17447 Parsons says counting is tagging as first, second, third..., and converting the last to a cardinal [Parsons,C, by Heck]
6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / c. Against mathematical empiricism
18201 General principles can be obvious in mathematics, but bold speculations in empirical science [Parsons,C]
6. Mathematics / C. Sources of Mathematics / 8. Finitism
13419 If functions are transfinite objects, finitists can have no conception of them [Parsons,C]
7. Existence / D. Theories of Reality / 11. Ontological Commitment / e. Ontological commitment problems
13417 If a mathematical structure is rejected from a physical theory, it retains its mathematical status [Parsons,C]
26. Natural Theory / D. Laws of Nature / 8. Scientific Essentialism / e. Anti scientific essentialism
10246 The limit of science is isomorphism of theories, with essences a matter of indifference [Weyl]