Ideas from 'Principia Mathematica' by B Russell/AN Whitehead [1913], by Theme Structure

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4. Formal Logic / B. Propositional Logic PL / 2. Tools of Propositional Logic / e. Axioms of PL
The best known axiomatization of PL is Whitehead/Russell, with four axioms and two rules
4. Formal Logic / F. Set Theory ST / 8. Critique of Set Theory
We regard classes as mere symbolic or linguistic conveniences
Russell denies extensional sets, because the null can't be a collection, and the singleton is just its element
5. Theory of Logic / E. Structures of Logic / 6. Relations in Logic
In 'Principia' a new abstract theory of relations appeared, and was applied
6. Mathematics / A. Nature of Mathematics / 3. Numbers / i. Reals from cuts
A real number is the class of rationals less than the number
6. Mathematics / B. Foundations for Mathematics / 4. Definitions of Number / a. Defining numbers
Russell takes numbers to be classes, but then reduces the classes to numerical quantifiers
6. Mathematics / C. Sources of Mathematics / 6. Logicism / a. Early logicism
Russell and Whitehead took arithmetic to be higher-order logic
Russell and Whitehead were not realists, but embraced nearly all of maths in logic
'Principia' lacks a precise statement of the syntax
6. Mathematics / C. Sources of Mathematics / 6. Logicism / b. Type theory
The ramified theory of types used propositional functions, and covered bound variables
The Russell/Whitehead type theory was limited, and was not really logic
6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
In 'Principia Mathematica', logic is exceeded in the axioms of infinity and reducibility, and in the domains
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / a. Constructivism
Russell and Whitehead consider the paradoxes to indicate that we create mathematical reality
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / d. Predicativism
To avoid vicious circularity Russell produced ramified type theory, but Ramsey simplified it
9. Objects / F. Identity among Objects / 7. Indiscernible Objects
An object is identical with itself, and no different indiscernible object can share that
12. Knowledge Sources / E. Direct Knowledge / 1. Intuition
Russell showed, through the paradoxes, that our basic logical intuitions are self-contradictory
19. Language / B. Meaning / 8. Meaning through Sentences
Only the act of judging completes the meaning of a statement