Principles of Mathematics

Russell, B. (1903). Principles of Mathematics. Cambridge University Press.

Type

Book

Intellectual base

Textbook

Year

1903

DOI

10.4324/9780203864760

Link

https://archive.org/details/principlesmathe00russgoog

Cited from

mathematics, philosophy, systems_cybernetics

Cited by

8 citations across 8 artifacts.

Each citation links to the sentence it supports in the citing article.

Primes

• Asymmetry
  • It is more than the mere absence of symmetry: it names a directed imbalance in which one side is privileged, larger, prior, default, or more endowed than the other, so that the relation reads differently from each end, a framing Russell (1903) made canonical in his treatment of asymmetric relations as primitive relational facts.

    This sourceChapter XXVI ("Asymmetrical Relations") and the surrounding treatment of order (esp. §§207–219, and the discussion of the "sense" of a relation) argue that asymmetrical relations are irreducible primitive relational facts — that an asymmetric relation aRb fixes a direction not recoverable from its terms or from any monadic predicate, so that swapping the relata changes the proposition.

• Infinity
  • The argument is structurally rich. (a) It is constructive in the negative direction — it constructs an explicit witness s to the failure of any proposed enumeration, so its conclusion holds even in constructive frameworks that reject classical excluded middle. (b) It generalizes to Cantor's theorem — for any set A, the power set 2^A has strictly larger cardinality than A; the reals' uncountability is the special case A = ℕ, 2^ℕ ↔ [0,1]. © It generalizes further to the cumulative hierarchy of cardinalities — ℵ₀ < 2^ℵ₀ < 2^(2^ℵ₀) < … — establishing that there is no largest infinite cardinality. (d) It is the prototype for self-reference and undecidability arguments — Russell's (1903) paradox in Principles of Mathematics
• Paradox
  • Logic and foundations of mathematics
• Reflexivity (Self-Reference)
  • Listed in the references but not attached to a specific claim.
• Set and Membership
  • In classical (ZFC) set theory, unrestricted intensional specification is constrained to avoid paradox (Russell's paradox
• Tolerance Paradox
  • The self-application structure parallels Russell, Curry, and Gödel constructions, in which a rule applied to its own negation generates the system-defining difficulty, and reasoning tools from those domains transfer: stratify the levels so the open rule cannot be turned on itself, or restrict the rule's domain so the self-negating element is excluded by construction.

    This sourcePresents Russell's paradox — the set of all sets that are not members of themselves — arising from unrestricted comprehension.

Domain-specific

• Reflexive relation
  • … the carrier set X, homogeneous binary relation R, universal self-relation formula, empty-set convention and distinction from irreflexive, coreflexive or locally reflexive variants are explicit. to An applied instance preserves the same invariant under changed scale, notation, dataset, jurisdiction, or implementation..
• Unity of the Proposition
  • Russell repeatedly confronts how a verb used as a verb contributes a unity lost when it is nominalized

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