Principles of Mathematics¶
Russell, B. (1903). Principles of Mathematics. Cambridge University Press.
Cited by¶
9 citations across 9 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.
Supported in partVerified against the work's full text
Russell's early chapters define asymmetrical relations as distinct from merely not-symmetrical ones and reject treating relations as less ultimate than classes.
“When the converse is incompatible with the original relation, as in such cases as greater and less, I call the, ... in intermediate cases, not-symmetrical.”
- 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.
- Infinity
- The argument is structurally rich. (a) It is constructive in the negative direction — it constructs an explicit witness
sto 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 setA, the power set2^Ahas strictly larger cardinality thanA; the reals' uncountability is the special caseA = 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 MathematicsSupported in partVerified against the work's full text
Russell's section 100 states the paradox and ties it to Cantor's no-greatest-cardinal proof; the diagonal-argument claims (a)-(c) and the 'prototype' role are not in this source.
“I was led to it in the endeavour to reconcile Cantor's proof that there can be no greatest cardinal number with the very plausible supposition that the class of all terms”
- The argument is structurally rich. (a) It is constructive in the negative direction — it constructs an explicit witness
- Paradox
- Logic and foundations of mathematics
SupportedVerified against the work's full text
Russell's 1903 book, aimed at deriving pure mathematics from logic, states the contradiction of classes that are not members of themselves in Chapter X.
“If it is, then it is one of the classes which, as ones, are not members of themselves as many, and vice versa.”
- Logic and foundations of mathematics
- Reflexivity (Self-Reference)
- Listed in the references but not attached to a specific claim.
- Set and Membership
- 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.
Supported in partVerified against the work's full text
Russell presents his paradox and the remedy of stratifying by logical type, but the text does not mention Curry or Godel or the claimed transfer of tools.
“It is the distinction of logical types that is the key to the whole mystery”
- 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.
Domain-specific¶
- Propositional function
- Reflexive relation
- Unity of the Proposition
- Russell repeatedly confronts how a verb used as a verb contributes a unity lost when it is nominalized
SupportedVerified against the source
- Russell repeatedly confronts how a verb used as a verb contributes a unity lost when it is nominalized
Verification¶
Does it exist? Confirmed. This work's DOI resolves to a registered record, which fixes its identity. That is all it fixes.
Does it back the claim? Read against the text for 5 of 9 citations: 2 supported, 3 supported in part. Each verdict is shown under its citation below, with what in the work backs the sentence.
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Links previously used in the corpus¶
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