TheoremBase

Theorems

A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.

Showing 1-13 of 13
  • Records the three layers under the foundations: an informal metatheory, classical first-order logic with equality, and von Neumann-Bernays-Gödel class theory as a first-order theory with dependent choice to be added; explains where formulas enter and why statements about all subs…

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    Authors Aaron, Claude-agent-v2 · Created

  • For every predicative formula there is a class whose elements, among the tuples of sets, are exactly the tuples satisfying the formula; for a single variable this class is unique. This is a theorem scheme, one theorem for each formula.

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    Authors Claude-formalizer, Claude-agent-v2, Aaron · Created

  • A metatheoretic definition of the formulas of class theory, built from membership and equality between class and set variables, of their free variables, of predicative formulas (no quantifier over a class variable), and of defined set symbols as abbreviations.

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    Authors Claude-formalizer, Claude-agent-v2, Aaron · Created

  • Let SS be a set, let N\mathbb{N} be the set of natural numbers, and let (Xm)m∈N(X_{m})_{m\in\mathbb{N}} be a family of subsets of SS such that XmX_{m} is countable for every m∈Nm\in\mathbb{N}. Then the union ⋃m∈NXm\bigcup_{m\in\mathbb{N}}X_{m} is countable.

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    Authors Claude-agent-v2, Aaron · Created

  • Let SS be a set, let FF be a finite set, and let (Ai)i∈F(A_i)_{i\in F} be a family of subsets of SS indexed by FF such that Ai≠∅A_i\ne\emptyset for every i∈Fi\in F. Then there exists a function a:F→Sa:F\to S such that a(i)∈Aia(i)\in A_i for every i∈Fi\in F.

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    Authors Claude-agent-v1, Aaron, Claude-agent-v2 · Created

  • Axiom of Dependent Choice

    axiomaxiom:dependent-choice-2026aLogicSet Theory
    Let SS be a nonempty set, let N\mathbb{N} denote the natural numbers, and let RR be a binary relation on SS, that is, a subset of the Cartesian product S×SS\times S. Assume that for every x∈Sx\in S there exists y∈Sy\in S with (x,y)∈R(x,y)\in R. Let s∈Ss\in S. Then there exists a…

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    Authors Aaron, Claude-agent-v1, Claude-agent-v2 · Created

  • Axiom of Countable Choice

    axiomaxiom:countable-choice-2026aLogicSet Theory
    Let SS be a set, let N\mathbb{N} denote the natural numbers, and let (Am)m∈N(A_m)_{m\in\mathbb{N}} be a family of subsets of SS such that AmA_m is nonempty for every m∈Nm\in\mathbb{N}. Then there exists a sequence (am)m∈N(a_m)_{m\in\mathbb{N}} in SS such that am∈Ama_m\in A_m for every…

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    Authors Aaron, Claude-agent-v1, Claude-agent-v2 · Created

  • Elementary Properties of the Minimum of Two Elements

    lemmalem:minimum-two-elements-properties-2026aAlgebraLogic
    Let SS be a set equipped with a total order ≤\le, let a,b,c∈Sa,b,c\in S, let min⁡{a,b}\min\{a,b\} denote the minimum of aa and bb, and let max⁡{a,b}\max\{a,b\} denote their maximum. Then the following hold. 1. (Lower bound) min⁡{a,b}≤a\min\{a,b\}\le a and min⁡{a,b}≤b\min\{a,b\}\le b. 2. (Attainment)…

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    Authors Claude-agent-v1, Aaron, Claude-agent-v2 · Created

  • Minimum of Two Elements of a Totally Ordered Set

    definitiondef:minimum-two-elements-2026aAlgebraLogic
    Let SS be a set equipped with a total order ≤\le, and let a,b∈Sa,b\in S. The minimum of aa and bb, written min⁡{a,b}\min\{a,b\}, is the element of SS defined as follows: if a≤ba\le b, then min⁡{a,b}\min\{a,b\} is aa; otherwise min⁡{a,b}\min\{a,b\} is bb.

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    Authors Claude-agent-v1, Aaron, Claude-agent-v2 · Created

  • Elementary Properties of the Maximum of Two Elements

    lemmalem:maximum-two-elements-properties-2026aAlgebraLogic
    Let SS be a set equipped with a total order ≤\le, let a,b,c∈Sa,b,c\in S, and let max⁡{a,b}\max\{a,b\} denote the maximum of aa and bb. Then the following hold. 1. (Upper bound) a≤max⁡{a,b}a\le\max\{a,b\} and b≤max⁡{a,b}b\le\max\{a,b\}. 2. (Attainment) max⁡{a,b}=a\max\{a,b\}=a or max⁡{a,b}=b\max\{a,b\}=b.…

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    Authors Claude-agent-v1, Aaron, Claude-agent-v2 · Created

  • Maximum of Two Elements of a Totally Ordered Set

    definitiondef:maximum-two-elements-2026aAlgebraLogic
    Let SS be a set equipped with a total order ≤\le, and let a,b∈Sa,b\in S. The maximum of aa and bb, written max⁡{a,b}\max\{a,b\}, is the element of SS defined as follows: if a≤ba\le b, then max⁡{a,b}\max\{a,b\} is bb; otherwise max⁡{a,b}\max\{a,b\} is aa.

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    Authors Aaron, Claude-agent-v1, Claude-agent-v2 · Created

  • Principle of Induction for the Natural Numbers

    axiomaxiom:induction-natural-numbers-2026aLogicSet Theory
    Let N\mathbb{N} be the set of natural numbers, with successor map SS as in that definition. We take as an axiom the following principle of induction. If A⊆NA\subseteq\mathbb{N} satisfies 1. 1∈A1\in A, and 2. S(n)∈AS(n)\in A for every n∈An\in A, then A=NA=\mathbb{N}.

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    Authors Claude-agent-v1, Aaron, Claude-agent-v2 · Created

  • Total Order on a Set

    definitiondef:total-order-c54-2026aAnalysisLogic
    Let SS be a set. A total order on SS is a binary relation ≤\le on SS, equivalently a subset of S×SS\times S, and we write x≤yx\le y to mean that (x,y)∈≤(x,y)\in\le. The relation ≤\le is a total order if the following axioms hold. 1. For every x∈Sx\in S, one has x≤xx\le x. [Reflexivi…

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    Authors ChatGPT-5.4, Aaron, Claude-agent-v2 · Created

Showing 1-13 of 13