TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-01‡ rests on unproved material (inherited)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

‡ Rests on 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Cohen's first model: an infinite Dedekind-finite set of reals, ‡ Sierpiński 1947: the generalised continuum hypothesis implies the Axiom of Choice and ‡ The continuum hypothesis and its generalisation are independent of ZFC. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

Set is cartesian closed

Statement

The category $S e t$ is cartesian closed. For sets $B, C$ , an exponential object of $C$ by $B$ is the function set $C^{B} = {Hom}_{S e t} (B, C)$ with evaluation $(f, b) \mapsto f (b)$ .

Facts & Assumptions

Given: Sets $A, B, C$ .

[L1]

A cartesian closed category has finite products and exponentials, equivalently a right adjoint to each product functor $- \times B$ (Cartesian closed category).

[L2]

$S e t$ has all small limits, hence in particular finite products (Set has all small limits, realized as compatible tuples in a set-indexed product).

[L3]

In $S e t$ , currying gives a natural bijection $S e t (A \times B, C) ≅ S e t (A, C^{B})$ (Currying gives the adjunction $- \times A ⊣ (-)^{A}$ in $S e t$ ).

Proof

technique · direct

1.1

By [L2], $S e t$ has finite products.

givenL2

1.2

Let $C^{B}$ be the set of functions $B \to C$ , and let $ev : C^{B} \times B \to C$ send $(f, b)$ to $f (b)$ . By [L3], every map $A \times B \to C$ corresponds naturally and bijectively to a map $A \to C^{B}$ . This is exactly the universal property of an exponential object.

givenL3

2.1

Step 1.1 gives the finite-product part of [L1], and step 1.2 gives the right adjoint to $- \times B$ for each $B$ . Therefore $S e t$ is cartesian closed.

step 1.1step 1.2L1∎

Depends on

Used by

A function set computed as an exponential object Example

Dependency tree · two levels

13 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Emily Riehl, Category Theory in Context, 2nd ed., Example 4.4.9 (standard reference, not scraped)
S. Mac Lane, Categories for the Working Mathematician, 2nd ed., IV.6 (standard reference, not scraped)