Sets, Functions, and Relations: Part I — Naïve Set Theory (Open Logic Project, 2026)

This is the Open Logic Project part artifact sets-functions-relations-complete, visible revision 9620cc7, built 2026-07-12 under CC BY 4.0. Its printed title is Part I — Naïve Set Theory. The file is a collaboratively authored production build rather than a polished standalone edition: source/debug identifiers and several unresolved ?? cross-references remain visible. Chapters 4–6 explicitly incorporate material from Tim Button's Open Set Theory.

Every one of the 79 physical pages was read through two independent native-text extractions and visually inspected at 1224×1584. Layout text, flow text, renders, and coverage each reconcile to the exact set 1..79. The final SHA-256 remained 6256aea8d23b6b90429a6607afeb3f8be65ccd3136574f348953275c61d493ae.

Concept and proof map #

Physical pages	Material	Proof strength in the artifact
3–12	Extensionality, subsets, power sets, operations, products, Russell's paradox	Definitions plus complete short proofs; empty-family intersection needs correction
13–23	Relations, property taxonomy, equivalence classes, orders, graphs, trees, relation operations and closures	Strong definitions; König's lemma is a sketch and closure minimality is omitted
24–32	Typed functions, injection/surjection/bijection, inverses, composition, partial functions	Strong core; left-inverse empty-domain edge case and Choice assumption matter
33–55	Enumerability, pairing, diagonalization, reductions, cardinal comparison, Cantor and Schröder–Bernstein	Several full proofs, but two configured enumeration conventions conflict slightly
56–70	Integers/rationals as quotients, Dedekind cuts, ordered fields, Cauchy-sequence construction	Selected proofs are strong; much well-definedness is exercised/deferred and Cauchy completeness is only a sketch
71–78	Dedekind infinity, closure, induction, and closure proof of Schröder–Bernstein	Full closure/induction chain and final proof, subject to explicit typing/foundational assumptions
79	Bibliography	Metadata only

The source is strongest where a precise distinction and a short proof reinforce each other: membership versus inclusion, relation versus graph, symmetry versus antisymmetry, range versus codomain, injectivity versus surjectivity, and equivalence classes versus quotient carriers.

Corrections and assumptions before reuse #

Location	Issue	Required correction
p. 8	Arbitrary intersection is stated for the empty family without an ambient universe	Require a nonempty family or specify the universe relative to which the intersection is taken
p. 17	Congruence is defined on integers but the displayed quotient changes the carrier to naturals	Keep the carrier fixed or explicitly restrict the relation
p. 22	Branch definition uses an undefined set symbol; König proof is informal	Repair the symbol and use a separately verified formulation/proof
p. 23	Positive powers are called the transitive closure without proving leastness	Prove both transitivity and minimality among transitive superrelations
pp. 29–30	“Every injection has a left inverse” fails for an empty domain and nonempty codomain	Add the nonempty-domain condition or split the empty case
p. 30	Arbitrary right inverses of surjections require selecting one preimage per output	Keep the Axiom of Choice assumption visible
p. 40	Countable union needs a uniform enumeration/choice hypothesis; inverse-pairing exercise is false as written	State assumptions and restrict decoding to the range or use a partial inverse
pp. 33–55	Enumeration first allows repetitions and later means a bijection	Choose one convention and translate the other explicitly
p. 48	Non-enumerability is equated with `N` being strictly smaller	Qualify by the required choice/infinity assumptions; Dedekind-finite infinite sets are a boundary case
p. 59	Rational-order prose reverses the difference sign	Use the correctly typed displayed definition and verify it from representatives
pp. 62, 66	Dedekind-cut arithmetic is stated more fully than it is proved	Do not treat the fragment as a complete field-construction proof
pp. 68–70	Theorems name Cauchy sequences where operations live on equivalence classes; completeness is a sketch	Correct the objects and supply representative-independence and omitted lemmas before any deep treatment
p. 72	Closure lemma leaves the carrier and function type implicit	State `f:A→A` and seed `o∈A` explicitly

Visual audit #

Forty-nine information-bearing visual/equation records were inspected. The high-value families are:

set-operation regions and a Cartesian-product grid;
relation matrices, a graph pair that differs only by an isolated vertex, and a rooted tree;
function, injection, surjection, bijection, and composition diagrams;
zig-zag/pairing tables and diagonal non-enumerability arrays;
quotient and Dedekind-cut equation chains;
the geometric √2 proof, Hilbert's Hotel, closure, induction, and the final Schröder–Bernstein construction.

No screenshot or source-debug annotation is retained. Diagrams used in GTK are independently reconstructed for article width, with explicit typing and the corrections above. Equation-heavy proofs are typeset locally rather than turned into raster images.

What GTK takes from it #

The source gives the house treatment a strong proof-oriented sequence:

state the typed object and its invariant;
show at least two representations;
derive the operation from membership or tuple semantics;
test a boundary case;
prove the claimed property or label it only as a sketch.

That sequence feeds Sets, Functions & Relations and Equivalence Relations & Partial Orders. The later cardinality and number-construction chapters are mapped here but are not forced into those two foundational entries.

Source claims audit #

Load-bearing source claim	Status	GTK treatment
Extensional equality can be proved by two inclusions	current	Retained with a direct element proof
Relations are subsets of products; functions add totality and uniqueness	current	Retained with typed domain/codomain and partial-function boundaries
Equivalence classes quotient a carrier; order notions require different axiom packages	current after typing repairs	Retained with partitions, quotients, covers, minimal/least distinctions, and counterexamples
Every injection has a left inverse	needs hypothesis	Empty-domain case made explicit
Every surjection has a right inverse	choice-sensitive	Choice assumption retained, not hidden
Positive relation powers form the transitive closure	incomplete in source	Minimality is independently proved
Pairing/diagonal constructions establish the presented cardinal results	mixed	Correct constructions retained only under one declared indexing/enumeration convention
Dedekind-cut and Cauchy fragments fully establish the real field	not established by this artifact alone	Mapped as future analysis material, not used as self-certifying proof
Closure supports induction and the final Schröder–Bernstein proof	current after typing repair	Retained as an optional deep proof chain

Concept and proof map #

Corrections and assumptions before reuse #

Visual audit #

What GTK takes from it #

Source claims audit #

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