Foundations

Summary

Foundations (Part VI: Defining) Chapters: 18, 19, 20, 21 Plane Position: (-0.6, 0.6) radius 0.35 Primitives: 55

Pure mathematical structure. Sets, groups, rings, fields, topology — the formal bedrock everything else rests on.

Key Concepts: Set Definition (ZFC), Topological Space, Group Definition and Axioms, Propositional Logic (Boolean Operations), Predicate Logic (Quantifiers)

Key Primitives

Set Definition (ZFC) (axiom): A set is a well-defined collection of distinct objects (elements). Membership is denoted x in S. Two sets are equal iff they have exactly the same elements (Axiom of Extensionality). Sets are the foundational objects of mathematics under ZFC.

• Define a collection of mathematical objects
• Establish the foundational objects for building mathematical structures
• Work with membership, inclusion, and equality of collections

Topological Space (axiom): A topological space (X, tau) is a set X with a collection tau of subsets (called open sets) satisfying: (1) emptyset and X are in tau. (2) Any union of sets in tau is in tau. (3) Any finite intersection of sets in tau is in tau.

• Define the concept of 'nearness' or 'openness' without a metric
• Study properties preserved under continuous deformation
• Generalize analysis to abstract settings

Group Definition and Axioms (axiom): A group (G, ) is a set G with a binary operation * satisfying: (1) Closure: ab in G for all a,b in G. (2) Associativity: (ab)c = a(bc). (3) Identity: exists e in G such that ea = ae = a. (4) Inverses: for each a, exists a^{-1} with a*a^{-1} = a^{-1}*a = e.

• Verify if a set with an operation forms a group
• Identify symmetries of objects as group elements
• Study algebraic structures with a single binary operation

Propositional Logic (Boolean Operations) (definition): Propositional logic deals with propositions (true/false statements) combined by logical connectives: AND (conjunction, p ^ q), OR (disjunction, p v q), NOT (negation, ~p), IMPLIES (conditional, p -> q), IFF (biconditional, p <-> q).

• Combine simple statements into complex logical expressions
• Determine the truth value of a compound proposition
• Formalize arguments and reasoning

Predicate Logic (Quantifiers) (definition): Predicate logic extends propositional logic with variables, predicates P(x), and quantifiers: universal (forall x, P(x)) meaning P holds for all x, and existential (exists x, P(x)) meaning P holds for some x. Negation: ~(forall x, P(x)) iff (exists x, ~P(x)).

• Express mathematical statements involving 'for all' or 'there exists'
• Negate quantified statements correctly
• Formalize mathematical definitions and theorems

Homomorphism (definition): A group homomorphism f: G -> H is a function satisfying f(a *_G b) = f(a) *_H f(b) for all a, b in G. It preserves the group operation. The kernel ker(f) = {a in G : f(a) = e_H} is a normal subgroup of G. The image im(f) is a subgroup of H.

• Define a structure-preserving map between groups
• Identify the kernel and image of a group map
• Classify groups up to homomorphic relationships

Open Set and Closed Set (definition): In a topological space (X, tau), a set U is open if U in tau. A set C is closed if X \ C is open. The closure cl(A) is the smallest closed set containing A. The interior int(A) is the largest open set contained in A. A set can be both open and closed (clopen).

• Determine if a set is open, closed, or neither in a given topology
• Compute the closure, interior, and boundary of a set
• Work with topological properties defined via open/closed sets

Cartesian Product (definition): The Cartesian product of A and B is A x B = {(a,b) : a in A, b in B}. For n sets: A_1 x ... x A_n = {(a_1,...,a_n) : a_i in A_i}. |A x B| = |A| * |B|. R^n = R x R x ... x R (n times).

• Form all possible pairs from two sets
• Construct the domain for relations and functions
• Build multi-dimensional spaces from one-dimensional sets

Relation (definition): A relation R from A to B is a subset of A x B. We write aRb or (a,b) in R. Properties: reflexive (aRa), symmetric (aRb => bRa), antisymmetric (aRb and bRa => a=b), transitive (aRb and bRc => aRc).

• Define a relationship between elements of two sets
• Check if a relation has special properties (reflexive, symmetric, transitive)
• Formalize order, equivalence, or other structural relationships

Equivalence Relation (definition): An equivalence relation ~ on set A is a relation that is reflexive (a ~ a), symmetric (a ~ b => b ~ a), and transitive (a ~ b and b ~ c => a ~ c). It partitions A into disjoint equivalence classes [a] = {x in A : x ~ a}.

• Classify elements into groups where they are considered equivalent
• Partition a set into disjoint equivalence classes
• Define modular arithmetic or congruence relations

Composition Patterns

• Set Definition (ZFC) + foundations-propositional-logic -> Set builder notation: {x in S : P(x)} uses logical predicates to define sets (parallel)
• Empty Set + foundations-set-definition -> Basis for inductive set construction: {}, {{}}, {{},{{}}}, ... (sequential)
• Set Union + foundations-set-intersection -> Boolean algebra of sets: union and intersection with complement form a complete Boolean algebra (parallel)
• Set Intersection + foundations-set-union -> Set algebra with distributive laws: A inter (B union C) = (A inter B) union (A inter C) (parallel)
• Set Complement + foundations-set-union -> De Morgan's laws for sets: (A union B)^c = A^c inter B^c and (A inter B)^c = A^c union B^c (parallel)
• Cartesian Product + foundations-relation -> Relations as subsets of Cartesian products: R subset A x B (sequential)
• Power Set + foundations-cardinality -> Cantor's theorem: |P(A)| > |A| for any set A, proving no largest cardinal (sequential)
• Relation + foundations-set-definition -> Relations as structured subsets of Cartesian products, enabling order theory (sequential)
• Equivalence Relation + foundations-group-definition -> Quotient group: G/N uses equivalence classes (cosets) as group elements (sequential)
• Equivalence Class / Partition Theorem + foundations-equivalence-relation -> Bijection between equivalence relations on A and partitions of A (parallel)

Cross-Domain Links

• structure: Compatible domain for composition and cross-referencing
• reality: Compatible domain for composition and cross-referencing
• mapping: Compatible domain for composition and cross-referencing
• unification: Compatible domain for composition and cross-referencing
• synthesis: Compatible domain for composition and cross-referencing

Activation Patterns

• set
• logic
• proof
• group
• ring
• field
• topology
• axiom
• formal
• abstract