Every lesson in the Foundations of Higher Mathematics slide course, in full text: 30 decks, 3326 slides.
Propositional Logic & the Architecture of ProofThis deck builds propositional logic from the ground up. It separates syntax from semantics, defines well-formed formulas by induction, and covers truth tables and the material conditional, tautologies and satisfiability, logical equivalence, and the canonical DNF and CNF forms. It then proves an impossibility result about functional completeness, works through natural deduction, states soundness and completeness, and closes with the Curry-Howard bridge. It targets the classic traps: reading vacuous truth as falsity, affirming the consequent, confusing a conditional with its converse, and treating "or" as exclusive.
Predicate Logic & QuantifiersThis deck covers predicates and open sentences, the universal and existential quantifiers over a stated domain, the order of nested quantifiers, and mechanical negation. It then adds the semantic layer: structures, satisfaction, validity as against satisfiability, Skolemization, and the decidability cliff. It targets the classic errors: swapping the order of quantifiers, negating "for all P" as "for all not P", forgetting the domain, and reading "there exists" as "exactly one".
Proof Techniques I: Direct, Contrapositive, ContradictionThis deck shows how to prove a theorem three ways - directly, by contrapositive, and by contradiction - and then covers vacuous and trivial proofs, biconditionals, arguing without loss of generality, and disproof by counterexample. It targets the misconceptions that examples prove universal statements, that the converse is the same as the contrapositive, and that assuming the conclusion is a valid argument.
Proof Techniques II: Induction, Strong Induction & Well-OrderingThis deck covers weak, strong, structural, and well-founded induction, and shows that they are equivalent to the well-ordering principle. It targets the misconceptions that induction is circular, that the base case is optional, and that weak induction always suffices, along with the all-horses-are-the-same-color fallacy.
Sets, Operations & the Boolean Algebra of SetsThis deck treats sets as the ambient language of mathematics. It separates membership from the subset relation and covers set-builder notation, the empty set, power sets, the Boolean-algebra laws for union, intersection, and complement - which are identical to those of propositional logic - and Cartesian products, then shows why naive comprehension collapses into Russell's paradox. It targets the classic confusion between element-of and subset-of, the belief that a universal set exists, and the slip between the empty set and the set containing the empty set.
Relations, Equivalence Relations & PartitionsThis deck defines relations as subsets of a product and covers the reflexive, symmetric, antisymmetric, and transitive properties, then equivalence relations and their classes, and the fundamental theorem that equivalence relations on a set correspond exactly to partitions of it. It targets the classic misconceptions: that symmetry and transitivity together imply reflexivity, that a partition is the same thing as an arbitrary cover, and that classes of related points are distinct.
Functions: Injections, Surjections, BijectionsThis deck presents functions as a special kind of relation and draws the distinction between the codomain and the image. It covers injective, surjective, and bijective maps and their characterizations in terms of inverses, the behavior of image and preimage under unions and intersections, and the pigeonhole principle. It targets the misconceptions that being one-to-one can be checked on a single pair, that the codomain has no bearing on whether a map is onto, that image distributes over intersection, and that the preimage notation requires an inverse function to exist.
Modular Arithmetic & CongruenceThis deck presents congruence modulo n as an equivalence relation that is also compatible with addition and multiplication, which is what gives the ring Z/nZ. It covers units and the gcd, modular inverses via the extended Euclidean algorithm, the theorems of Fermat and Euler, fast exponentiation, the Chinese Remainder Theorem, and RSA. It targets the traps of cancelling by something that is not a unit, misapplying Fermat's theorem when the gcd is not 1, reducing "mod" to a bare remainder operator, and using the Chinese Remainder Theorem with moduli that are not coprime.
Cardinality I: Countable SetsThis deck compares the sizes of infinite sets by means of bijections, covering equinumerosity, Dedekind-infinite sets, and countability. It builds explicit bijections for the naturals, the integers, pairs, and the rationals, proves that a countable union of countable sets stays countable, and uses the Schroeder-Bernstein theorem. It targets the misconceptions that a proper subset must be smaller, that the integers outnumber the naturals, and that countable means finite.
Cardinality II: Uncountability & the ContinuumThis deck covers Cantor's diagonal argument that the reals are uncountable and Cantor's theorem that every set is strictly smaller than its power set, then the cardinal arithmetic of the continuum, the Continuum Hypothesis, and its independence from ZFC. It ends with the collision between set theory and computation: there are only countably many programs but uncountably many reals, so most reals must be uncomputable. It targets the misconceptions that the diagonal number is already in the list, that uncountable simply means infinite, that the Continuum Hypothesis is merely unproven rather than independent, and that a bijection between the line and the plane is impossible.
Binary Operations, Semigroups & MonoidsThis deck builds the algebraic ladder from magma to semigroup to monoid from the ground up, treating closure, associativity, and identity as separate axioms that you must actually check. It targets the misconceptions that closure comes for free, that every operation is associative or commutative, and that an identity element is the same thing as an absorbing element. The free monoid of strings and the monoid of function composition anchor the ideas in computer science.
Groups: Axioms & First ExamplesThis deck presents the four group axioms as the upgrade from a monoid, then proves that the identity and inverses are unique and covers the cancellation law and the socks-and-shoes rule. It distinguishes the order of a group from the order of an element, and abelian from non-abelian groups, and works the standard first examples: the additive integers, the integers mod n, the units mod n, the symmetric group, and the dihedral symmetries of a square. It targets the real traps: forgetting closure or inverses when calling a subset a group, assuming every group is abelian, confusing the order of a group with the order of an element, and thinking that the integers mod n form a group under multiplication.
Subgroups, Cosets & Lagrange's TheoremThis deck covers subgroups and the one-step test, cyclic subgroups and the order of an element, left cosets as a partition into equal-sized tiles, and the index of a subgroup. That leads to Lagrange's theorem and its corollaries: the order of an element divides the order of the group, every group of prime order is cyclic, and Fermat's little theorem follows. It targets the traps of assuming that a closed subset is automatically a subgroup, that left and right cosets always coincide, and that the converse of Lagrange's theorem holds.
Group Homomorphisms & Quotient GroupsThis deck is about structure-preserving maps between groups. It gives the homomorphism equation and shows why the identity and inverses are preserved automatically, then covers the image and the kernel and why the kernel is always a normal subgroup. It treats isomorphism as sameness up to relabeling, builds the quotient group and explains why coset multiplication needs normality, and closes with the First Isomorphism Theorem as the master factorization. It targets the real traps: assuming that every subgroup is normal, thinking that coset multiplication is always well defined, confusing the image with the codomain, and treating an injective homomorphism as automatically surjective.
Rings, Integral Domains & IdealsThis deck presents rings as sets carrying two compatible operations, then splits the elements into units and zero-divisors. It covers integral domains and cancellation, fields as the best case, and ideals as the ring analog of normal subgroups, together with quotient rings and the First Isomorphism Theorem. It targets the misconceptions that every ring is commutative or has a 1, that no ring has zero-divisors, and that an ideal is just a subring.
Fields, Characteristic & Finite FieldsThis deck builds the field concept as the top of the ring ladder: division works everywhere, there are no zero divisors, and the characteristic is forced to be either zero or prime. It constructs the smallest genuinely new finite field, F_4, from scratch with full addition and multiplication tables, and hammers home the central trap that the field with four elements is NOT the ring of integers modulo four. It targets the misconceptions that the characteristic can be composite, that a finite ring with no obvious zero divisor must be a field, and that the multiplicative group need not be cyclic.
Vector Spaces: Axioms & ExamplesThis deck covers the eight vector-space axioms over a field, the consequences those axioms immediately force, and the zoo of genuine examples: tuples, matrices, polynomials, functions, the solution sets of homogeneous systems, and C regarded over different fields. It targets the misconceptions that a vector is an arrow in R^n, that any set of vectors is a space, that closure under scalar multiplication comes for free, and that the choice of scalar field does not affect dimension.
Subspaces, Sums & Direct SumsThis deck gives the subspace test, explains why an intersection of subspaces is a subspace but a union usually is not, and presents the sum as the smallest subspace containing both. It then covers the three equivalent faces of a direct sum and the Grassmann dimension formula. It targets the classic traps: taking unions to be subspaces, mistaking spanning for directness, and forgetting the intersection correction term.
Span & Linear IndependenceThis deck covers linear combinations and the span of a set as the smallest subspace containing it, then draws the sharp line between spanning and independence. It builds the Steinitz exchange lemma as the engine behind a well-defined notion of dimension, ties independence to unique representation and to row reduction, and dismantles the classic traps: confusing spanning with independence, testing three or more vectors a pair at a time, and slipping the zero vector into an independent set.
Basis & DimensionThis deck presents a basis as the fusion of independence and spanning, then covers the unique-coordinate theorem, extending an independent set and shrinking a spanning set, the invariance-of-dimension theorem proved via the exchange lemma, and the Grassmann dimension formula. It targets the classic traps: calling a spanning but dependent set a basis, assuming that any n vectors in an n-dimensional space form a basis, confusing the number of vectors with the dimension, and forgetting that infinite-dimensional spaces exist.
Linear MapsThis deck presents linear maps as the structure-preserving morphisms between vector spaces. It gives the two axioms, explains why the origin is fixed, shows that a basis determines the whole map, and covers the kernel and image as subspaces and injectivity as the kernel being trivial. It targets the classic traps of calling affine or squaring maps linear, thinking that one vector pins a map down, locating the kernel in the wrong space, and assuming that injective forces surjective.
The Rank–Nullity TheoremThis deck defines rank and nullity as the dimensions of the image and the kernel, then states and proves the Rank-Nullity Theorem by basis extension, computes both quantities by row reduction, and covers the finite-dimensional miracle that for an endomorphism injective and surjective mean the same thing. It targets the misconceptions of adding to the dimension of the codomain, confusing rank with the codomain, and expecting injective to force surjective in infinite dimensions.
Matrices, Coordinates & Change of BasisThis deck shows how an ordered basis turns abstract vectors into coordinate tuples, how a linear map becomes a matrix, why matrix multiplication is exactly composition, and how a change of basis - that is, similarity - re-coordinates one and the same map. It targets four real errors: confusing a vector with its coordinate tuple, multiplying matrices in the wrong order for a composition, mixing up P and P-inverse in the change-of-basis formula, and thinking that similar matrices are equal.
Invertibility, Isomorphism & DualityThis deck presents the Invertible Matrix Theorem as a single phenomenon with many faces, then covers isomorphism of vector spaces and their classification by dimension, the dual space and the dual basis, and the double dual with its natural isomorphism. It targets the traps of supposing that a non-square matrix could be invertible, that isomorphic spaces must share a common set, that the determinant is unrelated to injectivity, and that identifying a space with its dual is canonical.
Ordered Fields & the Completeness AxiomThis deck covers the ordered-field axioms and shows why the rationals form an ordered field that nevertheless has gaps. It introduces the supremum and infimum and the Least Upper Bound axiom, which pins down the real numbers as the unique complete ordered field, then derives the Archimedean property and the density of the rationals from completeness and constructs the reals by Dedekind cuts. It targets the traps of confusing a maximum with a supremum, assuming that the supremum lies in the set, and believing that the rationals are already complete.
Sequences & ConvergenceThis deck reads the epsilon-N definition of convergence as a nested-quantifier game: given a tolerance, produce a threshold. It covers the uniqueness of limits, the fact that a convergent sequence is bounded, the algebra of limits, divergence to infinity, and the squeeze theorem. It targets the classic traps: fixing N before epsilon, thinking that boundedness forces convergence, dividing by a limit that is zero, and the belief that terms bunching up is by itself enough, which is the difference between Cauchy and convergent.
Monotone Convergence, Bolzano-Weierstrass & CauchyThis deck covers the three theorems that make "the iterates settle down" rigorous: the Monotone Convergence Theorem, the Bolzano-Weierstrass theorem, and the Cauchy criterion. It targets the misconceptions that monotonicity alone gives convergence, that every bounded sequence converges, that Cauchy implies convergent in any space, and that a subsequential limit is THE limit.
Infinite SeriesA series is the limit of its partial sums, not a magical infinite addition, and this deck starts there. It builds the nth-term test, geometric and telescoping sums, the proof that the harmonic series diverges, the comparison, ratio, and root tests, and the distinction between absolute and conditional convergence. It targets the classic traps: thinking that terms shrinking to zero forces convergence, that |r| may be anything in the geometric formula, that the ratio test settles the case when its limit is 1, and that a conditionally convergent series keeps its sum under rearrangement.
Topology of R: Open, Closed & CompactThis deck builds the topology of the real line from its order. It covers open sets as wiggle room and closed sets as complements that contain their limit points, the line and the empty set as the clopen pair, the union and intersection laws and why an infinite intersection of open sets can fail to be open, compactness via open covers together with the Heine-Borel theorem, sequential compactness, and the connectedness of intervals. It targets the misconceptions that closed means not open, that an infinite intersection of open sets is open, that boundedness alone forces compactness, and that closed means finite or bounded.
Continuity & the Structural SynthesisThis deck presents the three faces of continuity - epsilon-delta, sequential, and the topological definition by preimages of open sets - and shows how continuity carries topological properties across, through the Intermediate Value and Extreme Value theorems. It closes with a synthesis that names the arc running from logic through algebra to topology by its three recurring threads: structure-preserving maps, quotients by congruences and equivalences, and completeness and compactness. It targets the classic traps: quantifier-order errors in epsilon-delta, the pen-lifting cartoon of continuity, confusing preimage-open with image-open, and misapplying the IVT or the EVT outside their required domains.
Want this taught 1-on-1? Alexander tutors Foundations of Higher Mathematics — $55/session, free consultation.