Proving 1+1=2: A Rigorous Foundation of Mathematics

This section presents the complete construction of the number systems from natural numbers. Integers (ℤ) are constructed as equivalence classes of pairs (m, n) representing m - n, with (m, n) ~ (m', n') iff m + n' = m' + n. Operations are defined component-wise. Rationals (ℚ) are constructed from integers as equivalence classes of pairs (m, n) with n ≠ 0, representing m/n, with (m, n) ~ (m', n') iff m × n' = m' × n. Addition uses common denominators. Reals (ℝ) are constructed using Dedekind cuts: subsets A ⊆ ℚ that are non-empty, proper, closed downward, and have no greatest element. The order is defined by set inclusion, and addition is component-wise. This construction ensures every non-empty set of reals bounded above has a supremum, distinguishing reals from rationals.

From R's algebraic structure, we derive three fundamental number systems: Natural numbers (N) from repeated addition of 1 (1, 2, 3, ...); Integers (Z) from adding additive inverses to N (...,-3,-2,-1,0,1,2,3,...); Rational numbers (Q) from combining integers with their multiplicative inverses (a/b for a ∈ Z, b ∈ Z\{0}). The second-level description introduces order on sets as a subset S of A × A satisfying reflexivity (a,a) ∈ S for all a), antisymmetry (if (a,b) and (b,a) ∈ S, then a = b), and transitivity (if (a,b) and (b,c) ∈ S, then (a,c) ∈ S). The order on R must be compatible with the first-level algebraic description, enabling definition of irrational numbers (√2, √3, √5) and transcendental numbers (e, π, log 2) that cannot be defined algebraically alone.

Number systems are constructed hierarchically: Natural numbers (N) use Peano axioms, Integers (Z) are equivalence classes of ordered pairs of naturals, Rationals (Q) are equivalence classes of ordered pairs of integers, and Reals (R) are Dedekind cuts. Each construction extends the previous system while preserving algebraic structure.

Numbers are constructed hierarchically from simpler to more complex systems: Natural numbers (counting) → Integers (adding additive inverses) → Rational numbers (fractions a/b where b≠0) → Real numbers (including irrationals like π and √2) → Complex numbers (a + bi where i²=-1). Each step introduces new algebraic structures while preserving previous properties, creating increasingly sophisticated mathematical frameworks.

Natural numbers are constructed from the empty set (∅) using set theory: 0 = ∅, 1 = {∅}, 2 = {∅, {∅}}, continuing inductively. Natural numbers satisfy five Peano axioms: existence of zero, injective successor function, zero not being a successor, and the principle of mathematical induction. Addition is defined recursively: n + 0 = n and n + S(k) = S(n + k). Multiplication is defined recursively: m × 0 = 0 and m × S(k) = (m × k) + m. Integers are constructed from natural numbers by considering pairs (m, n) representing m - n, with equivalence defined by m + n' = m' + n. Rational numbers are defined as equivalence classes of pairs of integers (p, q) with q ≠ 0, where (p, q) ~ (p', q') if p × q' = p' × q. Operations on rationals are defined by operating on representatives: addition is (p/q) + (r/s) = (ps + rq)/(qs), multiplication is (p/q) × (r/s) = (pr)/(qs). The inverse of p/q is q/p. The Greeks distinguished between numbers (countable) and magnitudes (measurable), delaying real number development for 1,500 years until infinite series and decimal representations resolved the problem of irrational numbers.

This section covers the two standard formal theories used in mathematics. Peano Arithmetic (PA) formalizes natural numbers with addition and multiplication, encoding all finite mathematical objects as natural numbers. Zermelo-Fraenkel Set Theory (ZF) formalizes mathematics using sets and membership, with variables ranging over all sets including infinite ones. The section explains how these theories relate: replacing ZF's axiom of infinity with its negation yields a theory equivalent to PA, demonstrating that PA captures all finite mathematics. The speaker also discusses Tarski's distinction between object language and metalanguage to avoid circularity in formalization.

A mathematical theory is a formal structure consisting of an ordered tuple (U, F, R, C) representing universe, functions, relations, and constants, combined with axioms that specify element behavior. Examples include group theory (identity element axioms) and real numbers (addition/multiplication axioms). ZFC (Zermelo-Fraenkel with Choice) serves as a foundational theory using set membership (∈). Peano Arithmetic formalizes natural numbers using N (universe), + and × (functions), S (successor predicate), and 0 (constant). Core axioms include: A1 (zero exists), A2 (successor closure), A3 (zero has no predecessor), A4 (successor injectivity), A5 (induction schema enabling proofs about all natural numbers), A6 (addition definition), and A7 (multiplication definition). These axioms establish how arithmetic operations emerge from simple successor-based rules.

In ZFC, natural numbers are constructed as von Neumann ordinals: 0=∅, 1={∅}, 2={∅,{∅}}, etc., where each number is the set of all smaller numbers. The successor function is n⁺ = n ∪ {n}. This construction makes every number a set, allowing the Peano axioms of arithmetic to become theorems within ZFC. All Peano axioms—including induction—are provable when interpreting the domain as ω (the minimal inductive set). The Axiom of Replacement enables construction of images under functions, while the Axiom of Choice enables selection from collections of non-empty sets. Despite ZFC being the accepted foundation for mathematics today, Gödel's incompleteness theorems prove no sufficiently powerful formal system can demonstrate its own consistency. This fundamental limitation means we cannot be certain ZFC doesn't contain hidden contradictions, though it has avoided known paradoxes and serves as the standard foundation for modern mathematics.

The classification of countable models of ZFC is as complex as possible within descriptive set theory, being Borel complete, which means it cannot be classified by any simpler equivalence relation; this result extends the analogous classification for Peano Arithmetic and relies on constructing canonical models using minimal types and indiscernible sequences, though the situation differs significantly for well-founded models due to the boundedness theorem.

This section establishes the foundational correspondences between systems of arithmetic and set theory. Robinson arithmetic (Q) interprets set theory with abduction, including extensionality, empty set, and closure under abduction. Peano arithmetic (PA) is synonymous with finite ZF plus transitive containment, as shown by Kripke and Aczel through mutual interpretability. Simpson's results show ATR₀ (one of the Big Five of reverse mathematics) is interpretable in countable set theory ATR, characterized by Mostowski collapse, though this is not a synonymy like PA and finite ZF. The tree interpretation method, using well-founded trees under bisimulation, provides a versatile framework for interpreting set theories satisfying regularity or anti-foundation axioms.

Kurt Gödel's incompleteness theorems, published in 1931, state two fundamental limitations of formal mathematical systems. First, any consistent formal system capable of expressing basic arithmetic is incomplete - there exist true statements within the system that cannot be proven using the system's own rules. Second, no such system can prove its own consistency; to establish that a system is free from contradictions, one must appeal to methods outside the system itself. These theorems shattered the dream of creating a complete, self-contained mathematical foundation.

Kurt Gödel's incompleteness theorems, published in 1931, demonstrate that in any sufficiently powerful axiomatic system capable of expressing basic arithmetic, there will always exist true statements that cannot be proven within that system (first incompleteness theorem), and that the consistency of such a system cannot be proven within the system itself (second incompleteness theorem); this means that no matter how comprehensive our mathematical axioms are, we will always encounter 'gray areas' of undecidable propositions, fundamentally limiting the power of formal mathematical systems.

Kurt Gödel's incompleteness theorems prove that in any sufficiently powerful mathematical system, there are true statements that cannot be proven within that system, demonstrating that mathematical truth extends beyond formal provability and establishing fundamental limits to human reason and logical deduction.

Gödel's First Incompleteness Theorem states that any consistent formal system capable of expressing basic arithmetic is incomplete, meaning there exist true statements within the system that cannot be proven from its axioms. Gödel's Second Incompleteness Theorem states that such a system cannot prove its own consistency. These theorems apply only to formal systems that are consistent, recursively axiomatized, and capable of expressing basic arithmetic, and they do not apply to natural language systems or non-formal frameworks.

Kurt Gödel, one of the most important mathematicians in history, proved that in any formal system (like mathematics), there are truths that cannot be proven within that system. In any formal system G, there exist truths that require axioms from a larger system G' to be proven, and G' in turn requires axioms from an even larger system, and so on infinitely. Gödel concluded that this chain must end at a spiritual entity - he stated that his work proves materialism is false.

Interactive theorem provers (ITPs) are programming languages that understand both computation and reasoning, allowing variables to represent theorem statements and proofs rather than just numbers. Examples include Lean, Isabelle, Metamath, and Coq. Formalization is the process of translating mathematics from informal human languages into formal languages that ITPs can understand. The speaker, a professor at Imperial College London, is working on formalizing Fermat's Last Theorem into Lean. By 2029, the goal is to reduce the proof to approximately 10 theorems known in the 1980s, representing about 10-15% of the total proof.

Interactive Theorem Provers (ITPs) are software tools that help mathematicians write and verify formal proofs. They perform two main functions: first, they check proof correctness using a 'kernel' - a small piece of code containing fundamental deduction rules of logic that is simple enough for humans to verify and stable over time; second, they help users write formal proofs faster by automating repetitive details through 'tactics.' Lean 4 is a modern ITP that stands out for being written in its own programming language, having strong metaprogramming capabilities, an extensive mathematical library (mathlib) containing most undergraduate-level mathematics, and using mathematical notation rather than purely computer-like syntax. The course will use Lean 4's web interface for coding exercises.

Interactive theorem provers are software systems based on type theory that allow mathematicians to write and verify proofs with computational rigor, preventing errors by enforcing correctness at compile time; these systems enable the formalization of mathematical knowledge into reusable libraries (like Mathlib for Lean), transforming proof verification from a human-intensive process into a collaborative, computationally-assisted endeavor that bridges the gap between mathematical theory and computer science.

Proof assistants like Lean enable rapid verification of mathematical proofs, reducing months of work to minutes. Lean treats propositions as types and proofs as terms, similar to how functions map inputs to outputs. The 'theorem' keyword defines a theorem with its proof, while 'sorry' serves as a placeholder. The type checker verifies both function types and theorem types using the same mechanism. Even false propositions are types but are empty since they have no proof.

Interactive Theorem Provers (ITPs) like Lean and Coq support proof development by allowing users to apply tactics that transform proof states with immediate feedback. A user study with 17 students in an introductory formal methods course found 75% reported positive experience, 50% found proofs easier than pen-and-paper, and 75% would recommend to peers. Positive features included visualization of proof states, clear feedback when stuck, and systematic proof development. Using Jacob Nielsen's usability heuristics, Lean better parallels pen-and-paper with infix notation and math symbols, while Coq requires more tactic-oriented approaches. Challenges include complex syntax creating steep learning curves, proof translation difficulties, and confusing error messages. Future work includes improving error messages, strengthening example-driven learning, and exploring LLM integration.