Foundations of Mathematics
by Ian Stewart, David Tall · Ian Stewart, David Tall
An undergraduate bridge text by two Warwick mathematicians covering sets, relations, functions, the construction of number systems, induction, cardinality and axiomatic reasoning. Readers learn to read and write rigorous proofs and see how real numbers are built from naturals.
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More resources on Logical Foundations of Mathematics
Wolfram MathWorld
MathWorld is an online mathematics encyclopedia from Wolfram Research offering detailed, browsable articles on topics across the math spectrum, including algebra, geometry, calculus, and number theory. Each entry includes definitions, theorems, formulas, diagrams, worked examples, and links to further reading.
plato.stanford.edu
The Stanford Encyclopedia of Philosophy is a peer‑reviewed online encyclopedia of philosophy featuring in‑depth, scholarly articles written and regularly updated by experts, including comprehensive coverage of logic topics.
nLab: Foundations of Mathematics
Wiki entry from the nLab, a collaborative category theory research wiki, surveying foundational systems for mathematics: ZFC and other set theories, structural set theory such as ETCS, type theory, and homotopy type theory. Readers learn how these foundations compare and relate.
Theory of Computation (MIT 18.404J)
Sipser's course on automata, computability and complexity: regular and context-free languages, decidability, reducibility, the recursion theorem, time and space complexity, NP-completeness, hierarchy theorems, probabilistic computation and interactive proofs. 25 lecture videos, slides, problem sets and exams support rigorous proof-based study.
Homotopy Type Theory: Univalent Foundations of Mathematics
Collaborative exposition written during the Institute for Advanced Study's Univalent Foundations year, presenting type theory as a foundation whose types behave like homotopy types. Covers identity types, the univalence axiom, higher inductive types, and formalised set and homotopy theory.
How to Prove It - Daniel J. Velleman
A clear, step-by-step transition from computational math to proof-based math, focusing on logic and set theory.