- Arose between end of 19th century and the beginning of the 20th century
- Several paradoxes have been discovered:
- parallel postulate cannot be proven
- Russell’s Paradox — R = {x s.t. x∉x} Implies R∈R ⟺ R∉R
- and others
- logic as a new area of mathematics
- Set theory with the Axiom of Choice - for each family of non-empty sets it exists a function f(S) in S became the standard foundation of modern mathematics
- One proposal to solve the crisis is known as Hilbert’s program was to formalize all of math, disproven by Gödel
- disproven by Gödel’s incompleteness theorems
- first of related theorems on the limitations of formal systems
- one of them is Turing’s theorem that there is no algorithm to solve the halting problem
- disproven by Gödel’s incompleteness theorems
Questions
What is the foundational crisis of mathematics?::Several paradoxes discovered around the same time. To solve it new paradigms, such as set theory, emerged
What is Hilbert’s program?::A proposal to formalize the whole field, disproven by Godel’s incompleteness theorem