Foundational Crisis of Mathematics

Select Connection: INPUT[inlineListSuggester(optionQuery(#permanent_note), optionQuery(#literature_note), optionQuery(#fleeting_note)):connections]

math

• 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
  • and others
• logic as a new area of mathematics
  • Set theory with the Axiom of Choice became the standard foundation of modern mathematics
• One proposal to solve the crisis is known as Hilbert’s program
  • 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

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