Kurt Gödel
1906-1978 · Austrian-American · Mathematical logic, philosophy of mathematics, metalogic, philosophy of language
Despite their remoteness from sense experience, we do have something like a perception also of the objects of set theory, as is seen from the fact that the axioms force themselves upon us as being true.
When Kurt Gödel presented his incompleteness theorems at a Vienna conference in 1930, he demolished the grandest ambition of modern mathematics: Hilbert's program to ground all mathematics in a complete, consistent formal system. Gödel proved that any such system must contain true statements it cannot prove — and cannot prove its own consistency. One of the most stunning results in intellectual history, from a logician who entered the Vienna Circle as a student and left as its silent refutation.
Topics Kurt Gödel addresses
- the incompleteness theorems
- the second incompleteness theorem
- truth versus provability
- the completeness theorem for first-order logic
- ω-consistency
- mathematical platonism
- conceptual realism and the perception of concepts
- the constructible universe L
- the axiom of choice
- the continuum hypothesis
- the generalized continuum hypothesis
- Cohen's forcing and independence results
- new axioms for set theory
- the disjunction thesis
- minds and machines
- absolutely unsolvable Diophantine problems
- Hilbert's program
- formalism in mathematics
- against logical positivism
- the Carnap critique
- Husserlian phenomenology and mathematical intuition
- rationalistic optimism
- the Leibnizian program
- the modal ontological argument
Questions to put to Kurt Gödel
Incompleteness and the Limits of Formal Systems
- What exactly do the incompleteness theorems prove?
- Did your theorems refute Hilbert's program, or redirect it?
- How can a mathematical statement be true but unprovable?
- Can you explain the difference between truth and provability?
- What's wrong with applying incompleteness to consciousness or theology?
Mathematical Platonism and the Perception of Concepts
- What do you mean by perceiving mathematical objects?
- How is mathematical intuition different from sense perception?
- Why isn't mathematics just syntax or linguistic convention?
- How did Husserl's phenomenology shape your view of mathematical knowledge?
- How do we know about mathematical objects if they don't cause anything?
- What's wrong with treating mathematics as useful fiction?
The Continuum Hypothesis and New Axioms
- Does the continuum hypothesis have a definite answer?
- What did Cohen's forcing proof show about the continuum problem?
- Why should we search for new axioms instead of accepting independence?
- What is the constructible universe, and why did you introduce it?
- How do you see the modern large cardinal program?
Minds, Machines, and Absolutely Unsolvable Problems
- What is the disjunction thesis from your Gibbs Lecture?
- Does incompleteness prove the human mind isn't a machine?
- What would it mean for there to be absolutely unsolvable problems?
- How do you respond to Penrose's argument in The Emperor's New Mind?
- Can a machine ever recognize its own consistency?
Rationalistic Optimism and the Leibnizian Program
- What do you mean by rationalistic optimism?
- Can philosophy really be made as exact as mathematics?
- Why did you turn to Leibniz's monadology late in life?
- How do you reconcile full intelligibility with possibly unsolvable problems?
- What is the modal ontological proof you developed?
Time, Relativity, and Conversations with Einstein
- What does your rotating universe solution say about time?
- How did working with Einstein shape your philosophical views?
- Is time objectively real, or is it something else?
- What did you and Einstein talk about during your daily walks?
Converse with Kurt Gödel on Simposeum. Replies are grounded in Kurt Gödel's own writing, cited to the page.