Ideas from 'What is Cantor's Continuum Problem?' by Kurt Gödel [1964], by Theme Structure

[found in 'Philosophy of Mathematics: readings (2nd)' (ed/tr Benacerraf/Putnam) [CUP 1983,0-521-29648-x]].

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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
We perceive the objects of set theory, just as we perceive with our senses
                        Full Idea: We have something like perception of the objects of set theory, shown by the axioms forcing themselves on us as being true. I don't see why we should have less confidence in this kind of perception (i.e. mathematical intuition) than in sense perception.
                        From: Kurt Gödel (What is Cantor's Continuum Problem? [1964], p.483), quoted by Michčle Friend - Introducing the Philosophy of Mathematics 2.4
                        A reaction: A famous strong expression of realism about the existence of sets. It is remarkable how the ingredients of mathematics spread themselves before the mind like a landscape, inviting journeys - but I think that just shows how minds cope with abstractions.
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / o. Axiom of Constructibility V = L
Gödel proved the classical relative consistency of the axiom V = L
                        Full Idea: Gödel proved the classical relative consistency of the axiom V = L (which implies the axiom of choice and the generalized continuum hypothesis). This established the full independence of the continuum hypothesis from the other axioms.
                        From: report of Kurt Gödel (What is Cantor's Continuum Problem? [1964]) by Hilary Putnam - Mathematics without Foundations
                        A reaction: Gödel initially wanted to make V = L an axiom, but the changed his mind. Maddy has lots to say on the subject.
5. Theory of Logic / L. Paradox / 5. Paradoxes in Set Theory / a. Set theory paradoxes
Set-theory paradoxes are no worse than sense deception in physics
                        Full Idea: The set-theoretical paradoxes are hardly any more troublesome for mathematics than deceptions of the senses are for physics.
                        From: Kurt Gödel (What is Cantor's Continuum Problem? [1964], p.271), quoted by Philip Kitcher - The Nature of Mathematical Knowledge 03.4
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / g. Continuum Hypothesis
The Continuum Hypothesis is not inconsistent with the axioms of set theory
                        Full Idea: Gödel proved that the Continuum Hypothesis was not inconsistent with the axioms of set theory.
                        From: report of Kurt Gödel (What is Cantor's Continuum Problem? [1964]) by Brian Clegg - Infinity: Quest to Think the Unthinkable Ch.15
If set theory is consistent, we cannot refute or prove the Continuum Hypothesis
                        Full Idea: Gödel proved that (if set theory is consistent) we cannot refute the continuum hypothesis, and Cohen proved that (if set theory is consistent) we cannot prove it either.
                        From: report of Kurt Gödel (What is Cantor's Continuum Problem? [1964]) by William D. Hart - The Evolution of Logic 10
6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / a. Mathematical empiricism
Basic mathematics is related to abstract elements of our empirical ideas
                        Full Idea: Evidently the 'given' underlying mathematics is closely related to the abstract elements contained in our empirical ideas.
                        From: Kurt Gödel (What is Cantor's Continuum Problem? [1964], Suppl)
                        A reaction: Yes! The great modern mathematical platonist says something with which I can agree. He goes on to hint at a platonic view of the structure of the empirical world, but we'll let that pass.