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8679 | 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. |
9942 | 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. |
18062 | 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 |
10868 | 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 |
13517 | 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 |
10271 | 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. |