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6408 | Russell needed three extra axioms to reduce maths to logic: infinity, choice and reducibility |
Full Idea: In order to deduce the theorems of mathematics from purely logical axioms, Russell had to add three new axioms to those of standards logic, which were: the axiom of infinity, the axiom of choice, and the axiom of reducibility. | |||
From: A.C. Grayling (Russell [1996], Ch.2) | |||
A reaction: The third one was adopted to avoid his 'barber' paradox, but many thinkers do not accept it. The interesting question is why anyone would 'accept' or 'reject' an axiom. |
6414 | Two propositions might seem self-evident, but contradict one another |
Full Idea: Two propositions might contradict each other despite appearing self-evident when considered separately. | |||
From: A.C. Grayling (Russell [1996], Ch.2) | |||
A reaction: Russell's proposal (Idea 5416) is important here, that self-evidence comes in degrees. If self-evidence was all-or-nothing, Grayling's point would be a major problem, but it isn't. Bonjour explores the idea more fully (e.g. Idea 3704) |