18189
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ZFC could contain a contradiction, and it can never prove its own consistency [MacLane]
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Full Idea:
We have at hand no proof that the axioms of ZFC for set theory will never yield a contradiction, while Gödel's second theorem tells us that such a consistency proof cannot be conducted within ZFC.
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From:
Saunders MacLane (Mathematics: Form and Function [1986], p.406), quoted by Penelope Maddy - Naturalism in Mathematics
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A reaction:
Maddy quotes this, while defending set theory as the foundation of mathematics, but it clearly isn't the most secure foundation that could be devised. She says the benefits of set theory do not need guaranteed consistency (p.30).
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18925
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If talking donkeys are possible, something exists which could be a talking donkey [Williamson, by Cameron]
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Full Idea:
Williamson's view on modality is that everything that could exist does exist: since there could exist a talking donkey there actually exists some thing that could be a talking donkey.
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From:
report of Timothy Williamson (Modal Logic as Metaphysics [2013], n20) by Ross P. Cameron - Truthmaking for Presentists n20
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A reaction:
Well that thing certainly isn't me, or Tim Williamson. I'm guessing that the thing is an actual donkey, probably a rather bright one. Actually, I think this is one of those views that invites the incredulous stare. (Barcan formulae).
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