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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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12729
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The cause of a change is not the real influence, but whatever gives a reason for the change [Leibniz]
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Full Idea:
That thing from whose state a reason for the changes is most readily provided is adjudged to be the cause. ...Causes are not derived from a real influence, but from the providing of a reason.
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From:
Gottfried Leibniz (Specimen inventorum [1689], A6.4.1620), quoted by Daniel Garber - Leibniz:Body,Substance,Monad 5
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A reaction:
Leibniz is not denying that there are real influences. He seems to be offering the thesis which I am pursuing, that the need for explanation is the crucial factor in shaping the structure of our metaphysics.
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