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On this webpage, the author argues:

The first problem, ”the paradox of omniscience”, is derived from Cantor’s proof that there is no set of all sets. Omniscience, it is said, entails knowledge of the set of all truths. Cantor’s proof, however, demonstrates that there is no such set. As there is no such set, it is argued, there can be no omniscient being.

How would a theist respond to this? Would they argue omniscience is not knowledge of all truths? Anything else?

  • Add to the invisible gardener's repetoire imperviousness to Cantor – Mr. Kennedy Nov 25 '16 at 4:18
  • @Mr.Kennedy. Flew says, "Just how does what you call an invisible, intangible, eternally elusive gardener differ from an imaginary gardener or even from no gardener at all?" Given that ~~Pa ⇔ Pa, his argument seems to suggest that the predication of divine qualities is the same as denying the predication of imaginary properties, as if to assert: ~~Ǝx[Px] ⇔ Pa. Since the latter is intuitively meaningless, he suggests that the predication of divine qualities is meaningless. Of course, ~~Pa is not equivalent to ~~Ǝx[Px], and the relation ~~Ǝx[Px] ⇔ Pa is invalid. – user3017 Nov 25 '16 at 12:57
  • Cantor did not "prove" that there is no set of all sets. That such a thing leads to contradictions was first in print suggested by Schroder in 1890, Cantor had similar misgivings around the same time. However, precise arguments were only offered by Zermelo and Russell a decade later, and more directly concerning the set of all sets not containing themselves, see How did Russell arrive at the paradox demonstrating the inconsistency of naive set theory? – Conifold Nov 25 '16 at 20:56
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Cantor's is only one way of looking at sets and containment. And even it, from a point of view like Goedel-Bernays-VonNeuman set theory this result only declares that the class of all sets is not a set, not that it does not exist. Variants of this solution provide models of mathematics that allow for a set of all sets (or at least a class of al classes, or a category of all categories), but they must sacrifice either some applications of negation or severely limit the degree to which self-reference makes sense in order to evade paradoxes like Russell's.

From the viewpoint of someone like Descartes, contradictions do not controvert omniscience. Instead, the fact that we cannot handle this apparent contradiction merely demonstrates the limitations imposed on our thinking by our nature as temporal, human animals -- including the very notion of contradiction. Applying this notion to mathematics (though through a Kantian lens) Brouwer decided negation is probably an aspect only of temporal thought, based entirely on our notion of before and after in time, and that we should not trust it completely: we should either limit it to finite cases or retain its temporal character.

  • Makes sense. Thank you very much. So is the Goedel-Bernays-VonNeuman set theory an extension of Cantor's set theory, or does it contain a completely different set of axioms? – APCoding Nov 25 '16 at 3:56
  • Looked at Goedel-Bernays-VonNeuman set theory in more detail, seems like it is an extension of ZFC, the most commonly accepted set theory. – APCoding Nov 25 '16 at 16:25
  • The ZF axioms do not allow for comprehension over the class of all sets, nor admit it is even possible for it to exist. GBvN does, and so does Category Theory. So if you ever do an open comprehension, without specifying the set you are selecting out of, you are not really working in ZF. The Axiom of Replacement, which builds comprehensions in ZF requires that you start with a set to begin with. – user9166 Nov 28 '16 at 15:45
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For the argument to hold, there's the implicit assumption that each 'truth' is a set, and each set is a truth. Then, since there is no set of all sets, there is no set of all truths.

However, a truth need not correspond to the mathematical notion of a set. If you just take a truth to be an element of a certain set A (with no assumption on the mathematical nature of truths except that not all sets are truths), then A could contains all truths, no problem, just as the set Z contains all integers.

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Just because something can be described does not make it an object of knowledge. An omniscient being does not need to know what a circle with corners looks like. An omniscient being does not need to know the answer to Bertrand Russell's question of whether the present king of France is bald or not. Omniscience means that if something can be known then it is known. An omniscient being will know all the interesting paradoxes in set theory, but they wouldn't have to know the contents of those sets.

For Christian theists, it does raise the interesting question of whether God can know things that are impossible for us to know. For example, the observer effect limits the precision of our measurements. But God, existing external to the universe, probably isn't bound by the observer effect. I don't understand the full distinction between the observer effect and the uncertainty principle but it could well be that God isn't bound by that either.

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