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If we define God's knowledge as a proper class (not a set, because that would create contradictions), that would mean he does not know he knows everything. Since a class cannot contain itself, he cannot know "I know everything" because everything is the class itself. Is there a way around this, or do theists accept God does not know he knows everything? Perhaps by defining God's knowledge not as a class, but as a collection? But this has problems as a collection is not formally definable. How would a theist answer to this?

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    Even sets can contain themselves, see anti-foundation. But why would God's knowledge be a set, class or collection? Dividing things into finite and discrete pieces is what human ("discursive") intellect does, God's presumably has no need for that. His "knowledge" would be more like a single all-encompassing act of comprehension, not a collection of pieces. Look at intensional logic and non-discursive intellect. – Conifold Nov 26 '16 at 0:35
  • @Conifold So you're saying that if we'd hypothetically "ask" God a question, he wouldn't be able to answer in a discursive way? For example, if we "ask" him 1+1, he wouldn't be able to answer 2, in our way of organizing knowledge? If he could answer, then if we asked him every possible question, the paradox would reemerge, correct? – APCoding Nov 26 '16 at 15:55
  • If we take the Bible as a model for how God would answer a question, I suspect the answer to 'What is 1+1?' would be ' Why do you ask?' – bradimus Nov 26 '16 at 16:50
  • @Conifold Also, is the idea of non-discursive intellect generally accepted as possible? For example, this forum post? – APCoding Nov 26 '16 at 20:33
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    Possible duplicate of How would a theist answer this argument against omniscience? – Chris Sunami Nov 28 '16 at 20:11
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This is a pile of category errors in search of some sense.

1) A reference to a thing is not the thing. The Library of Congress Catalog could have a shelf entry, then we would know that the Catalog is in the Library of Congress. But the entry is in the Catalog on a given page, the Catalog is not in the Catalog on that page -- it clearly would not fit. God's knowledge of his own omniscience would logically be referential in this way. I would not require that the contents of the omniscience should be spelled out in the statement of the omniscience itself.

2) A convention is not its domain. Set Theory is a model of human intuitions about containment, it is not a true description of how all language must be done. The fact that there are multiple versions of it, alone, prevent it from really being definitive in any argument about God.

3) Actual meaning does not go away because it fails to fit a model. The collection of all things other than Joan of Arc is not Joan of Arc, so she is in it. The Category of Categories is a Category. We can define the direct sum of Groups as an operator, and get a Group, in a real and meaningful sense, even if that violates our Set Theory. Collections contain themselves, referentially, all the time. You can quibble over whether 'set' is a specific kind of collection or not, but you can't just say collections cannot contain themselves.

There would be a problem with listing out everything God knows, but that is not required. We can formulate the rule that if there is something to be known, it is in the collection of things that he knows. That rule can be one of the things he knows. And even if the rule were to be represented by the contents it refers to, there is no contradiction between that statement and Set Theory, because this is not math.

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Your fault is in your definitions.

If God knows everything, God knows that God knows everything. If God does not know that God knows everything, then God does not know everything.

To answer your worry about infinite recursion, the answer when asking God the question "Do you know X?" is simply always "Yes.". There is no loop, no recursion, no processing; just "Yes.".

I'm not sure if you're computer-savvy, but perhaps a pseudocode programming answer might help you:

class Entity
  abstract func Boolean knows(Knowledge k)

class Person ; subclass of Entity
  func Boolean knows(Knowledge k) {
    // determine whether Person knows k
    return result
  }

class God ; subclass of Entity
  func Boolean knows(Knowledge k) {
    return true
  }
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If we look at God's knowledge as a set, where everything in the set is a separate piece of knowledge knowing one thing, then what is wrong with God having one of those items be awareness of the infinity of his knowledge set?

Christians believe God is all knowing, so he'd know he knows everything. God is also trans-infinite so he could handle any infinite loops that arise from that.

This question may also be relevant.

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Depends on the theist and their theism.

If God existed, would he know everything?

Again, depends on the theist and their theism.

If God existed, would he -

Why do you presume a gender?

If God existed -

What ever in the world do you mean by "God"?

Any answer other than deity is incoherent, imponderable nonsense begs the question and overlooks the obvious: deity does not exist and is only to be found in language.

That you may believe in or study unicorns makes neither speculation regarding their attributes or existence any more the case nor any more relevant to philosophy than presuming the number of angels dancing upon the head of a pin and proceeding from there to a conclusion regarding the tempo and meter to which they dance. And this despite how ever many may agree with said presumption or conclusion.

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If God exists and if Cantor is right, then God cannot know every real number, because there are uncountably many and hence there is no memory that can store and no intelligence that can define all. God would be overtaxed in the same way as when he had to name an even prime number beyond 2. This means, under the given constraints, God would know that he does not know everything.

  • But do you need memory to know something? For example, you might say I know the result of 3194+4379, yet it's not in my memory. – Keelan Aug 26 '17 at 17:15
  • But the two numbers or some other hint to obtain the result must be in the memory. But all hints from which you could obtain a real number will always lead to a real number that belongs to a countable set of real numbers. (For instance every diagonal number belongs to this countable set because it is defined by when and where who made the proof.) – Heinrich Aug 26 '17 at 21:50

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