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Kurt Goedel proved there may be some things that are true but that cannot be mathematically proven. But God, being infinite, can know everything, even things that cannot be proven.

Or can he?

Mauro ALLEGRANZA
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moonman239
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  • If you define God as omnisapient, it should know. If you define it as the image of men, he might not know. Spinoza's God (see: pantheism) don't, because it's not human and don't know math as such. – RodolfoAP Aug 08 '22 at 01:34
  • even we can know things that cannot be proven, for example that in classic logic, Godels sentence is true. More generally, Godels result does not clearly have implications for God's knowledge. – emesupap Aug 08 '22 at 01:44
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    Does this answer your question? [Is the "omniscient-omnipotent-omnipresent" definition of God consistent?](https://philosophy.stackexchange.com/questions/300/is-the-omniscient-omnipotent-omnipresent-definition-of-god-consistent) –  Aug 08 '22 at 03:27
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    Goedel proved that in a *recursively axiomatizable* first order system that can simulate arithmetic there are some true but unprovable facts. But we only resort to such systems because we can only contemplate finitely many axiom schemas and rules in our heads, and everything else has to be derived from them to be proven. If one can contemplate *all* true facts directly there is no need for derivations, and Goedel's theorem becomes irrelevant. The facts themselves are then the "axioms", and are trivially proven by themselves. Such a system is not recursively axiomatizable. – Conifold Aug 08 '22 at 05:25
  • @Conifold, you should make that comment an answer. – David Gudeman Aug 08 '22 at 06:19
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    Nothing to do with Godel. If God is omniscient he knows everything. – Mauro ALLEGRANZA Aug 08 '22 at 10:14
  • Godel proved that consistency offormal arithmetic cannot be proved in the formal system itself. – Mauro ALLEGRANZA Aug 09 '22 at 08:51
  • Gentzen proved the consistency of formal arithmetic in a different (stronger) system. – Mauro ALLEGRANZA Aug 09 '22 at 08:52
  • Conclusion: this example show that it is not true - without further specificatioms - that there are math facts that ccannot be proven. – Mauro ALLEGRANZA Aug 09 '22 at 08:54
  • And God knows every facts, not necessario using proofs. – Mauro ALLEGRANZA Aug 09 '22 at 08:55
  • It seems like OP's real question is whether there is any class of knowledge that is necessarily unreachable, and only offers up God as an example of a potentially limiting case. But the answers (so far) seem to be mostly wrapped up in the concept of God specifically, rather than analysis of what knowledge is or whether there are limits that would necessarily apply to any kind of "knower." – Tom Aug 11 '22 at 03:18

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In direct answer to the title question, here's a little discourse from an encyclopedia article on omniscience:

Another recent concern is whether it really is possible to know all truths. Grim (1988) has objected to the possibility of omniscience on the basis of an argument that concludes that there is no set of all truths. The argument (by reductio) that there is no set T of all truths goes by way of Cantor’s Theorem. Suppose there were such a set. Then consider its power set, ℘(T), that is, the set of all subsets of T. Now take some truth t1. For each member of ℘(T), either t1 is a member of that set or it is not. There will thus correspond to each member of ℘(T) a further truth, specifying whether t1 is or is not a member of that set. Accordingly, there are at least as many truths as there are members of ℘(T). But Cantor’s Theorem tells us that there must be more members of ℘(T) than there are of T. So T is not the set of all truths, after all. The assumption that it is leads to the conclusion that it is not. Now Grim thinks that this is a problem for omniscience because he thinks that a being could know all truths only if there were a set of all truths. In reply, Plantinga (Plantinga and Grim 1993) holds that knowledge of all truths does not require the existence of a set of all truths. Plantinga notes that a parallel argument shows that there is no set of all propositions, yet it is intelligible to say, for example, that every proposition is either true or false. A more technical reply in terms of levels of sets has been given by Simmons (1993), but it goes beyond the scope of this entry. See also Wainwright (2010: 50–51) and Oppy (2014: 223–244).

The parallel entry on omnipotence offers that using class-much an amount of power, of "more energy or force than can be quantified by any transfinite cardinal," is impossible, so saying God knows the class of all truths rather than some purported set of them all, either stands or falls with a counterpart claim about divine might, perhaps.

As far as the incompleteness problem goes, we might construct a similar argument such as:

  1. God knows how to prove anything whatsoever.
  2. So God knows how to prove a theory to be complete/consistent.
  3. Therefore, God's knowledge violates the incompleteness protocols.
  4. It is impossible to violate the incompleteness protocols. Ergo...

However, as far as (2) goes, God might prove any lower theory complete/consistent by adverting to a higher theory. Worse, and to this day I don't know how this really works, a model of (at least some) set theory, of uncountable cardinality κ, can be compressed into a countable model. Yet at the same time, worldly cardinals are (relatively) uncountable cardinals such that the smallest of them is meant to be the smallest cardinal that yields a transitive model of (some) set theory (or theories). So there is some sense of "model" that is ambiguous over the option of reducing infinity to countability, or requiring that it be inflatable over uncountable systems to boot. And don't get me started on how proof theory sounds like it flies in the face of the incompleteness protocols: it doesn't, not really (again, consider relative-consistency analysis), but understanding why this is so will not be obvious without a better sense of what is at stake respecting those protocols in the first place.

So for all that, God might know how to prove anything, yet might not need to know anything by proving it (but rather by intellectual intuition), or can use a divine version of model reduction to compress any given set of truths into a knowable form, or who knows what.

Kristian Berry
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  • In the second part of the answer you construct an argument on the premise "God knows everything implies he can prove anything". I think knowing the truth value of a proposition and being able to prove it are two different things. – kutschkem Aug 08 '22 at 12:00
  • @kutschkem, no doubt, but at the same time, if God's unity is reflected in the form of Its knowledge (say), or then if the divine intellect's virtually differentiable functions are materially united, God's knowledge-that will be of a piece with Its knowledge-how, alongside knowledge-whether; and so God might know *that* **X** *is* knowledge-*how* to know *whether* **Y**, and yet then God also justs knows **X** and **Y** themselves. – Kristian Berry Aug 08 '22 at 12:10
  • But that assumes there is a "how to know whether Y", which you show there is not. That's like saying an Omniscient being knows where the unicorns are, and concluding Omniscience can't exist because there are no Unicorns. When in fact the real knowledge here is that there are no Unicorns, or in other words, knowing the Unprovability of a proposition( but still its truth value). – kutschkem Aug 08 '22 at 12:20
  • @kutschkem, I'm not denying that, it's just not relevant enough, here. An axiom that could never be turned into a theorem of a greater axiom would be absolutely unprovable, but incompleteness is more about relative unprovability. – Kristian Berry Aug 08 '22 at 13:41
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First, what does it mean to know?

Knowledge can be placed into two broad categories, particulars and universals. Particulars are raw sense impressions, universals are abstractions from particulars. According to Avicenna we do not know particulars, we only know them through their abstractions into universals. This is broadly Kant's view given his theory of mind in his Critique of Pure Reason.

Avicenna goes on further to say that Allah/God also only knows particulars through universals but for a different reason: He is not in time. This was controversial as it appeared to place limitations on Allah/Gods omniscience with in particular, al-Ghazali being quite forceful in his critique.

Averroes criticised both in the basis that its does away with the distinction between divine and human knowledge. Allah/Gods knowledge is neither particular nor universal. It is not particular because it is not gained through sense impressions and nor is it universal as it is not abstracted from particulars. Allah/God's knowledge is radically different.

Now mathematics and mathematical truths are universals abstracted from particulars. In particular they are truths tied closely with the logical structure of neccessity in our universe. Our understanding of this is tied in with the rational structure of our minds. But this seen from the divine perspective is contingent as Allah/God is the only neccessary being. So whilst Goedels result is a necessary truth, it is still only a necessary truth in so far as it is comprehended by rational beings, which are all possible beings in this universe and also in so far it is about this universe. From sub species aeternitas (in relation to eternity) it is not necessary.

So to know, is to know as human beings. In this sense Allah/God does not know, as he is not human. He does not have human knowledge, what He has is divine knowledge and this is generally taken to be omniscient and hence, yes, He does know, but not in ways that we can comprehend.

Mozibur Ullah
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  • Particulars, universals, sense impressions, Avicenna, Kant. It's like the 20th century has never happened. The idea sb would read that and think it's a philosophical answer up to date is kind of sad. –  Aug 09 '22 at 07:15
  • @Mr.White: The 20C is in love with itself. Like Pound said, "make it new". The riposte should be from what? This was also Simone Wel's response. It's not possible to make things *ex nihilo*. Thus there is a continuity with the past. Take for example, Deleuze, a preeminently modern philosopher; he borrowed a lot of concepts from traditional metaphysics and renamed them. Thats nice work if you can get it. However, it disguises the origins of concepts and just makes tge task of understanding what is said that much harder. Its like I renamed a square to "brxyiz". It's hard to pronounce, shows ... – Mozibur Ullah Aug 20 '22 at 12:08
  • @Mr. White: ... no relation to other parallel concepts and abandons any continuity with the past. Does that make it really modern? Don't you find that "sad"? – Mozibur Ullah Aug 20 '22 at 12:09
  • An answer that starts with "Knowledge can be placed into two broad categories, particulars and universals." was up to a good start in 1912 and is hopelessly outdated, today. Now, you can make a drama about this simple fact about epistemology. Or write about fact and "fact"... I don't care. Just, if you had studied the field or you would listen to people who have, you would get it. So long. –  Aug 20 '22 at 12:15
  • @Mr. White: Really? For all your evocation of "being up to date" and criticising me for not "being up to date", you seem to offer no other critique of substance. Can you state an actual critique from your knowledge of the "up-to-date" other than mouthing platitudes? If you "don't care", I certainly don't see why I should. – Mozibur Ullah Aug 20 '22 at 12:22