A Defense of Divine Omniscience for KeithB

[Guest]

First,

This is just me trying to describe the omniscience is a language that is coherent and systematically definable to try and get around some things brought up by keithb. This is nothing but a puzzle that I’m trying to work out, it’s not a proof for or against God, simply an attempt to find a way to talk about a divine property.


Second, some definitions:

X + Y = The unions of X and Y

X ~> Y = X is a member of Y

For any property O, that is, if O is the property of being omniscient, then God has that property to the maximal degree. The best way to define O is to use transfinite recursion to define a series of degrees of O, which approach the absolutely maximal degree of O.

For every ordinal n, we define an nth degree of omniscience. Omniscience will be understood as divine knowledge of some propositions. So I propose to use transfinite recursion to define an endless series of degrees of divine knowledge of some propositions and each degree of divine knowledge with some collection of unique propositions that can be known (shown as C).

The Hoops Rule: For the initial ordinal 0, there exists an initial degree of omniscience O. This is O(God, 0). 0 degrees of omniscience is the null degree. O(God,0) = {}

The Blixa Rule: For every successor ordinal, n+1 on the maximal ordinal line, there exists a successor degree of omniscience. To extend O(God, n) to O(God, n+1) we just add a collection of unique propositions that can be known C(n).

O(God, 1) = O(God,0) + {C(0)} or more precisely: {C(0)}

So O(God, 3) can be seen as {C(0), C(1), U(2)}

Now, O(God, n+1) = {C(0),…C(n)}, I’ll assert that since n+1 = {0,…n}, it follows that for any successor ordinal n+1, O(God, n+1) = {C(i)| i ~> n+1}

The Tarski Rule: For every limit ordinal L on the maximal ordinal line, there exists a limit degree of omniscience. This is O(God, L). Just as the ordinal L is defined in terms of all the ordinals less than L, so O(God, L) is defined in terms of O(God, i) for all i less than L.

Hence: O(God, L) = {C(i) | i ~> L]}

For any ordinal x:

O(God, x) = {C(i) | i ~> x}

Now the sequence of degrees indexed by ordinals rises to a maximal degree of omniscience, but this degree is not indexed by any ordinal, just, O(God). To express this better, I'll use the collection of all ordinals be denoted as OMEGA. This collection of ordinals is too general to be a set, but OMEGA is a proper class. O(God) has the rank of OMEGA. The proper class of ordinals is absolutely infinite.

O(God) = {C(i) | i ~> OMEGA}

[_MrStakhanovite]

HAHAHAHAHAHAHAH THIS IS WHAT I DO ON A SATURDAY AFTERNOON

GOD IS DEAD

DEATH IS CERTAIN

[_zeezrom]

Good God, Stak. Is this how our frolicking will go? I was looking forward to more of a Sacrates type, not Leonhard Euler!

Recursion?! You should start programming in Scheme to figure the Divine Omniscience.

[_MrStakhanovite]

Good God, Stak. Is this how our frolicking will go?

no sir. But I was thinking about Keith's comments about power sets, and thought one could avoid those issues by casting it terms of ordinals instead of cardinals.

[_just me]

The answer is sex.

[_MrStakhanovite]

The answer is sex.

Heroin is better than sex, hence my gif.

[_keithb]

Here are a few of my initial thoughts.

First, awesome job outlining this into a more formal mathematical language.

Second, at the end of the day, I think that you are left with the same problem -- that there is no largest set of knowledge (no supremum) for your recursively defined set of C{(n+1)} = U{ C(0), C(1), C(2), ... C(n+1)} if you consider each C(n) to be the set of all possible subsets of the (possibly infinite) set of knowledge C(n-1). To me, the proof of this would be as follows. Perhaps someone with more mathematical knowledge than I have can check my work?

Theorem 1: For a set of discrete statements about any system S, a larger set of knowable statements can be derived from unions of the statements in S that are also true and also knowable to an omniscient being.

Proof: Take any system of discrete statements {s_1, s_2, s_3, ... s_n} in S. Assume that the union of a subset of these statements is not in S -- i.e. that for some combination of s_n in S, there exists {s_n | s_n is in S} such that U(s_n | s_n is in S) is not a member of S. However, this statement can be rewritten as U{s_k}U{s_n | s_n is in S}/s_k. One of these statements must not lie in S for their union not to lie in S. However, we know by definition that s_k lies in S. Therefore, U{s_n | s_n is in S}/s_k is not in S, which can be redefined as another, smaller subset of U(s_m | s_m is in S) where s_m = s_n/s_k.

However, this process can be repeated for the new set U(s_m) and a new proposition s_j. We could repeat this process (infinitely many times if necessary) to systematically exclude all statements s_n until we arrive at the null set. However, this contradicts the initial proposition that s_n is non-empty. Q.E.D.

The fact that these combinations would be knowable to a god follows from the definition of omniscience.

Theorem 2: For all n > 0, the cardinality of the set C(n) is greater than or equal to the cardinality of the natural numbers.

Proof: The simplest proof of this comes through construction. Let sqrt(n) be the square roots of all of the natural numbers [0, infinity). It follows that the set of all these values comprises a system that has the same cardinality as the natural numbers.

Theorem 3: For any system of statements with the same cardinality as the natural numbers, there exists (assuming the axiom of choice) a set of facts -- also knowable by a god -- that has a cardinality greater than the natural numbers. Further, the existence of a superset with higher cardinality can be extended to infinities with a higher cardinality than the natural numbers.

Proof: From Theorem 1, we know that it is possible to construct a union of subset of any discrete system that must also be true, even an infinite set of them. Also, it is trivial to prove that the real numbers are the set of all possible subsets of the naturals.

Further, we can use Cantor's proofs about attempts to map the real numbers onto the natural numbers to show that the set of all possible subsets of the reals comprises a set with a higher cardinality than that of the naturals.

Theorem 4: There is no supremum of the set C(n) = U{C(n-1), C(n-2),... C(1)}.

Proof: Choose a value of N such that C(N) is the supremum of the set defined above. However, it is possible by Theorems 1 and Theorem 3, it is possible to construct a set from the superset of C(N) that has a higher cardinality than the original set C(N). Therefore the assumption of a supremum is contradicted.

So, there are my thoughts on the matter. I am not sure if my proofs are correct or not, but they made sense when I wrote them. And, if there are mistakes, they are the mistakes of men, blah, blah, blah. Wherefore, reject not these things ... :D

To me, this leaves believers in an omniscient god with an uncomfortable choice: either believe in a god who's knowledge can't even be defined by an infinite set, no matter how large, or believe in a god that is in fact not omniscient, and whom by definition doesn't know an entire class of infinite statements that are true.

For me, it's just easier to reject the idea of omniscience.

I also wanted to explain more about what I meant by a set of knowledge that is not "defined" by an infinite set. Basically, it would mean that you could never have a god that could even precisely define what it was that he knows, even in an infinitely large set. The moment that he/she did, then you could construct a superset from the original set, meaning that god would cease to be omniscient.

Awesome thread!

[_MrStakhanovite]

Thanks Keith, it's going to take me a bit to work through all that, but I appreciate the time you put into responding.

[_MrStakhanovite]

Hey Keith,

What did you make of this:

Now the sequence of degrees indexed by ordinals rises to a maximal degree of omniscience, but this degree is not indexed by any ordinal, just, O(God). To express this better, I'll use the collection of all ordinals be denoted as OMEGA. This collection of ordinals is too general to be a set, but OMEGA is a proper class. O(God) has the rank of OMEGA. The proper class of ordinals is absolutely infinite.

This was my move to get around your orginal observation about super sets, by shifting from cardinal to ordinal.