Sets vs Omniscience

[ydoaPs]

Let A be the set of all things known by God. If God knows it, it's in A. It doesn't matter what it is; if it's a piece of God's knowledge, then it's in A.

Now, let's take A and construct what's called the "Power Set" of A [we'll use "P(A)" for short]. The power set is just the set of all subsets of the set. 

So, if our set is {1, 2, 3}, then it's power set is {∅, {1}, {2}, {3}, {1, 2}, {1, 3}, {2,3}, {1, 2, 3}}, where ∅ is the set of nothing. An important fact to note is that there is a lot more in the power set than there is in the original set. In fact, for any set, its power set has a higher cardinality than it. This is a famous result called "Cantor's Theorem". Power sets are always bigger than the sets from which they are generated.

If we take P(A), we can make a new set B = {"x is a subset of A"|x∈P(A)}. So we take an element of P(A) and the statement that this element is a subset of A is an element of B. And we do that for all elements of P(A). Since there is nothing else in B, this relation between P(A) and B is a bijection. So we know B and P(A) have the same size, which means B is bigger than A. Since the power set of A set just is the collection of subsets of the set, and B just is collection of statements asserting that each element of of P(A) is a subset of A, we know that every statement in B is true.

Since B is bigger than A, we can conclude that there are truths that God cannot know. If God knows infinitely many things, in fact, there are infinitely many truths that God does not know.

[MigL]

An example of another type of logic which doesn't apply to the concept of God ?
( there are many )

[studiot]

2 hours ago, ydoaPs said:

Power sets are always bigger than the sets from which they are generated.

Not always.

For example the power set of the empty set is the empty set.

I am not actually sure that we can create a set of "what God knows" within set theory in any case. We may be in 'type theory country'.

 

 

[ydoaPs]

32 minutes ago, studiot said:

Not always.

For example the power set of the empty set is the empty set.

I am not actually sure that we can create a set of "what God knows" within set theory in any case. We may be in 'type theory country'.

 

 

Yes, that should have been "nonempty set from which it is generated". Good catch.

 

I'm not sure how type theory would help here. Could you elaborate?

[studiot]

1 hour ago, ydoaPs said:

Yes, that should have been "nonempty set from which it is generated". Good catch.

 

I'm not sure how type theory would help here. Could you elaborate?

Glad you saw it that way.

 

Pure set theory is not my prime area, but straight thinking certainly is, which is why I said I am not sure.

I am a great fan of Russell.

Russell introduced type theory to cope with/get around sets which were to ornery to fit standard set theory and kept throwing up paradoxes.
So I wondered if there was an underlying paradox similar to the "An all powerful God can set himself the task of finding a task he cannot perform." for an all knowing God.

If wtf is in a good mood perhaps he might have something to say here.
I know that terminology has moved on in this subject.

[Sensei]

In Multiverses and parallel Universes any answer to any question (about state of the Universe, pure mathematics excluded) is true in "infinite" quantity of parallel Universes, and is also false in "infinite" quantity of parallel Universes.

 

[ydoaPs]

2 hours ago, Sensei said:

In Multiverses and parallel Universes any answer to any question (about state of the Universe, pure mathematics excluded) is true in "infinite" quantity of parallel Universes, and is also false in "infinite" quantity of parallel Universes.

 

I'm not sure why you thought that was a relevant response (wrong thread, maybe?), but it's trivially wrong. That would imply that there exist infinitely many universes in which the multiverse theory is false

[Sensei]

11 minutes ago, ydoaPs said:

I'm not sure why you thought that was a relevant response (wrong thread, maybe?),

No. It's response in the correct thread.

12 minutes ago, ydoaPs said:

but it's trivially wrong.

..or you didn't understand my reply..

Suppose so human being meets omniscience entity, and asks some important for him/her question e.g. "when will I die?".. Omniscience entity can tell exact date of death only exclusively when there is existing one and only one perfectly deterministic Universe.. In non-deterministic parallel Universes human being is dying at any picosecond/femtosecond/fraction of second of his/her life.. so any given answer will be true in some parallel Universe. Many people would treat such convoluted reply as sign of not being omniscience, because they learned that there is just one possible answer (it would be true in the single perfectly deterministic Universe).

27 minutes ago, ydoaPs said:

That would imply that there exist infinitely many universes in which the multiverse theory is false

If multiverse/parallel Universes theory is false, then there must be only one Universe. This one. Then there can't be infinite many universes if there is just 1.

Maybe you misinterpreted what I called "question about state of Universe".

 

[uncool]

10 hours ago, studiot said:

Not always.

For example the power set of the empty set is the empty set.

I am not actually sure that we can create a set of "what God knows" within set theory in any case. We may be in 'type theory country'.

 

 

The powerset of the empty set is the set of the empty set, P({}) = {{}}. It has one element: the empty set. 

[studiot]

5 hours ago, uncool said:

The powerset of the empty set is the set of the empty set, P({}) = {{}}. It has one element: the empty set. 

You are completely right of course, which makes me wrong. Thank you for the correction. +1

But your comment goes further since it raises the important fact that the members of the power set are sets, not the original members of the original set.

Perhaps you would like to comment further ?

[ydoaPs]

3 hours ago, studiot said:

But your comment goes further since it raises the important fact that the members of the power set are sets, not the original members of the original set.

Perhaps you would like to comment further ?

That's the reason for set B. It takes the sets and transforms them into truths

[dimreepr]

19 hours ago, ydoaPs said:

Yes, that should have been "nonempty set from which it is generated". Good catch.

God would have known.

[ydoaPs]

On 11/28/2018 at 2:51 PM, studiot said:

Glad you saw it that way.

 

Pure set theory is not my prime area, but straight thinking certainly is, which is why I said I am not sure.

I am a great fan of Russell.

Russell introduced type theory to cope with/get around sets which were to ornery to fit standard set theory and kept throwing up paradoxes.
So I wondered if there was an underlying paradox similar to the "An all powerful God can set himself the task of finding a task he cannot perform." for an all knowing God.

If wtf is in a good mood perhaps he might have something to say here.
I know that terminology has moved on in this subject.

This argument came up again recently, and I realized that appealing to proper classes or conglomerates won't actually help. See, the argument itself is a proof of negation, so it works in any nondegenerate topos. And it came to my attention that Cantor's Theorem generalizes from Set to an arbitrary topos. For any object Y in an arbitrary nondegenerate topos, there is no surjection f: Y -> 2^Y. So you can run the exact same argument swapping out talk of sets with talk of objects and talk of subsets and members with talk of subobjects. So, the argument can be formulated for an arbitrary topos.