MrStakhanovite wrote:keithb wrote:Not only this, but I think that the whole idea of the hierarchy of infinite sets with different cardinalities also seems to pose problems to this notion, as I noted in one of my earliest posts on this forum.
Suppose that a god exists with some sort of knowledge set that is infinite in scope. What I argued in my post (and what I believe can be easily shown) is that there exists a super set that can be constructed from this original set (assuming the axiom of choice) that has a higher cardinality than the original set (and thus can not be mapped onto the original set with a one-to-one correspondence).
I am sure that Tarski could probably talk in more depth about these types of theoretical mathematical topics than I could, but the very idea of an omniscient god existing is for me problematic on a philosophical and mathematical level.
That was a cool thread by the way, I forget if this was answered, but couldn't one define God's knowledge as the set of all sets?
I have so many thoughts on this that I just need a time to sit and write them down. I've just been so busy lately that it's hard to even post on interesting threads, let alone write the several pages of text that I would need to try and express some of the ideas that I've thought about lately.
My thoughts on this would be that you would be unable to really write down a set of all possible knowledge for a god, no matter how big the set is. Mathematicians realize this fact about the set of all supersets, which is why they define another mathematical object in its place called a "class". However, let's think about what this would imply a bit for a god in terms of being omniscient.
If we define a set of knowledge for god, no matter how large, we can construct a larger set of knowledge from the original set by constructing the superset of the original set (defined as the set of all possible subsets of the original and permissible as long as the axiom of choice holds in ZF set theory).
However, this would be unsettling in terms of a supposedly omniscient god. Why?
Well, because there is no largest set of knowledge. No matter how large we make the original set (even a very high cardinality of infinity), there will always exist a larger infinite set of knowledge. Period.
Now, I guess someone could argue that the knowledge of this god is progressing infinitely fast and has been doing so for an infinitely long time, so somehow this god has acquired all of the knowledge possible through this process, somewhat like the Mormon idea of eternal progression. But, if we relate knowledge to sets of statements about the real numbers (I guess taking a Platonist approach to numbers -- that true statements exist about numbers independent of the human mind) even this brings up a few questions:
1. How is this god expanding its knowledge base forever? First, as Turing showed in a paper (I don't have the reference handy but can look it up on request), there are infinitely many numbers that are
not computable. He showed this by enumerating the possible computer programs as computing some set of numbers and then taking the diagonal numbers of this enumerated list. So, this would raise the question of how exactly this god is expanding its knowledge base if no computer programs exist (or can exist) to compute certain true statements.
2. Where is this god storing all the information? The only way that we humans know to store information is ultimately through energy, and there can't be an infinite density of energy, for obvious reasons (i.e. the Big Bang). Also, we have only seen particles that have a limited number of accessible quantum states (i.e. spin), so it doesn't seem reasonable to say that a god could store infinitely much information on a single particle, since this would presumably mean an infinite number of quantum states accessible by each particle to store each piece of information. So, does this god have a universe somewhere (or infinitely many of them) that it is using to store all of this ever increasing information that it is acquiring?
3. How is this god retrieving information? If a god knows everything that it is possible to ever know but can't retrieve the information, then can we really say that the god is omniscient? It's kind of like me saying, "I know what's going to happen a year from now today, but it will take me five years to retrieve the information." So, does the god have some sort of a super retrieval system to sort through the multiple universes that he is using as a hard disk in order to find pertinent information instantaneously?
There are other questions I have about the supposed idea of omniscience and omnipotence, but I guess this makes a good start. Just to give an example of the scope of the problem facing a being who claims to be (or thinks mistakenly) that it is god, here are a few of the situations a god should already know everything about:
1. What the universe will look like in 10 minutes.
2. What the universe would look like with one star less.
3. What the universe would look like with one star more.
4. The fastest possible draw in a chess game if both sides removed one pawn at random.
5. The first occurrence of a string of 1 million 2's in a row in the number Pi (or if such a string never occurs).
6. What the earth would look like if it were suddenly to turn into a giant cheese burger.
And so on ... for infinitely many possibilities.
Does this mean that there is no god? No. Possibly there are solutions to these problems that I am not smart enough to see. After all, there are infinitely many things that I don't know about the universe, as I just explained at the first of my post.
However, this does raise several serious questions in my mind as to how some being could be omniscient. It's an easy thing to say ... much harder to actually imagine or explain logically.