That’s your crux? Lesser interpretations than E won’t do?
I am not convinced that E is logically coherent. It’s as meaningless as “married bachelor”.
Suppose that God’s memory is the set of “all facts” O.
The set of all subsets (or powerset) of O, we’ll call p(O).
Then, for any given fact f, there is a further fact f ′ stating that it’s either in or not in each subset of O in p(O).
Thus, there must be at least as many facts as there are elements of p(O), which, being the powerset of O, by Cantor’s Theorem must have a strictly greater cardinality than O.
But we assumed that O contains all facts. Contradiction!
And Cantor’s Theorem holds even for infinite sets! Q.E.D.
Well, your argument should be able to kill the concept of Tegmark mathematical multiverse then, so you can guess it is not a “silver bullet” :) Two possible answers:
1. You can not just change the word “mathematical universe” to the word “fact” in my definition E. “Mathematical universe stating that...” makes no sense for me.
However, there are different set theory axiomatics. Some of them allow universal sets
OK, that’s a good point. I had not heard of the universal sets that contain themselves, which I thought would lead to contradictions.
should be able to kill the concept of Tegmark mathematical multiverse
I’m really not persuaded by the MUH, but at least it’s based on reasoned a priori arguments. Do you have similar a priori arguments for God? There’s no way for evidence to ever be enough establish omniscience by itself.
Yeah, given New Foundations, I’m no longer confident that “omniscience” is a logical contradiction, but neither am I confident that it isn’t. And I still think it would take an infinite amount of evidence to prove inductively, so you would need some kind of a priori argument for it instead (or why believe it at all?). That’s one obstacle down, but still a long way to go.
That’s your crux? Lesser interpretations than E won’t do?
I am not convinced that E is logically coherent. It’s as meaningless as “married bachelor”.
Suppose that God’s memory is the set of “all facts” O.
The set of all subsets (or powerset) of O, we’ll call p(O).
Then, for any given fact f, there is a further fact f ′ stating that it’s either in or not in each subset of O in p(O).
Thus, there must be at least as many facts as there are elements of p(O), which, being the powerset of O, by Cantor’s Theorem must have a strictly greater cardinality than O.
But we assumed that O contains all facts. Contradiction!
And Cantor’s Theorem holds even for infinite sets! Q.E.D.
Did I just disprove God?
Well, your argument should be able to kill the concept of Tegmark mathematical multiverse then, so you can guess it is not a “silver bullet” :) Two possible answers:
1. You can not just change the word “mathematical universe” to the word “fact” in my definition E. “Mathematical universe stating that...” makes no sense for me.
2. Cantor’s theorem is based on particular set axiomatic. However, there are different set theory axiomatics. Some of them allow universal sets https://en.wikipedia.org/wiki/Universal_set
OK, that’s a good point. I had not heard of the universal sets that contain themselves, which I thought would lead to contradictions.
I’m really not persuaded by the MUH, but at least it’s based on reasoned a priori arguments. Do you have similar a priori arguments for God? There’s no way for evidence to ever be enough establish omniscience by itself.
″ OK, that’s a good point. I had not heard of the universal sets than contain themselves, which I thought would lead to contradictions. ”
Great, the update of belief :)
Yeah, given New Foundations, I’m no longer confident that “omniscience” is a logical contradiction, but neither am I confident that it isn’t. And I still think it would take an infinite amount of evidence to prove inductively, so you would need some kind of a priori argument for it instead (or why believe it at all?). That’s one obstacle down, but still a long way to go.