At some point a few months ago, someone (sadly I lost their name and email) sent me a link to yet another Cantor crank. At the time, I didn’t feel like writing another Cantor crankery post, so I put it aside. Now, having lost it, I was using Google to try to find the crank in question. I didn’t, but I found something really quite remarkably idiotic.
(As a quick side-comment, my queue of bad-math-crankery is, sadly, empty. If you’ve got any links to something yummy, please shoot it to me at markcc@gmail.com.)
The item in question is this beauty. It’s short, so I’ll quote the whole beast.
MYTH: Cantor’s Set Theorem disproves divine omniscience
God is omniscient in the sense that He knows all that is not impossible to know. God knows Himself, He knows and does, knows every creature ideally, knows evil, knows changing things, and knows all possibilites. His knowledge allows free will.
Cantor’s set theorem is often used to argue against the possibility of divine omniscience and therefore against the existence of God. It can be stated:
- If God exists, then God is omniscient.
- If God is omniscient, then, by definition, God knows the set of all truths.
- If Cantor’s theorem is true, then there is no set of all truths.
- But Cantor’s theorem is true.
- Therefore, God does not exist.
However, this argument is false. The non-existence of a set of all truths does not entail that it is impossible for God to know all truths. The consistency of a plausible theistic position can be established relative to a widely accepted understanding of the standard model of Cantorian set theorem. The metaphysical Cantorian premises imply that Cantor’s theorem is inapplicable to the things that God knows. A set of all truths, if it exists, must be non-Cantorian.
The attempted disproof of God’s omniscience is, from a meta-mathematical standpoint, is inadequate to the extent that it doesn’t explain well-known mathematical contexts in which Cantor’s theorem is invalid. The “disproof” doesn’t acknowledge standard meta-mathematical conceptions that can analogically be used to establish the relative consistency of certain theistic positions. The metaphysical assertions concerning a set of all truths in the atheistic argument above imply that Cantor’s theorem is inapplicable to a set of all truths.
This is an absolute masterwork of crankery! It’s remarkably silly argument on so many levels.
- The first problem is just figuring out what the heck he’s talking about! When you say “Cantor’s theorem”, what I think of is one of Cantor’s actual theorems: “For any set S, the powerset of S is larger than S.” But that is clearly not what he’s referring to. I did a bit of searching to make sure that this wasn’t my error, but I can’t find anything else called Cantor’s theorem.
-
So what the heck does he mean by “Cantor’s set theorem”? From his text, it appears to be a statement something like: “there is no set of all truths”. The closest actual mathematical statement that I can come up with to match that is Gödel’s incompleteness theorem. If that’s what he means, then he’s messed it up pretty badly. The closest I can come to stating incompleteness informally is: “In any formal mathematical system that’s powerful enough to express Peano arithmetic, there will be statements that are true, but which cannot be proven”. It’s long, complex, not particularly intuitive, and it’s still not a particularly good statement of incompleteness.
Incompleteness is a difficult concept, and as I’ve written about before, it’s almost impossible to state incompleteness in an informal way. When you try to do that, it’s inevitable that you’re going to miss some of its subtleties. When you try to take an informal statement of incompleteness, and reason from it, the results are pretty much guaranteed to be garbage – as he’s done. He’s using a mis-statement of incompleteness,and trying to reason from it. It doesn’t matter what he says: he’s trying to show how “Cantor’s set theorem” doesn’t disprove his notion of theism. Whether it does or not doesn’t matter: for any statement X, no matter what X is, you can’t prove that “Cantor’s set theorem” or Gödel’s incompleteness theorem, or anything else disproves X if you’re arguing against something that isn’t X.
-
Ignoring his mis-identification of the supposed theorem, the way that he stated it is actually meaningless. When we talk about sets, we’re using the word set in the sense of either ZFC or NBG set theory. Mathematical set theory defines what a set is, using first order predicate logic. His version of “Cantor’s set theorem” talks about a set which cannot be a set!
He wants to create a set of truths. In set theory terms, that’s something you’d define with the axiom of specification: you’d use a predicate ranging over your objects to select the ones in the set. What’s your predicate? Truth. At best, that’s going to be a second-order predicate. You can’t form sets using second-order predicates! The entire idea of “the set of truths” isn’t something that can be expressed in set theory.
- Let’s ignore the problems with his “Cantor’s theorem” for the moment. Let’s pretend that the “set of all truths” was well-defined and meaningful. How does his argument stand up? It doesn’t: it’s a terrible argument. It’s ultimately nothing more than “Because I say so!” hidden behind a collection of impressive-sounding words. The argument, ultimately, is that the set of all truths as understood in set theory isn’t the same thing as the set of all truths in theology (because he says that they’re different), therefore you can’t use a statement about the set of all truths from set theory to talk about the set of all truths in theology.
- I’ve saved what I think is the worst for last. The entire thing is a strawman. As a religious science blogger, I get almost as much mail from atheists trying to convince me that my religion is wrong as I do from Christians trying to convert me. After doing this blogging thing for six years, I’m pretty sure that I’ve been pestered with every argument, both pro- and anti-theistic that you’ll find anywhere. But I’ve never actually seen this argument used anywhere except in articles like this one, which purport to show why it’s wrong. The entire argument being refuted is a total fake: no one actually argues that you should be an atheist using this piece of crap. It only exists in the minds of crusading religious folk who prop it up and then knock it down to show how smart they supposedly are, and how stupid the dirty rotten atheists are.
I think he, in a rather muddled and confused fashion, was referring to the exchange bewteen Plantinga and Grim, i.e. non-denumerablity of power set => non-existence of omniscience:
Patrick 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 (Grim and Plantinga, 1993) holds that knowledge of all truths does not require the existence of a set of all truths.
He 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).
http://www.apologeticsinthechurch.com/20/post/2012/04/does-cantorian-set-theory-make-omniscience-impossible-a-dialoguedebate-between-patrick-grim-alvin-plantinga.html
It’s fundamentally impossible to prove anything at all, one way or the other, about the nature of the real world. This is fundamentally because any proof requires certain assumptions, and it may be the case that those assumptions don’t actually apply to the real world. This is, of course, barring inconsistencies: a proposal that includes inconsistencies clearly cannot be true, provided those inconsistencies cannot be hidden in some fashion.
In this case, even if we were to accept that his argument that omniscience is mathematically impossible were a valid argument, it would be trivially-easy for a religious person to simply state that that’s not what they meant by omniscience. Or, alternatively, it may be the case that the mathematical framework under which omniscience were disproved turned out to not accurately describe reality.
For example, this person might be talking about Cantor’s first uncountability proof, in which case if real numbers describe something like the physical distance between two objects, then one could claim it is physically impossible to know that physical distance perfectly because there is no way to represent the number. But I could as easily point out that, in principle, it is possible that real numbers don’t actually describe anything about reality, and quantum mechanics makes it so that only integers (and rational numbers) are actually necessary at a fundamental quantum level (this isn’t necessarily the case, but it’s a possible way out).
I think you can prove certain things. For example you could always show that Newtonian mechanics don’t describe the universe perfectly just by doing certain experiments. Also, “humans can’t prove things about the real world” is a statement about the real world and if true is unprovable so you are not justified in making that statement. Even proving: “this is the only provable fact about the world” is a pretty tall order.
By “prove” here I meant in the strong sense, as in mathematical proof. Obviously we can obtain very high degrees of confidence about the natural world through experimental observation.
But the post was about somebody sitting in a room and thinking about how to prove whether or not there is a god. And that is a fool’s errand. It can’t be done.
Presumably the charitable interpratation of the argument would take all relations between sets to express truths. Thus, the set of all truths of set theory would, in this sense, essentially be the set of sentences true of V in a language augmented with constants for every member of V. Cantor’s theorem does tell us that such a creature could not be a set (though it’s more accurate to say that Russell’s paradox gives us such a result).
Of course the whole thing is silly as you suggest since there is no reason to assume that the collection of “truths” a being can know must be a set in the hierarchical sense (as opposed to say an NF set).
nek, good catch! That must be it, it’s got “Cantor’s Theorem” in it and everything.
I also agree that “the whole thing is silly.” I don’t see why a defender of “omniscience” couldn’t just say: whenever it makes sense to ask “Does God know X?” then the answer is “yes”, and leave the “set of all truths” – whatever that might mean – out of it.
I should also mention that some people (I bet Plantinga is one of them) *want* the questionable assumptions necessary for there to be a “set of all truths” – so while the whole thing is indeed silly, maybe Plantinga started it (so Grim isn’t so off base after all … ?). Not like I care, just don’t want to jump to conclusions.
Well, Platinga might be right about that one. Consider a set theory strong enough to reason about power sets and formulas in formal axiomatic systems. Now:
* the class of all strings over a countable alphabet is a countable set,
* the set of all well-formed formulas is a subset of the above (not necessarily proper, if we get the language right)
* the set of provable formulas is recursively enumerable (by mechanical listing of all possible derivations of the axioms)
Now, what is “the set of all truths”? An intuitive definition would be: a quotient set of the set of provable formulas over the relation of mutual derivability. But that is a set, and even smaller one than the one we started with.
Question: if we define things like this, what does Grim’s argument show? Because it is obvious it does not show that all “truths” cannot be contained in a set. My gut feeling is that it has something to do with the relation between first- and second-/higher-order logic, or between logical theorems and rules of inference.
“Now, what is “the set of all truths”? An intuitive definition would be: a quotient set of the set of provable formulas over the relation of mutual derivability.”
Not in the sense in which it would be interesting or important (or historically relevant) to say of God that he “knew all of them”. YMMV though.
The title is completely misleading. This might be a confused argument but it in no way purports to be a refutation of atheism. It is a (confused) refutation of one argument against theism.
I wonder if he is thinking of Godel’s incompleteness theorem, rather than Cantor.
Is there a loophole in the power-set argument? I *think* that if you take a countable set, and then take the power set of that infinitely many times, then the result is a set that’s identical to its power set. The only catch I can think of is that you have to avoid Russel’s paradox by saying that no set contains itself. So, the infinitely-deep set “{{{…}}}” exists, and it looks like it contains itself but it doesn’t really. Am I on the right track here or am I missing something?
To prove that God isn’t omniscient, I’d go the Liar’s Paradox route: We know (through faith) that God is perfectly consistent, so He doesn’t believe that anything is true and false simultaneously. So we know that, “God does not know that this sentence is true” must be true, but God can’t know that. In fact, God can’t even know that He’s perfectly consistent.
“Am I on the right track here or am I missing something?”
You are missing something.
Theorem (Cantor): If A is a set, the P(A) has cardinality strictly greater than A.
WRT the first point: There is no such thing as “the” infinitely deep set. The process of taking supersets never ends. There is no ultimate limit to the process. It just keeps going.
WRT the second: What you’re trying to do is play a game with word definitions. Word-games don’t prove much of anything except that people like playing word-games.
When you talk about “taking the powerset infinitely many times”, you must be talking about iterating the operation of powerset over some infintie ordinal. The set you end up with depends on the ordinal you choose.
If you iterate the process of taking the powerset of the natural numbers omega-many times and form the union, the resulting set U is not identical to its powerset. For example, the set {N, PN, PPN, …} is a subset of U, but not a member of U.
If you repeat the process over all the ordinals, and take the union, you end up with a proper class V, which is indeed identical with its power class. That is, V is identical with the class of all subsets of V. There is no paradox, though, as V is not a set. (NB: V is not identical to the collection of all subclasses of V.)
Doesn’t Spinoza use what looks like mathematical reasoning but isn’t to establish his theories on the divine? Using this kind of thinking isn’t always cranky. And a French Marxist Philosopher called Badiou is using set theory to rescue philosophy & politics from post-modern excess.