God Paradox: Infinite Sets and Omniscience

[Speakpigeon]

According to his book, God had trouble computing pi.

What, because the guy didn't want his book to be infinitely long it means he must not know infinitely much? If the basin was 30 cubits in circumference, was God supposed to write that it was 9.5492965855... cubits across? So he rounded off, so sue him.

Nah, it would take an infinite amount of time and I don't know yet if I have that much to live.
EB

 

[Speakpigeon]

Now for the omniscience paradox.

Since there is no largest set, there can be no such thing as complete omniscience. That is because if one takes the set of everything that one knows about, then one can construct a larger set from it by taking its power set.

Nah, you are just moving the goal posts, literally.

To know something is to know something that exists in actual reality. You cannot know something that doesn't exist. So God being omniscient doesn't imply that He should know all conceivable sets. Instead He will know any set He cares to actually conceive of. Once He decides to imagine the power set of A then He knows it.

That's the kind of omniscience God does know anyway.

Although we can do it too.

But God can do it faster.
EB

 

[Angra Mainyu]

Now for the omniscience paradox.

Since there is no largest set, there can be no such thing as complete omniscience. That is because if one takes the set of everything that one knows about, then one can construct a larger set from it by taking its power set.

Nah, you are just moving the goal posts, literally.

To know something is to know something that exists in actual reality. You cannot know something that doesn't exist. So God being omniscient doesn't imply that He should know all conceivable sets. Instead He will know any set He cares to actually conceive of. Once He decides to imagine the power set of A then He knows it.

That's the kind of omniscience God does know anyway.

Although we can do it too.

But God can do it faster.
EB

He's not moving the goal posts. Theist philosophers usually hold that God knows all necessary truths, including all truths about sets, etc., not only about sets of concrete objects, or sets he imagined, etc. The target is okay. He doesn't hit the target because as Inhabiting Void pointed out theists can just reply there is no set of everything God knows.

 

[Phil Scott]

Nah, you are just moving the goal posts, literally.

To know something is to know something that exists in actual reality. You cannot know something that doesn't exist. So God being omniscient doesn't imply that He should know all conceivable sets. Instead He will know any set He cares to actually conceive of. Once He decides to imagine the power set of A then He knows it.

That's the kind of omniscience God does know anyway.

Although we can do it too.

But God can do it faster.
EB

He's not moving the goal posts. Theist philosophers usually hold that God knows all necessary truths, including all truths about sets, etc., not only about sets of concrete objects, or sets he imagined, etc. The target is okay. He doesn't hit the target because as Inhabiting Void pointed out theists can just reply there is no set of everything God knows.

Thanks. I'm not sure why that post got ignored.

I first read the argument from HRG way back here. HRG had this to say at the time with regards to the set/class distinction.

Personally, I consider the whole omni-debate as very much like a game on words, and akin to mental masturbation . But it is fun!"

Hey. Anything to get people talking about set theory, eh?

 

[Angra Mainyu]

He's not moving the goal posts. Theist philosophers usually hold that God knows all necessary truths, including all truths about sets, etc., not only about sets of concrete objects, or sets he imagined, etc. The target is okay. He doesn't hit the target because as Inhabiting Void pointed out theists can just reply there is no set of everything God knows.

Thanks. I'm not sure why that post got ignored.

I first read the argument from HRG way back here. HRG had this to say at the time with regards to the set/class distinction.

Personally, I consider the whole omni-debate as very much like a game on words, and akin to mental masturbation . But it is fun!"

Hey. Anything to get people talking about set theory, eh?

You're welcome.

I read an argument somewhere that all true propositions do not even form a proper class (and so, the same might apply to God's knowledge, if it's propositional, which some theists claim and others reject), but I don't remember where I found it. It depends on how you define "proper class", though.
In re: omniscience+omnipotence: I think theists would likely deny that it has to be a set, and say that from the fact that his mind is a unified, coherent whole it does not follow that its contents form a set. They might say that all truths, or the ordinals, etc., are in a sense a unified coherent whole, and if there is a claim that they're not, the person making the claim should explain why not. At any rate, they might say that it's unclear how the set/not set distinction would play a role on whether a mind is a unified, coherent whole.

 

[Phil Scott]

Thanks. I'm not sure why that post got ignored.

I first read the argument from HRG way back here. HRG had this to say at the time with regards to the set/class distinction.

Personally, I consider the whole omni-debate as very much like a game on words, and akin to mental masturbation . But it is fun!"

Hey. Anything to get people talking about set theory, eh?

You're welcome.

I read an argument somewhere that all true propositions do not even form a proper class (and so, the same might apply to God's knowledge, if it's propositional, which some theists claim and others reject), but I don't remember where I found it. It depends on how you define "proper class", though.
In re: omniscience+omnipotence: I think theists would likely deny that it has to be a set, and say that from the fact that his mind is a unified, coherent whole it does not follow that its contents form a set. They might say that all truths, or the ordinals, etc., are in a sense a unified coherent whole, and if there is a claim that they're not, the person making the claim should explain why not. At any rate, they might say that it's unclear how the set/not set distinction would play a role on whether a mind is a unified, coherent whole.

If, for any ordinal number w, there is a true proposition that w belongs to its successor, then the true propositions are not a set.

The actual formal logic of this, however, is trickier. A proper class in formal ZFC set theory is really just a formula φ with a free-variable x. The class is implicitly taken to be the totality of sets which satisfy the formula. NBG set theory goes further and actually includes proper classes as objects of the theory, but it ends up much the same: a class gets picked out by a formula. Higher-order logics do similar, but extend the classes to an infinite tower of ever larger totalities.

Now here's the thing: we know from Tarski's Theorem that there can be no formula in any of these theories which says "x is a true proposition." If there were, you could straightforwardly construct the paradox of the liar, and show the theory to be inconsistent. Thus, formally, there is no class of true propositions. Indeed, formally, there is no concept of "truth".

Instead, "true proposition" seems to be a concept belonging to yet another sort of higher domain, most likely Tarski's tower of meta-theories. There is no formula to say "true proposition of ZFC" in ZFC, but you can extend ZFC so there is. This avoids the diagonal/self-reference arguments, because statements involving "true" in the extended theory cannot refer to themselves. The predicate "true" ranges only over the original theory, just as proper classes of sets which do not contain themselves cannot refer to themselves, because sets cannot contain proper classes.

The similarity of the two situations is interesting to me, and has another angle: if you want to define "true in theory T" by extending theory T, you often do it by adding a large cardinal axiom: you extend the theory by requiring it to have a totality of objects that is bigger than anything you had previously. Inside this totality, you can carve out the intended model of your original theory, hook it up to a definition of your theory's syntax, and thus define "true proposition" as a relation between the syntax and model.

A working example of this is from Simple Type Theory (basically higher-order logic). This theory has a working implementation in a computerised theorem prover called "HOL Light", which, in order to confer trust in its system, also contains a proof of its own consistency. This might seem objectionable, as it runs counter to Goedel's Second Theorem, but actually, the proof makes use of a single additional axiom asserting that there is a very large type in which one can construct a model of the original theory. It's not a trivial proof, by any means, and really does go a long way to show that the fiddly algorithms used to implement simple type theory are bug free.

 

[Speakpigeon]

Nah, you are just moving the goal posts, literally.

To know something is to know something that exists in actual reality. You cannot know something that doesn't exist. So God being omniscient doesn't imply that He should know all conceivable sets. Instead He will know any set He cares to actually conceive of. Once He decides to imagine the power set of A then He knows it.

That's the kind of omniscience God does know anyway.

Although we can do it too.

But God can do it faster.
EB

He's not moving the goal posts. Theist philosophers usually hold that God knows all necessary truths, including all truths about sets, etc., not only about sets of concrete objects, or sets he imagined, etc. The target is okay.

Oh so we're not talking about God's omniscience after all, we're just discussing the flaws in the arguments from feeble human beings. Sure, they're wrong.

He doesn't hit the target because as Inhabiting Void pointed out theists can just reply there is no set of everything God knows.

It depends on what kind of knowledge God has. The ridiculous mistake is to project the prevailing theory of human knowledge on to God, as if it was anything but a non-starter. But that's not how God knows things. God has an intimate knowledge of things, what Russell called acquaintance knowledge. And it's not propositional so any notions of 'set' just won't ever apply. It's a bit like ants disputing that it's impossible for human beings to visit Mars because that would require an intelligence inconceivably greater than what ants are familiar with.



Yeah, Ok, human beings do possess acquaintance knowledge and, well, it's the only knowledge they possess, that's true, but, still, God does it better. Always.
EB

 

[Speakpigeon]

I read an argument somewhere that all true propositions do not even form a proper class (and so, the same might apply to God's knowledge, if it's propositional, which some theists claim and others reject), but I don't remember where I found it. It depends on how you define "proper class", though.
In re: omniscience+omnipotence: I think theists would likely deny that it has to be a set, and say that from the fact that his mind is a unified, coherent whole it does not follow that its contents form a set. They might say that all truths, or the ordinals, etc., are in a sense a unified coherent whole, and if there is a claim that they're not, the person making the claim should explain why not. At any rate, they might say that it's unclear how the set/not set distinction would play a role on whether a mind is a unified, coherent whole.

Yeah. Not bad.
I like this notion of "unified, coherent whole". It's so... quaint.
EB

 

[Angra Mainyu]

Instead, "true proposition" seems to be a concept belonging to yet another sort of higher domain, most likely Tarski's tower of meta-theories. There is no formula to say "true proposition of ZFC" in ZFC, but you can extend ZFC so there is. This avoids the diagonal/self-reference arguments, because statements involving "true" in the extended theory cannot refer to themselves. The predicate "true" ranges only over the original theory, just as proper classes of sets which do not contain themselves cannot refer to themselves, because sets cannot contain proper classes.

Right, but I think you were using a broader concept before - if I get this right?

Nowadays, we call these totalities "proper classes." Cantor called them "actual infinities", and attributed to them divine significance.

And therein lies the counterargument. God's omniscience is not thwarted by these paradoxes, because the totality of God's knowledge forms a proper class.

I got the impression that you were using "proper class" in that context to mean something like "any totality too large to be a set", and that would make God's knowledge (if he existed) a proper class.

At any rate, and regardless of how we define "proper class", the set-theoretical argument against omniscience doesn't succeed.

 

[Angra Mainyu]

Oh so we're not talking about God's omniscience after all, we're just discussing the flaws in the arguments from feeble human beings. Sure, they're wrong.

No, we're talking abot God's omniscience. The word "God", however, is used by different people to mean different things, so talking about God's omniscience might be talk about different things. However, since the target for debunking is what (most) theist philosophers call "God", going by their concept(s) seems proper.

Now, even among Christian philosophers (who make up the vast majority of theist philosophers), there are different usages of "God". But the usage of nearly all of them does seem to have some things in common. In particular, they all require that for a person to be God, he must know all truths - including all necessary truths.

So, the target is fine.

If you want to use the word "God" to mean something else, that's fine too. But that would not affect Ipetrich's argument. It's his argument, and so it's his choice what to target (which he did not specify, but I reckon it's implicit in the context: he's trying to debunk the existence of God, using "God" in a way that covers what nearly all theist philosophers, not what you might mean by that word).

It depends on what kind of knowledge God has. The ridiculous mistake is to project the prevailing theory of human knowledge on to God, as if it was anything but a non-starter. But that's not how God knows things. God has an intimate knowledge of things, what Russell called acquaintance knowledge. And it's not propositional so any notions of 'set' just won't ever apply. It's a bit like ants disputing that it's impossible for human beings to visit Mars because that would require an intelligence inconceivably greater than what ants are familiar with.

Actually, different theists have different beliefs about whether God's knowledge is propositional (I already pointed that out), and sometimes even define "God" in a way that implies propositional knowledge.
But regardless, the concept of set might still apply to non-propositional knowledge, as long as it weren't so big.
After all, if it were not for size constraints, if one is justified in saying that if A is God, then A knows what you had for breakfast, that 2+2=4, etc. (which is implicit in the way theist philosophers talk), then whether there is some sophisticated way in which the knowledge can be called "not propositional", an argument would succeed: one would just use another word instead of "proposition".

Yeah, Ok, human beings do possess acquaintance knowledge and, well, it's the only knowledge they possess, that's true, but, still, God does it better. Always.

But as long as it's coherent and justified to talk about God knowing that such-and-such, etc., whether there is some sophisticated objection to the claim that God's knowledge is propositional is immaterial: one can make an argument as if it were. Now, a theist might say it's justified to talk about God knowing that such-and-such, etc., in some cases, but not in the case in which one is making an argument against omniscience. But that would be blatant special pleading, unless the theist can give some reason to make the distinction.

 

[Speakpigeon]

No, we're talking abot God's omniscience. The word "God", however, is used by different people to mean different things, so talking about God's omniscience might be talk about different things. However, since the target for debunking is what (most) theist philosophers call "God", going by their concept(s) seems proper.

Now, even among Christian philosophers (who make up the vast majority of theist philosophers), there are different usages of "God". But the usage of nearly all of them does seem to have some things in common. In particular, they all require that for a person to be God, he must know all truths - including all necessary truths.

So, the target is fine.

If you want to use the word "God" to mean something else, that's fine too. But that would not affect Ipetrich's argument. It's his argument, and so it's his choice what to target (which he did not specify, but I reckon it's implicit in the context: he's trying to debunk the existence of God, using "God" in a way that covers what nearly all theist philosophers, not what you might mean by that word).

It depends on what kind of knowledge God has. The ridiculous mistake is to project the prevailing theory of human knowledge on to God, as if it was anything but a non-starter. But that's not how God knows things. God has an intimate knowledge of things, what Russell called acquaintance knowledge. And it's not propositional so any notions of 'set' just won't ever apply. It's a bit like ants disputing that it's impossible for human beings to visit Mars because that would require an intelligence inconceivably greater than what ants are familiar with.

Actually, different theists have different beliefs about whether God's knowledge is propositional (I already pointed that out), and sometimes even define "God" in a way that implies propositional knowledge.
But regardless, the concept of set might still apply to non-propositional knowledge, as long as it weren't so big.
After all, if it were not for size constraints, if one is justified in saying that if A is God, then A knows what you had for breakfast, that 2+2=4, etc. (which is implicit in the way theist philosophers talk), then whether there is some sophisticated way in which the knowledge can be called "not propositional", an argument would succeed: one would just use another word instead of "proposition".

Yeah, Ok, human beings do possess acquaintance knowledge and, well, it's the only knowledge they possess, that's true, but, still, God does it better. Always.

But as long as it's coherent and justified to talk about God knowing that such-and-such, etc., whether there is some sophisticated objection to the claim that God's knowledge is propositional is immaterial: one can make an argument as if it were. Now, a theist might say it's justified to talk about God knowing that such-and-such, etc., in some cases, but not in the case in which one is making an argument against omniscience. But that would be blatant special pleading, unless the theist can give some reason to make the distinction.

Let me say it differently.

The argument may work against propositional knowledge but this requires assuming God only possess propositional knowledge, so it's a conditional: If God only possesses propositional knowledge then He is not omniscient, but then who said God possessed only propositional knowledge? Sure, if there are philosophers of religion who have a notion of God as only possessing propositional knowledge then too bad for them. As I already said, sure, they're wrong. But not if, as indeed already explained, they have a different concept of God's form of knowledge, then they're immune to the argument against omniscience. I'm not arguing in support of dumb philosophers of religion. I'm satisfied that they can get it wrong.I'm arguing from the premise that God is taken to be omniscient and I say there's no contradiction. There's a contradiction only if you move the goal posts by sneaking in the ridiculous assumption that His knowledge is only propositional, premise which has not been made explicit in the OP and cannot be demonstrated to be true. Of course, assuming that non-propositional knowledge somehow magically reduces to propositional knowledge is a non-starter. It is equivalent to assuming from the start that God is not omniscient, a straightforward begging of the question. It's wrong for the same reason that integers cannot match with reals, or any non-empty continuous interval of reals, despite being infinite themselves. There are different kinds of infinities which are not commensurate to each other. And God would have to be incommensurate to all but the largest.
EB

 

[Angra Mainyu]

The argument may work against propositional knowledge but this requires assuming God only possess propositional knowledge, so it's a conditional: If God only possesses propositional knowledge then He is not omniscient, but then who said God possessed only propositional knowledge?

The argument could not work for the reasons already explained by Inhabiting Void, but if it did, it would not require that God possessed only propositional knowledge. For example, if God knew all true propositions, that would be enough to make the argument run regardless of whether God somehow also knew something else apart from propositions - using the totality of God's propositional knowledge instead of the totality of God's knowledge.

Sure, if there are philosophers of religion who have a notion of God as only possessing propositional knowledge then too bad for them.

That's not the notion they have. The notion some theistic philosophers or religion have entails that God knows all true propositions. It does not entail that that is the totality of God's knowledge, though it would be the sort of knowledge relevant to the matters at hand. For example, experiences might be another kind of knowledge, but not relevant to the matters at hand, but their notion of omniscience entails that God's knowledge of, say, the fact that I ate a banana less than 2 hours ago is propositional. Regardless, without that notion, the argument would run anyway if it didn't fail for other reasons, just using the theist's own language when they talk about God's knowing such-and-such stuff.

But not if, as indeed already explained, they have a different concept of God's form of knowledge, then they're immune to the argument against omniscience.

They are, but not for the reason you give, as I already explained, because they are committed to the coherence and even justification of claims like "God knows that 2+1=3", etc. Whether one calls it "proposition" or something else, and whether there is some sophisticated metaphysical distinction saying it's not a proposition when God knows it, is irrelevant for the purposes of the argument, since it could be run on the same basis, just using the same language theists themselves use to talk about God's knowledge. Again, if a theist might say it's justified to talk about God knowing that such-and-such, etc., in some cases, but not in the case in which one is making an argument against omniscience. But that would be blatant special pleading, unless the theist can give some reason to make the distinction.

That said, the argument fails for the reason already explained by Inhabiting Void.

I'm arguing from the premise that God is taken to be omniscient and I say there's no contradiction.

You haven't given a good argument for that. Inhabiting Void gave a good argument showing that a particular argument meant to establish that God's omniscience would be contradictory, fails.

There's a contradiction only if you move the goal posts by sneaking in the ridiculous assumption that His knowledge is only propositional, premise which has not been made explicit in the OP and cannot be demonstrated to be true.

I don't know that there is a contradiction in that case (I've not seen it), but that aside, you have not given a good argument.

Of course, assuming that non-propositional knowledge somehow magically reduces to propositional knowledge is a non-starter.

No such assumption is made, but it's irrelevant anyway. Make the assumption that , and you won't be able to derive a contradiction (if you think otherwise, I'd like to ask you to prove the contradiction from the premise).

It is equivalent to assuming from the start that God is not omniscient, a straightforward begging of the question. It's wrong for the same reason that integers cannot match with reals, or any non-empty continuous interval of reals, despite being infinite themselves.

No, that is not true. In order to derive a contradiction by means of that sort of diagonal argument, one would require an assumption that God's knowledge is a set, or that God's propositional knowledge is a set, or something along those lines, not the assumption that God's knowledge is only propositional. That assumption would not do it (if you think otherwise, I'll ask you to assume that Bob has only propositional knowledge and knows all true propositions, and then derive a contradiction by means of a diagonal argument).

There are different kinds of infinities which are not commensurate to each other.

Sure.

And God would have to be incommensurate to all but the largest.

What does that mean?
When I read about infinities which are not commensurate to each other, I understand that we're talking about the non-existence of bijections. That is what it means for two infinities not to commensurate to each other in the sort of arguments put forth in this thread. Of course, in that sense, it would be contradictory incoherent to say that God would have to be incommensurate to all but the largest. God is not the sort of entity, thing, or whatever you call it (saying God is not an entity won't help) for which it makes sense to talk about whether it commensurates to some infinity.
So, I would like to ask you what you mean.

 

[Phil Scott]

At any rate, and regardless of how we define "proper class", the set-theoretical argument against omniscience doesn't succeed.

To my mind, there is only this one definition of proper class. The stuff about "true propositions" is independent, but structurally similar: you get ramified hierarchies of totalities to avoid self-referential paradoxes, just as you get ramified hierarchies of metatheories.

The argument doesn't succeed unless you can defend the idea that God's knowledge forms a set. bd-from-kg tries that in the thread I linked, but I can't say I follow the argument. I'm pretty suspicious of that phrase "unified, coherent whole." But then, I also think we're taking serious liberties when we assume that "knowledge" is a sufficiently coherent notion that we can go straight to discrete mathematics for its analysis. I might be with Speakpigeon here. But then, where's the fun in that?

 

[Angra Mainyu]

At any rate, and regardless of how we define "proper class", the set-theoretical argument against omniscience doesn't succeed.

To my mind, there is only this one definition of proper class. The stuff about "true propositions" is independent, but structurally similar: you get ramified hierarchies of totalities to avoid self-referential paradoxes, just as you get ramified hierarchies of metatheories.

The argument doesn't succeed unless you can defend the idea that God's knowledge forms a set. bd-from-kg tries that in the thread I linked, but I can't say I follow the argument. I'm pretty suspicious of that phrase "unified, coherent whole." But then, I also think we're taking serious liberties when we assume that "knowledge" is a sufficiently coherent notion that we can go straight to discrete mathematics for its analysis. I might be with Speakpigeon here. But then, where's the fun in that?

What would be the only one definition?
I'm asking because you said in your first post (where you explained why the OP argument failed) that we call totalities too big to be a set "proper classes.", and you gave the example of the totality of all truths as a totality too big to be a set. Is the only definition you have in mind that a proper class is any totality too big to be a set?

I'm not sure on what you'd agree with Speakpigeon. Have you read my latest reply to him?

 

[Phil Scott]

What would be the only one definition?
I'm asking because you said in your first post (where you explained why the OP argument failed) that we call totalities too big to be a set "proper classes.", and you gave the example of the totality of all truths as a totality too big to be a set. Is the only definition you have in mind that a proper class is any totality too big to be a set?

Yep. That's the informal definition. Formally, you might have to say something like "a proper class is any class which is not a member of any class", but such things are still exactly what Cantor had in mind when he talked of totalities that are just too big to be sets. The proper classes are always big. They sit at the top of the lattice defined by the subclass relation.

I'm not sure on what you'd agree with Speakpigeon. Have you read my latest reply to him?

Let's say I'm "sympathetic", not to anything specific. To the extent that the argument is to be taken seriously, I agree on all the points you've raised, but I would also have no problem with a theist dismissing it out-of-hand, saying that it presumes a metaphysical/epistemological analysis of knowledge and propositions that they haven't bought into, or saying that they take God's omniscience as a divine mystery beyond the ken of us mortals.

 

[cpollett]

Yep. That's the informal definition. Formally, you might have to say something like "a proper class is any class which is not a member of any class", but such things are still exactly what Cantor had in mind when he talked of totalities that are just too big to be sets. The proper classes are always big.

It might not hurt to tack on the definition of class... Since its short here goes: a class consists of those sets which satisfy some fixed formula of set theory with one free parameter. I.e., given a formula of set theory, F(x), the class corresponding to it is \(x \in C\) iff the F(x) formula holds. This is often written like \(\{x | F(x)\}\). Some classes are sets. For example, \(\{x | x \not\in x\}\) is the empty set. Some classes are not sets, for example, \(\{x | x = x\}\) is the universal class consisting of all sets. Classes which are not sets are called proper. This doesn't contradict anything above, though, I thought it might be useful for the discussion. I suppose an Omniscient god, if it desired, could serve as an oracle for formulas involving the \(\in\) relation of the "real set universe", so that if any finite being asked some fixed finite question expressed as a set theory sentence about the \(\in\) relation, it could answer it.

 

[Phil Scott]

Yep. That's the informal definition. Formally, you might have to say something like "a proper class is any class which is not a member of any class", but such things are still exactly what Cantor had in mind when he talked of totalities that are just too big to be sets. The proper classes are always big.

It might not hurt to tack on the definition of class... Since its short here goes: a class consists of those sets which satisfy some fixed formula of set theory with one free parameter. I.e., given a formula of set theory, F(x), the class corresponding to it is \(x \in C\) iff the F(x) formula holds. This is often written like \(\{x | F(x)\}\). Some classes are sets. For example, \(\{x | x \not\in x\}\) is the empty set. Some classes are not sets, for example, \(\{x | x = x\}\) is the universal class consisting of all sets. Classes which are not sets are called proper. This doesn't contradict anything above, though, I thought it might be useful for the discussion. I suppose an Omniscient god, if it desired, could serve as an oracle for formulas involving the \(\in\) relation of the "real set universe", so that if any finite being asked some fixed finite question expressed as a set theory sentence about the \(\in\) relation, it could answer it.

This is cool, but it's all fairly specific to theories like ZF where proper classes aren't in the domain of the theory, but have to be defined externally by talking about formulas with a free variable. If you take NBG set theory, you have a true theory of classes, where sets are just those classes which are elements of other classes, and proper classes are those classes which are not. You end up proving the theorem that the Russell class R is such that \(\forall X. R \not\in X\).

(I left set theory behind some time ago when I discovered type theory, which has whole dimensions of towers of stratification, which leaves me even less impressed by the original argument )