No, it’s not. Maybe it blows your mind to imagine space stretching away without limit, but if space is there independent of you, and if it has no edge, and if it doesn’t close back on itself, then it’s an actually realized infinity.
The second independent clause is true, but if (as I contend) actually realized infinities are incoherent, the proper conclusion is that the three assumptions cannot all hold.
Of course, having one’s mind blown doesn’t prove the concept entertained in incoherent; I must demonstrate that the concept really contains a logical contradiction. The contradiction in actual infinity is revealed by a question such as this one:
Assume there are an infinite number of quarks in the universe. Then, are there any quarks that aren’t contained in the set of all the quarks in the universe?
Suggestion:Answer the question thoughtfully for yourself before proceeding to my answer.
By definition, they’re all in the set. But, you can add a finite number to an infinite set and not change the number of elements. So, there are at the same time other quarks than are contained in the set of all quarks.
(I accept that Cantor demonstrated that infinities are consistent. The incoherence doesn’t lie in the mathematics of infinity but in conceiving of them as actually realized. This was also the stance of mathematician and philosopher of mathematics David Hilbert, who devised the Hilbert’s Hotel thought experiment to bring out the absurdity of actually realized infinities—while warmly welcoming Cantor’s achievements in infinity taken strictly mathematically. Or as we might say, infinity as a limit rather than as a number).
But, you can add a finite number to an infinite set and not change the number of elements. So, there are at the same time other quarks than are contained in the set of all quarks.
Could you clarify this inference, please? How does the second sentence follow from the first?
Here’s my interpretation of what you’re saying: Let the set of all quarks be Q, and assume Q has infinite elements. Now pick a particular quark, let’s call it Bob, and remove it from the set Q. Call the new set thus formed Q\Bob. Now, it’s true that Q\Bob has the same number of elements as Q. But your claim seems to be stronger, that Q\Bob is in fact the same set as Q. If that is the case, then Q\Bob both is and isn’t the set of all quarks and we have a contradiction. But why should I believe Q\Bob is identical to Q?
I agree that belief in the existence of actually infinite sets leads to all sorts of very counterintuitive scenarios, and perhaps that is adequate reason to be an infinite set atheist like Eliezer (although I’m unconvinced). But it does not lead to explicit contradiction, as you seem to be claiming.
Here’s my interpretation of what you’re saying: Let the set of all quarks be Q, and assume Q has infinite elements. Now pick a particular quark, let’s call it Bob, and remove it from the set Q. Call the new set thus formed Q\Bob. Now, it’s true that Q\Bob has the same number of elements as Q. But your claim seems to be stronger, that Q\Bob is in fact the same set as Q. If that is the case, then Q\Bob both is and isn’t the set of all quarks and we have a contradiction. But why should I believe Q\Bob is identical to Q?
Because there is no difference between Q and Q/Bob besides that Q/Bob contains Bob, a difference I’m trying to bracket: distinctions between individual quarks.
Instead of quarks, speak of points in Platonic heaven. Say there are infinitely many of them, and they have no defining individuality. The set Platonic points and the set of Platonic points points plus one are different sets: they contain different elements. Yet, in contradiction, they are the same set: there is no way to distinguish them.
Platonic points are potentially problematic in a way quarks aren’t. (For one thing, they don’t really exist.) But they bring out what I regard as the contradiction in actually realized infinite sets: infinite sets can sometimes be distinguished only by their cardinality, and then sets that are different (because they are formed by adding or subtracting elements) are the same (because they subsequently aren’t distinguishable).
If Q genuinely has infinite cardinality, then its members cannot all be equal to one another. If you take, at random, any two purportedly distinct members of Q u and w, then it has to be the case that u is not equal to w. If the members were all equal to each other, then Q would have cardinality 1. So the members of Q have to be distinguishable in at least this sense—there needs to be enough distinguishability so that the set genuinely has cardinality infinity. If you can actually build an infinite set of quarks or Platonic points, it cannot be the case that any arbitrary quark (or point) is identical to any other. If one accepts the principle of identity of indistinguishable, then it follows that quarks or points must be distinguishable (since they can be non-identical). But you need not accept this principle; you just need to agree with me that the members of the set Q cannot all be identical to one another.
Now, the criterion for identity of two sets A and B is that any z is a member of A if and only if it is a member of B. In other words, take any member of A, say z. If A = B you have to be able to find some member of B that is identical to z. But this is not true of the sets Q and Q\Bob. There is at least one member of Q which is not identical to any member of Q\Bob—the member that was removed when constructing Q\Bob (which, remember, is not identical to any other member of Q). So Q is not identical to Q\Bob. There is no separate criterion for the identity of sets which leads to the conclusion that Q is identical to Q\Bob, so we do not have a contradiction.
Believe me, if there was an obvious contradiction in Zermelo-Fraenkel set theory (which includes an axiom of infinity), mathematicians would have noticed it by now.
If one accepts the principle of identity of indistinguishable, then it follows that quarks or points must be distinguishable (since they can be non-identical)
I accept the principle, but I think it isn’t relevant to this part of the problem. I can best elaborate by first dealing with another point.
There is no separate criterion for the identity of sets which leads to the conclusion that Q is identical to Q\Bob, so we do not have a contradiction
True, but my claim is that there is a separate criterion for identity for actually realized sets. It arises exactly from the principle of the identity of indistinguishables. Q and Q/Bob are indistinguishable when the elements are indistinguishable; they should be distinguishable despite the elements being indistinguishable.
What justifies “suspending” the identity of indistinguishables when you talk about elements is that it’s legitimate to talk about a set of things you consider metaphysically impossible. It’s legitimate to talk about a set of Platonic points, none distinguishable from another except in being different from one another. We can easily conceive (but not picture) a set of 10 Platonic points, where selecting Bob doesn’t differ from selecting Sam, but taking Bob and Sam differs from taking just Bob or just Sam. So, the identity of indistinguishables shouldn’t apply to the elements of a set, where we must represent various metaphysical views. But if you accept the identity of indistinguishables, an infinite set containing Bob where Bob isn’t distinguishable from Sam or Bill is identical to an infinite set without Bob.
Believe me, if there was an obvious contradiction in Zermelo-Fraenkel set theory (which includes an axiom of infinity), mathematicians would have noticed it by now.
I’ll take your word on that, but I don’t think it’s relevant here. I think this is an argument in metaphysics rather than in mathematics. It deals in the implications of “actual realization.” (Metaphysical issues, I think, are about coherence, just not mathematical coherence; the contradictions are conceptual rather than mathematical.) I don’t think “actual realization” is a mathematical concept; otherwise—to return full circle—mathematics could decide whether Tegmark’s right.
Among metaphysicians, infinity has gotten a free ride, the reason seeming to be that once you accept there’s a consistent mathematical concept of infinity, the question of whether there are any actually realized infinities seems empirical.
Could you clarify this inference, please? How does the second sentence follow from the first?
Let me restate it, as my language contained miscues, such as “adding” elements to the set. Restated:
If there are infinitely many quarks in the universe, then I can form an infinite set of quarks. That set includes all the quarks in the universe, since there can be no set of the same cardinality that’s greater and because, from the bare description, “quarks,” I have no basis for establishing a subset/superset relationship (HT JoshuaZ) within the set of quarks. But that set does not include all the quarks in the universe because finding other quarks is consistent with the set’s defining [added 9⁄02] requirement that it contain infinitely many elements.
I agree that belief in the existence of actually infinite sets leads to all sorts of very counterintuitive scenarios, and perhaps that is adequate reason to be an infinite set atheist like Eliezer (although I’m unconvinced). But it does not lead to explicit contradiction, as you seem to be claiming.
Could you (or anyone else) possibly provide me with a clue as to how I might find E.Y.’s opinions on this subject or on what you base that he’s an infinite set atheist?
I’m also interested in how E.Y. avoids infinite sets when endorsing Tegmarkism or even the Many Worlds Interpretation of q.m. [In another thread, one poster explained that “worlds” are not ontologically basic in MWI, but I wonder if that’s correct for realist versions (as opposed to Hawking-style fictitional worlds).]
If intuitions have any relevance to discussions of the metaphysics of infinity, I think they would have to be intuitions of incoherence: incomplete glimmerings of explicit contradiction. The contradiction that seems to lurk in actually realized infinities is between the implications of absence of limit provided by infinity and the implications of limit implied by its realization.
If there are infinitely many quarks in the universe, then I can form an infinite set of quarks. That set includes all the quarks in the universe, since there can be no set of the same cardinality that’s greater and because, from the bare description, “quarks,” I have no basis for establishing a subset/superset relationship (HT JoshuaZ) within the set of quarks. But that set does not include all the quarks in the universe because more quarks can be found and still be consistent with the only requirement that there be infinitely many quarks.
I’m still confused by this argument. Are you arguing in the second sentence that “any infinite set of quarks must be the set of all quarks”? But for example I could form the set of all up quarks, which is an infinite set of quarks, yet does not include any down quarks, and so is not the set of all quarks.
Are you implicitly using the following idea? “Suppose A and B are two sets of the same cardinality. Then A cannot be a proper subset of B.” This is true for finite sets but false for infinite sets: the set of even integers has the same cardinality as the set of all integers, but the even integers are a proper subset of the set of all integers.
The key is the qualification “from the bare description, ‘quarks.’”
To elaborate—JoshuaZ’s comment brought this home—you can distinguish infinite sets by their cardinality or by their subset/superset relationship, and these are independent. The reasoning about quarks brackets all knowledge about the distinctions between quarks that could be used to establish a set/superset relationship.
By default, sets are different. You can’t argue “two sets are the same because they have the same cardinality and we don’t know anything else about them” which I think is what you’re doing.
If there are infinitely many quarks, then we can form infinite sets of quarks. One of these sets is the set of all quarks. This set is infinite, includes all quarks, and there are no quarks it doesn’t include, and saying anything else is patent nonsense whether you’re talking about quarks, integers, or kittens.
By default, sets are different. You can’t argue “two sets are the same because they have the same cardinality and we don’t know anything else about them”
Sets with different elements are different. But, unfortunately for actually realized infinities, you can argue that two sets with different elements are the same when those infinite sets are actually realized—but only because actually realized infinities are incoherent. That you can argue both sides, contradicted only by the other side, is what makes actual infinity incoherent.
You can’t defeat an argument purporting to show a contradiction by simply upholding one side; you can’t deny me the argument that the two sets are the same (as part of that argument to contradiction) simply based on a separate argument that they’re different.
for actually realized infinities, you can argue that two sets with different elements are the same
Suppose I restate your argument for integers instead of quarks:
“If there are infinitely many integers, then I can form an infinite set of integers. That set includes all the integers, since there can be no set of the same cardinality that’s greater and because, from the bare description, “integers,” I have no basis for establishing a subset/superset relationship (HT JoshuaZ) within the set of integers. [I don’t follow this sentence, so I’ve just copied it.]. But that set does not include all the integers because the existence of other integers outside the set is consistent with the set’s defining requirement that it contain infinitely many elements.”
As I mentioned above, we can form infinite sets of integers that do not include all integers, for example the set of even numbers, so the argument cannot be valid when it’s made about integers. What about the argument makes it valid for quarks but not for integers? I imagine it must have to do with your distinction between an abstract infinity and an “actually realized” infinity. Perhaps you can clarify where you are using this distinction in your argument?
To help us better understand what you’re claiming, suppose the universe is infinite and I form an infinite set of quarks, any infinite set of quarks. Is it your contention that we can prove that this set of quarks equals the set of all quarks?
Also, regarding this key sentence:
That set includes all the quarks in the universe, since there can be no set of the same cardinality that’s greater and because, from the bare description, “quarks,” I have no basis for establishing a subset/superset relationship (HT JoshuaZ) within the set of quarks.
I don’t follow this sentence, I didn’t follow the clarification you made three posts up. Perhaps you could expand this sentence into a paragraph or two that a five year old could understand?
Suppose I restate your argument for integers instead of quarks...
We don’t need to assume there are infinitely many integers, only that integers are unlimited. Some Platonists may think that an infinite set of integers is realized, and I think the arguments does pertain to that claim.
As I mentioned above, we can form infinite sets of integers that do not include all integers, for example the set of even numbers, so the argument cannot be valid when it’s made about integers. What about the argument makes it valid for quarks but not for integers? I imagine it must have to do with your distinction between [a potential] infinity and an “actually realized” infinity. Perhaps you can clarify where you are using this distinction in your argument?
The distinction is relevant to why I have no quarrel with potential infinities as such.
To help us better understand what you’re claiming, suppose the universe is infinite and I form an infinite set of quarks, any infinite set of quarks. Is it your contention that we can prove that this set of quarks equals the set of all quarks?
No. It’s only the case if (per stipulation) you know nothing about properties that distinguish one quark from another. Then, the only way you can form an infinite set of quarks is by taking all of them. So, I’m not assuming that any infinite set of quarks I can form is the only infinite set of quarks I can form; I’m setting up the problem so there’s only one way to form an infinite set of quarks. Any set conforming to that description “should” be the only set.
Perhaps you could expand this sentence:
That set includes all the quarks, since there can be no set of the same cardinality that’s greater and because, from the bare description, “quarks,” I have no basis for establishing a subset/superset relationship (HT JoshuaZ) within the set of integers.
The only way you can form an infinite set of quarks—given that you can’t distinguish one quark from another—is by selecting for inclusion all quarks indiscriminately. This is because there are only two ways that infinite subsets can be distinguished from their supersets: 1) the subset is of lower cardinality than the superset or 2) the elements are distinguishable to create a logical superset/set relationship (such as exists in quarks/upside-down quarks).
The only way you can form an infinite set of quarks—given that you can’t distinguish one quark from another—is by selecting for inclusion all quarks indiscriminately.
OK, suppose I grant this. I now feel like I might be able to formulate your argument in my own words. Here’s an attempt; let me know if and when it diverges from what you’re actually arguing.
--
“Suppose I have sworn to give up the hateful practice of discriminating between quarks based on their differences. Henceforth I shall treat all quarks as utterly indistinguishable from one another. Having made this solemn vow, I now ask you to bring me an infinite set of quarks (note that I do not specify which quarks, for that would violate my vow!). You oblige, and provide me with a set called S.
“I inspect the set S and try to see whether it’s different from the set of all quarks, which we call Q. First I look at the cardinalities of S and Q. If their cardinalities were different, then obviously S and Q would be different sets. But their cardinalities are the same. Next I look for a quark that is contained in Q, but not contained in S. If there were such an element, then obviously S and Q would be different sets. But in order to successfully find such an element, I would have to make use of the distinctions between quarks. After all, how would I know that a given quark was in Q, but not in S? I would have to show that the quark in Q was distinct from each quark in S, but I have agreed to regard all quarks as indistinguishable. Therefore my search for an element of Q that is not in S will fail. I conclude that the set S is the same as the set Q. That is the set you gave me must be the set of all quarks.
“But this conclusion is obviously wrong. All I asked you for was an infinite set of quarks. There are many infinite sets of quarks, not all of which are the same as Q, the set of all quarks. You might have left some quarks out of S, and still provided me with an infinite set of quarks, which was all I asked for.
“Therefore we have a contradiction: I have proved something that is not necessarily true. Therefore the set of quarks cannot be infinite.”
--
The response to this argument is that because I’ve blinded myself to the differences between quarks, I’ve lost the ability to show that Q and S are different. That does not mean that I’m entitled to conclude that Q and S are the same! After all, if I did allow myself to see the differences between quarks, such as their different positions in space, I might notice that Q contained a quark located at the position (3, 4, 5), but that S contained no quark at that position. This would let me see that Q and S are in fact distinct sets.
I take issue with your translation at only a single point:
Having made this solemn vow, I now ask you to bring me an infinite set of quarks (note that I do not specify which quarks, for that would violate my vow!). You oblige, and provide me with a set called S.
My version contains a further constraint: When you ask me to bring you an infinite set of quarks, you instruct me to be as blind as you to the features that distinguish between quarks.
The response to this argument is that because I’ve blinded myself to the differences between quarks, I’ve lost the ability to show that Q and S are different. That does not mean that I’m entitled to conclude that Q and S are the same! After all, if I did allow myself to see the differences between quarks, such as their different positions in space, I might notice that Q contained a quark located at the position (3, 4, 5), but that S contained no quark at that position. This would let me see that Q and S are in fact distinct sets. [emphasis added.]
The_Duck tells metaphysicist to gather together an infinite set of quarks while remaining blind to their individuality. Metaphysicist, having no distinctions on which to carve infinite subsets, can respond to this request in only one way; include every quark. (I want to resist calling this the “set of all quarks,” because the incoherence of that concept with infinite quarks is what I argue.) The_Duck then goes out and finds another quark, and scolds metaphysicist, “You missed one.”
The_Duck is unjustified in criticizing metaphysicist, who must have picked “all the quarks,” given that metaphysicist succeeded—without knowing of any proper subsets—in assembling an infinite set . Having “selected all the quarks” doesn’t preclude finding another when they’re infinite in number and the only criterion for success is the number.
You will say that there is a fact of the matter as to whether the first set I assembled was all the quarks. Unblind yourself to the quarks’ individuating features, you say, and you get an underlying reality where the sets are different. I agree, but I think a more limited point suffices. When I follow the same procedure—gather all the quarks—I will be equally justified in gathering a set and in gathering a superset consisting of one other quark. There’s no way for me to distinguish the two sets. The contradiction is that following the procedure “gather all the quarks” should constrain me to a single set, “all the quarks,” rather than allowing a hierarchy of options consisting of supersets.
I take issue with your translation at only a single point:
I’m making progress then. :)
When I follow the same procedure—gather all the quarks—I will be equally justified in gathering a set and in gathering a superset consisting of one other quark.
No. If what you gathered is a proper subset of what you could have gathered, then you didn’t gather all the quarks, and you’re not justified in claiming that you did. How did you decide to leave out that one other quark? You must have made a distinction between it and the others that you did gather.
There’s no way for me to distinguish the two sets.
Of course there is. The superset contains a quark that the subset doesn’t. If you refuse to notice the differences that single that quark out from the others, that’s your loss.
I think that maybe you’re trying not to distinguish between quarks, but are implicitly distinguishing between “quarks that you know about” and “quarks that you don’t know about.” So you might assemble all the quarks you know about—an infinite number—and not have any evidence that this isn’t all the quarks there are. But later, you worry, you might find some other quarks that you didn’t know about before, so that your original set didn’t actually contain all quarks. This is not contradictory. If there was a chance that there existed quarks you didn’t know about, then you weren’t justified in saying that you had gathered all the quarks.
following the procedure “gather all the quarks” should constrain me to a single set, “all the quarks,” rather than allowing a hierarchy of options consisting of supersets.
It does. If you’re not at the top of the hierarchy, you haven’t gathered all the quarks. And you can’t justify claiming that you’re at the top of the hierarchy by blinding yourself to evidence that would prove otherwise.
Could you (or anyone else) possibly provide me with a clue as to how I might find E.Y.’s opinions on this subject or on what you base that he’s an infinite set atheist.
But, you can always add a finite number to an infinite set and not change the number of elements. So, there are more quarks than are contained in the set of all quarks. (I accept that Cantor demonstrated that infinities are consistent. The incoherence doesn’t lie in the mathematics of infinity but in conceiving of them as actually realized. I understand that was Hilbert’s position.)
No, then there are the same number of quarks in both cases in the sense of cardinality. Your intuition just isn’t very good for handling how infinite sets behave- adding more to the an infinite set in some sense doesn’t necessarily make it larger. Failure at having a good intuition for such things shouldn’t be surprising; we didn’t evolve to handle infinite sets.
No, then there are the same number of quarks in both cases in the sense of cardinality.
Yes, I understand that; in fact, it was my express premise: “You can always add a finite number to an infinite set and not change the number of elements.” That is, not change the number of quarks from one case to another.
Please read it again more carefully. My argument may be wrong, but it’s really not that naive.
Added.
I see what you might be responding to: “So, there are more quarks than are contained in the set of all quarks.” The second sentence, not the first. It’s stated imprecisely. It should read, “So, there are other quarks than are contained in the set of all quarks.” Now changed in the original.
So, there are other quarks than are contained in the set of all quarks.
You’ve collapsed the distinction between two possible worlds. You started out by saying, consider a universe containing infinitely many quarks. Then you say, consider a universe which has all the quarks from the first universe, plus a finite number of extra quarks. The set of all quarks in the second scenario indeed contains quarks that aren’t in the set of all quarks in the first scenario, but that’s not a contradiction.
It’s like saying: Consider the possible world where Dick Cheney ended up as president for the last two years of Bush’s second term. Then that would mean that there was a president who wasn’t an element of the set of all presidents.
Replying separately to this now added comment. it still seems like this is an issue of ambiguous language. It isn’t that there are other quarks that aren’t contained in the set of all quarks.” Is is that there’s a set of quarks and a superset that have the same cardinality.
I think you responded before my correction, where I came to the same conclusion that my use of “more” was imprecise.
Added
I remember reading an essay maybe five years ago by Eliezer Yudkowsky where he maintained that the early Greek thinkers had been right to reject actual infinities for logical reasons. I can’t find the essay. Has it been recanted? Is it a mere figment of my imagination? Does anyone recall this essay?
The second independent clause is true, but if (as I contend) actually realized infinities are incoherent, the proper conclusion is that the three assumptions cannot all hold.
Of course, having one’s mind blown doesn’t prove the concept entertained in incoherent; I must demonstrate that the concept really contains a logical contradiction. The contradiction in actual infinity is revealed by a question such as this one:
Assume there are an infinite number of quarks in the universe. Then, are there any quarks that aren’t contained in the set of all the quarks in the universe?
Suggestion: Answer the question thoughtfully for yourself before proceeding to my answer.
By definition, they’re all in the set. But, you can add a finite number to an infinite set and not change the number of elements. So, there are at the same time other quarks than are contained in the set of all quarks.
(I accept that Cantor demonstrated that infinities are consistent. The incoherence doesn’t lie in the mathematics of infinity but in conceiving of them as actually realized. This was also the stance of mathematician and philosopher of mathematics David Hilbert, who devised the Hilbert’s Hotel thought experiment to bring out the absurdity of actually realized infinities—while warmly welcoming Cantor’s achievements in infinity taken strictly mathematically. Or as we might say, infinity as a limit rather than as a number).
Important changes for clarity Sept. 2.
Could you clarify this inference, please? How does the second sentence follow from the first?
Here’s my interpretation of what you’re saying: Let the set of all quarks be Q, and assume Q has infinite elements. Now pick a particular quark, let’s call it Bob, and remove it from the set Q. Call the new set thus formed Q\Bob. Now, it’s true that Q\Bob has the same number of elements as Q. But your claim seems to be stronger, that Q\Bob is in fact the same set as Q. If that is the case, then Q\Bob both is and isn’t the set of all quarks and we have a contradiction. But why should I believe Q\Bob is identical to Q?
I agree that belief in the existence of actually infinite sets leads to all sorts of very counterintuitive scenarios, and perhaps that is adequate reason to be an infinite set atheist like Eliezer (although I’m unconvinced). But it does not lead to explicit contradiction, as you seem to be claiming.
Because there is no difference between Q and Q/Bob besides that Q/Bob contains Bob, a difference I’m trying to bracket: distinctions between individual quarks.
Instead of quarks, speak of points in Platonic heaven. Say there are infinitely many of them, and they have no defining individuality. The set Platonic points and the set of Platonic points points plus one are different sets: they contain different elements. Yet, in contradiction, they are the same set: there is no way to distinguish them.
Platonic points are potentially problematic in a way quarks aren’t. (For one thing, they don’t really exist.) But they bring out what I regard as the contradiction in actually realized infinite sets: infinite sets can sometimes be distinguished only by their cardinality, and then sets that are different (because they are formed by adding or subtracting elements) are the same (because they subsequently aren’t distinguishable).
If Q genuinely has infinite cardinality, then its members cannot all be equal to one another. If you take, at random, any two purportedly distinct members of Q u and w, then it has to be the case that u is not equal to w. If the members were all equal to each other, then Q would have cardinality 1. So the members of Q have to be distinguishable in at least this sense—there needs to be enough distinguishability so that the set genuinely has cardinality infinity. If you can actually build an infinite set of quarks or Platonic points, it cannot be the case that any arbitrary quark (or point) is identical to any other. If one accepts the principle of identity of indistinguishable, then it follows that quarks or points must be distinguishable (since they can be non-identical). But you need not accept this principle; you just need to agree with me that the members of the set Q cannot all be identical to one another.
Now, the criterion for identity of two sets A and B is that any z is a member of A if and only if it is a member of B. In other words, take any member of A, say z. If A = B you have to be able to find some member of B that is identical to z. But this is not true of the sets Q and Q\Bob. There is at least one member of Q which is not identical to any member of Q\Bob—the member that was removed when constructing Q\Bob (which, remember, is not identical to any other member of Q). So Q is not identical to Q\Bob. There is no separate criterion for the identity of sets which leads to the conclusion that Q is identical to Q\Bob, so we do not have a contradiction.
Believe me, if there was an obvious contradiction in Zermelo-Fraenkel set theory (which includes an axiom of infinity), mathematicians would have noticed it by now.
I accept the principle, but I think it isn’t relevant to this part of the problem. I can best elaborate by first dealing with another point.
True, but my claim is that there is a separate criterion for identity for actually realized sets. It arises exactly from the principle of the identity of indistinguishables. Q and Q/Bob are indistinguishable when the elements are indistinguishable; they should be distinguishable despite the elements being indistinguishable.
What justifies “suspending” the identity of indistinguishables when you talk about elements is that it’s legitimate to talk about a set of things you consider metaphysically impossible. It’s legitimate to talk about a set of Platonic points, none distinguishable from another except in being different from one another. We can easily conceive (but not picture) a set of 10 Platonic points, where selecting Bob doesn’t differ from selecting Sam, but taking Bob and Sam differs from taking just Bob or just Sam. So, the identity of indistinguishables shouldn’t apply to the elements of a set, where we must represent various metaphysical views. But if you accept the identity of indistinguishables, an infinite set containing Bob where Bob isn’t distinguishable from Sam or Bill is identical to an infinite set without Bob.
I’ll take your word on that, but I don’t think it’s relevant here. I think this is an argument in metaphysics rather than in mathematics. It deals in the implications of “actual realization.” (Metaphysical issues, I think, are about coherence, just not mathematical coherence; the contradictions are conceptual rather than mathematical.) I don’t think “actual realization” is a mathematical concept; otherwise—to return full circle—mathematics could decide whether Tegmark’s right.
Among metaphysicians, infinity has gotten a free ride, the reason seeming to be that once you accept there’s a consistent mathematical concept of infinity, the question of whether there are any actually realized infinities seems empirical.
Let me restate it, as my language contained miscues, such as “adding” elements to the set. Restated:
If there are infinitely many quarks in the universe, then I can form an infinite set of quarks. That set includes all the quarks in the universe, since there can be no set of the same cardinality that’s greater and because, from the bare description, “quarks,” I have no basis for establishing a subset/superset relationship (HT JoshuaZ) within the set of quarks. But that set does not include all the quarks in the universe because finding other quarks is consistent with the set’s defining [added 9⁄02] requirement that it contain infinitely many elements.
Could you (or anyone else) possibly provide me with a clue as to how I might find E.Y.’s opinions on this subject or on what you base that he’s an infinite set atheist?
I’m also interested in how E.Y. avoids infinite sets when endorsing Tegmarkism or even the Many Worlds Interpretation of q.m. [In another thread, one poster explained that “worlds” are not ontologically basic in MWI, but I wonder if that’s correct for realist versions (as opposed to Hawking-style fictitional worlds).]
If intuitions have any relevance to discussions of the metaphysics of infinity, I think they would have to be intuitions of incoherence: incomplete glimmerings of explicit contradiction. The contradiction that seems to lurk in actually realized infinities is between the implications of absence of limit provided by infinity and the implications of limit implied by its realization.
I’m still confused by this argument. Are you arguing in the second sentence that “any infinite set of quarks must be the set of all quarks”? But for example I could form the set of all up quarks, which is an infinite set of quarks, yet does not include any down quarks, and so is not the set of all quarks.
Are you implicitly using the following idea? “Suppose A and B are two sets of the same cardinality. Then A cannot be a proper subset of B.” This is true for finite sets but false for infinite sets: the set of even integers has the same cardinality as the set of all integers, but the even integers are a proper subset of the set of all integers.
The key is the qualification “from the bare description, ‘quarks.’”
To elaborate—JoshuaZ’s comment brought this home—you can distinguish infinite sets by their cardinality or by their subset/superset relationship, and these are independent. The reasoning about quarks brackets all knowledge about the distinctions between quarks that could be used to establish a set/superset relationship.
By default, sets are different. You can’t argue “two sets are the same because they have the same cardinality and we don’t know anything else about them” which I think is what you’re doing.
If there are infinitely many quarks, then we can form infinite sets of quarks. One of these sets is the set of all quarks. This set is infinite, includes all quarks, and there are no quarks it doesn’t include, and saying anything else is patent nonsense whether you’re talking about quarks, integers, or kittens.
Sets with different elements are different. But, unfortunately for actually realized infinities, you can argue that two sets with different elements are the same when those infinite sets are actually realized—but only because actually realized infinities are incoherent. That you can argue both sides, contradicted only by the other side, is what makes actual infinity incoherent.
You can’t defeat an argument purporting to show a contradiction by simply upholding one side; you can’t deny me the argument that the two sets are the same (as part of that argument to contradiction) simply based on a separate argument that they’re different.
Suppose I restate your argument for integers instead of quarks:
“If there are infinitely many integers, then I can form an infinite set of integers. That set includes all the integers, since there can be no set of the same cardinality that’s greater and because, from the bare description, “integers,” I have no basis for establishing a subset/superset relationship (HT JoshuaZ) within the set of integers. [I don’t follow this sentence, so I’ve just copied it.]. But that set does not include all the integers because the existence of other integers outside the set is consistent with the set’s defining requirement that it contain infinitely many elements.”
As I mentioned above, we can form infinite sets of integers that do not include all integers, for example the set of even numbers, so the argument cannot be valid when it’s made about integers. What about the argument makes it valid for quarks but not for integers? I imagine it must have to do with your distinction between an abstract infinity and an “actually realized” infinity. Perhaps you can clarify where you are using this distinction in your argument?
To help us better understand what you’re claiming, suppose the universe is infinite and I form an infinite set of quarks, any infinite set of quarks. Is it your contention that we can prove that this set of quarks equals the set of all quarks?
Also, regarding this key sentence:
I don’t follow this sentence, I didn’t follow the clarification you made three posts up. Perhaps you could expand this sentence into a paragraph or two that a five year old could understand?
We don’t need to assume there are infinitely many integers, only that integers are unlimited. Some Platonists may think that an infinite set of integers is realized, and I think the arguments does pertain to that claim.
The distinction is relevant to why I have no quarrel with potential infinities as such.
No. It’s only the case if (per stipulation) you know nothing about properties that distinguish one quark from another. Then, the only way you can form an infinite set of quarks is by taking all of them. So, I’m not assuming that any infinite set of quarks I can form is the only infinite set of quarks I can form; I’m setting up the problem so there’s only one way to form an infinite set of quarks. Any set conforming to that description “should” be the only set.
That set includes all the quarks, since there can be no set of the same cardinality that’s greater and because, from the bare description, “quarks,” I have no basis for establishing a subset/superset relationship (HT JoshuaZ) within the set of integers.
The only way you can form an infinite set of quarks—given that you can’t distinguish one quark from another—is by selecting for inclusion all quarks indiscriminately. This is because there are only two ways that infinite subsets can be distinguished from their supersets: 1) the subset is of lower cardinality than the superset or 2) the elements are distinguishable to create a logical superset/set relationship (such as exists in quarks/upside-down quarks).
OK, suppose I grant this. I now feel like I might be able to formulate your argument in my own words. Here’s an attempt; let me know if and when it diverges from what you’re actually arguing.
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“Suppose I have sworn to give up the hateful practice of discriminating between quarks based on their differences. Henceforth I shall treat all quarks as utterly indistinguishable from one another. Having made this solemn vow, I now ask you to bring me an infinite set of quarks (note that I do not specify which quarks, for that would violate my vow!). You oblige, and provide me with a set called S.
“I inspect the set S and try to see whether it’s different from the set of all quarks, which we call Q. First I look at the cardinalities of S and Q. If their cardinalities were different, then obviously S and Q would be different sets. But their cardinalities are the same. Next I look for a quark that is contained in Q, but not contained in S. If there were such an element, then obviously S and Q would be different sets. But in order to successfully find such an element, I would have to make use of the distinctions between quarks. After all, how would I know that a given quark was in Q, but not in S? I would have to show that the quark in Q was distinct from each quark in S, but I have agreed to regard all quarks as indistinguishable. Therefore my search for an element of Q that is not in S will fail. I conclude that the set S is the same as the set Q. That is the set you gave me must be the set of all quarks.
“But this conclusion is obviously wrong. All I asked you for was an infinite set of quarks. There are many infinite sets of quarks, not all of which are the same as Q, the set of all quarks. You might have left some quarks out of S, and still provided me with an infinite set of quarks, which was all I asked for.
“Therefore we have a contradiction: I have proved something that is not necessarily true. Therefore the set of quarks cannot be infinite.”
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The response to this argument is that because I’ve blinded myself to the differences between quarks, I’ve lost the ability to show that Q and S are different. That does not mean that I’m entitled to conclude that Q and S are the same! After all, if I did allow myself to see the differences between quarks, such as their different positions in space, I might notice that Q contained a quark located at the position (3, 4, 5), but that S contained no quark at that position. This would let me see that Q and S are in fact distinct sets.
I take issue with your translation at only a single point:
My version contains a further constraint: When you ask me to bring you an infinite set of quarks, you instruct me to be as blind as you to the features that distinguish between quarks.
The_Duck tells metaphysicist to gather together an infinite set of quarks while remaining blind to their individuality. Metaphysicist, having no distinctions on which to carve infinite subsets, can respond to this request in only one way; include every quark. (I want to resist calling this the “set of all quarks,” because the incoherence of that concept with infinite quarks is what I argue.) The_Duck then goes out and finds another quark, and scolds metaphysicist, “You missed one.”
The_Duck is unjustified in criticizing metaphysicist, who must have picked “all the quarks,” given that metaphysicist succeeded—without knowing of any proper subsets—in assembling an infinite set . Having “selected all the quarks” doesn’t preclude finding another when they’re infinite in number and the only criterion for success is the number.
You will say that there is a fact of the matter as to whether the first set I assembled was all the quarks. Unblind yourself to the quarks’ individuating features, you say, and you get an underlying reality where the sets are different. I agree, but I think a more limited point suffices. When I follow the same procedure—gather all the quarks—I will be equally justified in gathering a set and in gathering a superset consisting of one other quark. There’s no way for me to distinguish the two sets. The contradiction is that following the procedure “gather all the quarks” should constrain me to a single set, “all the quarks,” rather than allowing a hierarchy of options consisting of supersets.
I’m making progress then. :)
No. If what you gathered is a proper subset of what you could have gathered, then you didn’t gather all the quarks, and you’re not justified in claiming that you did. How did you decide to leave out that one other quark? You must have made a distinction between it and the others that you did gather.
Of course there is. The superset contains a quark that the subset doesn’t. If you refuse to notice the differences that single that quark out from the others, that’s your loss.
I think that maybe you’re trying not to distinguish between quarks, but are implicitly distinguishing between “quarks that you know about” and “quarks that you don’t know about.” So you might assemble all the quarks you know about—an infinite number—and not have any evidence that this isn’t all the quarks there are. But later, you worry, you might find some other quarks that you didn’t know about before, so that your original set didn’t actually contain all quarks. This is not contradictory. If there was a chance that there existed quarks you didn’t know about, then you weren’t justified in saying that you had gathered all the quarks.
It does. If you’re not at the top of the hierarchy, you haven’t gathered all the quarks. And you can’t justify claiming that you’re at the top of the hierarchy by blinding yourself to evidence that would prove otherwise.
Well, “site:lesswrong.com ‘infinite set atheist’” is a clue, but http://lesswrong.com/lw/mp/0_and_1_are_not_probabilities/hkd is also a place to start.
No, then there are the same number of quarks in both cases in the sense of cardinality. Your intuition just isn’t very good for handling how infinite sets behave- adding more to the an infinite set in some sense doesn’t necessarily make it larger. Failure at having a good intuition for such things shouldn’t be surprising; we didn’t evolve to handle infinite sets.
Yes, I understand that; in fact, it was my express premise: “You can always add a finite number to an infinite set and not change the number of elements.” That is, not change the number of quarks from one case to another.
Please read it again more carefully. My argument may be wrong, but it’s really not that naive.
Added.
I see what you might be responding to: “So, there are more quarks than are contained in the set of all quarks.” The second sentence, not the first. It’s stated imprecisely. It should read, “So, there are other quarks than are contained in the set of all quarks.” Now changed in the original.
You’ve collapsed the distinction between two possible worlds. You started out by saying, consider a universe containing infinitely many quarks. Then you say, consider a universe which has all the quarks from the first universe, plus a finite number of extra quarks. The set of all quarks in the second scenario indeed contains quarks that aren’t in the set of all quarks in the first scenario, but that’s not a contradiction.
It’s like saying: Consider the possible world where Dick Cheney ended up as president for the last two years of Bush’s second term. Then that would mean that there was a president who wasn’t an element of the set of all presidents.
Replying separately to this now added comment. it still seems like this is an issue of ambiguous language. It isn’t that there are other quarks that aren’t contained in the set of all quarks.” Is is that there’s a set of quarks and a superset that have the same cardinality.
The problem seems to be that you are using the word “more” in a vague way that reflects more intuition than mathematical precision.
I think you responded before my correction, where I came to the same conclusion that my use of “more” was imprecise.
Added
I remember reading an essay maybe five years ago by Eliezer Yudkowsky where he maintained that the early Greek thinkers had been right to reject actual infinities for logical reasons. I can’t find the essay. Has it been recanted? Is it a mere figment of my imagination? Does anyone recall this essay?