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.
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.