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.
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.