OK, you say you don’t accept that sort of uncomputable leap to the end. The problem is that, AIUI, you’re already accepting it as soon as you accept the power set of N. (Of the various “axioms of power” of ZFC, power set is the only one needed here. And if you just want omega_1, you don’t need arbitrary power sets, just that of N. I mean really you want P(N x N), but since N is in an easily-described bijection with N x N, it shouldn’t make a difference; just use a pairing function instead of proper ordered pairs.) The construction of omega_1 from P(N) is pretty straightforward, really, and doesn’t use any of ZFC’s other powerful axioms. Maybe you can somehow have the reals without P(N)? I.e. without binary expansions? shrug This is getting rather far away from what I know. Constructivists—well, not the milder ones who just reject excluded middle, but the stricter ones who don’t like impredicativity (whatever that might be, don’t ask me) -- don’t accept the axiom of power sets; they consider it just as much an unjustified leap to the end.
Of course you could always try summoning TobyBartels and ask him how the constructivists do it. When you say these sorts of things I’m a little of surprised you haven’t gone constructivist already. But I guess you like classical logic. :)
(By the “axioms of power”, I mean replacement, power set, and choice; the ones anyone might object to. Well, foundation is objectionable too, but it’s more of an axiom of weakness. Healing Salve as opposed to Ancestral Recall. :P Also looking things up apparently the no-impredicativity constructivists insist on weakening axiom of separation as well? Well, I think their weaker version should suffice here. Again, I am saying these things without carefully checking them because hopefully TobyBartels will show up and correct me if I am wrong. :) )
The construction of omega_1 from P(N) is pretty straightforward, really, and doesn’t use any of ZFC’s other powerful axioms.
You either need P(P(N)) or something like an axiom of quotient sets to take the equivalence classes that are the actual elements of this version of omega_1. I presume (but haven’t checked) that this is why J_2 has R but not omega_1 (although J_2 is not written in set-theoretic language, so you have to encode these).
Maybe you can somehow have the reals without P(N)?
Assuming you accept classical logic, then P(N) may be constructed as a subset of R: that famous fractal the Cantor set.
I am saying these things without carefully checking them because hopefully TobyBartels will show up and correct me if I am wrong.
Just about everything that I know about predicative mathematics is distilled here. There I describe two schools, and the constructive one (which is less predicative than the classical one!) is the only one that I know well.
You either need P(P(N)) or something like an axiom of quotient sets to take the equivalence classes that are the actual elements of this version of omega_1.
Crap, looks like I should have checked that after all! OK, I guess if Eliezer accepts R but not P(R) then there’s less of a problem here than I thought. :P
I presume (but haven’t checked) that this is why J_2 has R but not omega_1 (although J_2 is not written in set-theoretic language, so you have to encode these).
Edit: Nevermind, this line was asking what J_2 was, you’ve given a reference elsewhere.
Maybe you can somehow have the reals without P(N)?
Assuming you accept classical logic, then P(N) may be constructed as a subset of R: that famous fractal the Cantor set.
Oh, that works. Should have thought of that.
The constructive one (which is less predicative than the classical one!)
Huh, so there’s two separate things going on here. Constructivism in the sense of no-excluded-middle, and I guess “predicativism” in the sense of, uh, things should be predicative? I probably should have realized those were largely independent, but didn’t. How is the constructive version less predicative? Is it just the function set issue?
OK, you say you don’t accept that sort of uncomputable leap to the end. The problem is that, AIUI, you’re already accepting it as soon as you accept the power set of N. (Of the various “axioms of power” of ZFC, power set is the only one needed here. And if you just want omega_1, you don’t need arbitrary power sets, just that of N. I mean really you want P(N x N), but since N is in an easily-described bijection with N x N, it shouldn’t make a difference; just use a pairing function instead of proper ordered pairs.) The construction of omega_1 from P(N) is pretty straightforward, really, and doesn’t use any of ZFC’s other powerful axioms. Maybe you can somehow have the reals without P(N)? I.e. without binary expansions? shrug This is getting rather far away from what I know. Constructivists—well, not the milder ones who just reject excluded middle, but the stricter ones who don’t like impredicativity (whatever that might be, don’t ask me) -- don’t accept the axiom of power sets; they consider it just as much an unjustified leap to the end.
Of course you could always try summoning TobyBartels and ask him how the constructivists do it. When you say these sorts of things I’m a little of surprised you haven’t gone constructivist already. But I guess you like classical logic. :)
(By the “axioms of power”, I mean replacement, power set, and choice; the ones anyone might object to. Well, foundation is objectionable too, but it’s more of an axiom of weakness. Healing Salve as opposed to Ancestral Recall. :P Also looking things up apparently the no-impredicativity constructivists insist on weakening axiom of separation as well? Well, I think their weaker version should suffice here. Again, I am saying these things without carefully checking them because hopefully TobyBartels will show up and correct me if I am wrong. :) )
You either need P(P(N)) or something like an axiom of quotient sets to take the equivalence classes that are the actual elements of this version of omega_1. I presume (but haven’t checked) that this is why J_2 has R but not omega_1 (although J_2 is not written in set-theoretic language, so you have to encode these).
Assuming you accept classical logic, then P(N) may be constructed as a subset of R: that famous fractal the Cantor set.
Just about everything that I know about predicative mathematics is distilled here. There I describe two schools, and the constructive one (which is less predicative than the classical one!) is the only one that I know well.
Crap, looks like I should have checked that after all! OK, I guess if Eliezer accepts R but not P(R) then there’s less of a problem here than I thought. :P
Edit: Nevermind, this line was asking what J_2 was, you’ve given a reference elsewhere.
Oh, that works. Should have thought of that.
Huh, so there’s two separate things going on here. Constructivism in the sense of no-excluded-middle, and I guess “predicativism” in the sense of, uh, things should be predicative? I probably should have realized those were largely independent, but didn’t. How is the constructive version less predicative? Is it just the function set issue?