Uncountability of the Rationals?
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From: Tony Orlow on 14 Jun 2010 18:02 On Jun 14, 3:23 pm, David R Tribble <da...(a)tribble.com> wrote: > Brian Chandler wrote: > >> I certainly don't think it would be correct to say that the reciprocal > >> of zero is the empty set. > > Tony Orlow wrote: > > In standard mathematics it is "undefined", both positively AND > > negatively infinite. It doesn't exist. The set of solutions at 0 is > > empty, ot perhaps a pair of hyperreals. > > A hyperreal can be returned by a real-valued function? Only possibly where it is undefined. Don't ask me for a method. I said, "perhaps". Tony
From: Tony Orlow on 14 Jun 2010 18:02 On Jun 14, 3:49 pm, Virgil <Vir...(a)home.esc> wrote: > In article > <49f89dd7-4c7f-434d-af8c-391194e15...(a)a30g2000yqn.googlegroups.com>, > Tony Orlow <t...(a)lightlink.com> wrote: > > > Nope. They simply say that IF N has a size, it cannot be a member of > > N. > > Note that for the von Neumann naturals, the 'size' of a natural is never > a member of that natural. So why should it be any different for the set > of all of them? How many elements in the null set?
From: Virgil on 14 Jun 2010 19:41 In article <357d9d42-f0ac-4fdc-bf9a-8092183cae64(a)3g2000vbg.googlegroups.com>, Tony Orlow <tony(a)lightlink.com> wrote: > On Jun 14, 3:49�pm, Virgil <Vir...(a)home.esc> wrote: > > In article > > <49f89dd7-4c7f-434d-af8c-391194e15...(a)a30g2000yqn.googlegroups.com>, > > �Tony Orlow <t...(a)lightlink.com> wrote: > > > > > Nope. They simply say that IF N has a size, it cannot be a member of > > > N. > > > > Note that for the von Neumann naturals, the 'size' of a natural is never > > a member of that natural. So why should it be any different for the set > > of all of them? > > How many elements in the null set? Is the size of the set with no elements an ELEMENT of that set? If so that would make it not empty after all! Read more carefully, TO!
From: Virgil on 14 Jun 2010 19:46 In article <23e0c345-10ff-4172-8a1e-0c230c1707c6(a)f8g2000vbl.googlegroups.com>, Tony Orlow <tony(a)lightlink.com> wrote: > On Jun 14, 3:23�pm, David R Tribble <da...(a)tribble.com> wrote: > > Brian Chandler wrote: > > >> I certainly don't think it would be correct to say that the reciprocal > > >> of zero is the empty set. > > > > Tony Orlow wrote: > > > In standard mathematics it is "undefined", both positively AND > > > negatively infinite. It doesn't exist. The set of solutions at 0 is > > > empty, ot perhaps a pair of hyperreals. What is undefined or does not exist is neither positively nor negatively infinite. 1/x does not have a value, or a solution, at x = 0. > > > > A hyperreal can be returned by a real-valued function? > > Only possibly where it is undefined. Don't ask me for a method. I > said, "perhaps". > > Tony It is certainly not standard for a function all of whose values all reals to have a value which is not a real, whether hyperreal or unreal in some other way.
From: Virgil on 14 Jun 2010 19:55
In article <b95c48f1-4244-4d6b-b218-5dcebc16605b(a)30g2000vbf.googlegroups.com>, Tony Orlow <tony(a)lightlink.com> wrote: > On Jun 14, 3:08�pm, David R Tribble <da...(a)tribble.com> wrote: > > Tony Orlow wrote: > > >> Actually, in Bigulosity, there are much larger countably infinite > > >> sets. The square roots of the naturals have a higher "density" than > > >> the naturals. > > > > David R Tribble wrote: > > >> How is that? The two sets should be exactly the same size > > >> (your size(), not just cardinality). Every natural has a square root, > > >> and every natural square root has a corresponding natural. > > >> (I assume you're only dealing with the positive roots.) > > > > Tony Orlow wrote: > > > Holy cannoli! There exists a bijection, f:N->S and g:S->N, so indeed > > > they have the same cardinality. They do not share the same bigulosity, > > > since f and g are not the identity function. The number of square > > > roots over omega is omega^2. > > > > Sorry, but it's not clear what you mean by "the number of square > > roots over omega is omega^2". Do you mean: > > 1. the number of naturals in omega that are square roots is omega^2? > > 2. the number of naturals in omega that have square roots is > > � omega^2? > > 3. the set of the squares roots of all the naturals in omega has > > �omega^2 members? > > 4. or something else? > > Number 3. The set of square roots of naturals has a Bigulosity of > omega^2. Since every natural has 2 square roots, why wouldn't it be 2*aleph_0? |