Cantor's Diagonal?
in [Math]
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From: MoeBlee on 4 Feb 2010 19:33 On Feb 4, 5:59 pm, zuhair <zaljo...(a)gmail.com> wrote: > ANY is EVERY. In a certain sense in logic, yes. So? Do you have any remaining question or doubt that in Z set theory proves there is no bijection between w and {f | f: w -> {0 1}}? MoeBlee
From: Virgil on 4 Feb 2010 20:17 In article <20fc2ead-8f0e-4214-91b5-51b15b5a53a0(a)u9g2000yqb.googlegroups.com>, zuhair <zaljohar(a)gmail.com> wrote: > On Feb 4, 3:45�pm, Virgil <Vir...(a)home.esc> wrote: > > In article > > <4a86efc3-0a32-4e3e-85dc-3f65b7645...(a)q4g2000yqm.googlegroups.com>, > > > > �zuhair <zaljo...(a)gmail.com> wrote: > > > > > It seems to me that a modification of this argument can actually work > > > > > for every well orderable set, however I don't know if a modification > > > > > of this argument can be made general enough to prove that the power of > > > > > every non well orderable set is bigger than it. > > > > > > The standard proof that the cardinality of a power set is greater than > > > > that of the base set in no way requires that either of the sets be > > > > ordered, much less well ordered. > > > > > Agreed, provided what you mean by the standard proof the one in which > > > prove that *every* function from a set to its power is not surjective. > > > > It is enough to show that ANY such function fails to be surjective, > > which one can do by considering for any function a set determined by > > that function which is not in its range. > > ANY is EVERY. Thus EVERY is already proven.
From: zuhair on 4 Feb 2010 20:20 On Feb 4, 7:33 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > On Feb 4, 5:59 pm, zuhair <zaljo...(a)gmail.com> wrote: > > > ANY is EVERY. > > In a certain sense in logic, yes. So? I just wanted to clarify to Virgil that in logic ANY is EVERY, apparently Virgil thought they are different, so he iterated my argument replacing ANY (as he emphasized it by writing in CAPITAL letters) instead of EVERY, so my reply to him was a clarification that ANY is EVERY, that's all. > > Do you have any remaining question or doubt that in Z set theory > proves there is no bijection between w and {f | f: w -> {0 1}}? > > MoeBlee No. Zuhair
From: zuhair on 4 Feb 2010 20:21 On Feb 4, 8:17 pm, Virgil <Vir...(a)home.esc> wrote: > In article > <20fc2ead-8f0e-4214-91b5-51b15b5a5...(a)u9g2000yqb.googlegroups.com>, > > > > > > zuhair <zaljo...(a)gmail.com> wrote: > > On Feb 4, 3:45 pm, Virgil <Vir...(a)home.esc> wrote: > > > In article > > > <4a86efc3-0a32-4e3e-85dc-3f65b7645...(a)q4g2000yqm.googlegroups.com>, > > > > zuhair <zaljo...(a)gmail.com> wrote: > > > > > > It seems to me that a modification of this argument can actually work > > > > > > for every well orderable set, however I don't know if a modification > > > > > > of this argument can be made general enough to prove that the power of > > > > > > every non well orderable set is bigger than it. > > > > > > The standard proof that the cardinality of a power set is greater than > > > > > that of the base set in no way requires that either of the sets be > > > > > ordered, much less well ordered. > > > > > Agreed, provided what you mean by the standard proof the one in which > > > > prove that *every* function from a set to its power is not surjective. > > > > It is enough to show that ANY such function fails to be surjective, > > > which one can do by considering for any function a set determined by > > > that function which is not in its range. > > > ANY is EVERY. > > Thus EVERY is already proven. Yep. Zuhair
From: Virgil on 4 Feb 2010 22:43
In article <f3e811e3-c4b0-4309-97b2-9d4771ed6ab5(a)u41g2000yqe.googlegroups.com>, zuhair <zaljohar(a)gmail.com> wrote: > On Feb 4, 7:33�pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > > On Feb 4, 5:59�pm, zuhair <zaljo...(a)gmail.com> wrote: > > > > > ANY is EVERY. > > > > In a certain sense in logic, yes. So? > > I just wanted to clarify to Virgil that in logic ANY is EVERY, > apparently Virgil > thought they are different, so he iterated my argument replacing ANY > (as he > emphasized it by writing in CAPITAL letters) instead of EVERY, so > my reply to him was a clarification that ANY is EVERY, that's all. > > > > Do you have any remaining question or doubt that in Z set theory > > proves there is no bijection between w and {f | f: w -> {0 1}}? > > > > MoeBlee > > No. > > Zuhair You missed my point. When there is a proof, as there is, covering ANY instance, it automatically covers EVERY instance too. Thus one does not need a separate proof for both. |