Another AC anomaly?
in [Math]
|
Prev: Dumb Question on Inference For Regression (Ho:= No Linear Relation)
Next: Is this a valid statement?
From: Virgil on 1 Dec 2009 17:17 In article <f37ab501-a998-446b-8aaf-e88059d16a8d(a)z41g2000yqz.googlegroups.com>, WM <mueckenh(a)rz.fh-augsburg.de> wrote: > On 1 Dez., 20:46, Virgil <Vir...(a)home.esc> wrote: > > > What makes you think that the elements of a limit set can be counted? > > If the set exists, its elements exist and can be counted. That depends on what one means by "counting". Outside of Wolkenmuekenheim, there are sets which are not images of the naturals under any function, and such sets are uncountable.
From: Ross A. Finlayson on 1 Dec 2009 18:22 On Dec 1, 2:17 pm, Virgil <Vir...(a)home.esc> wrote: > In article > <f37ab501-a998-446b-8aaf-e88059d16...(a)z41g2000yqz.googlegroups.com>, > > WM <mueck...(a)rz.fh-augsburg.de> wrote: > > On 1 Dez., 20:46, Virgil <Vir...(a)home.esc> wrote: > > > > What makes you think that the elements of a limit set can be counted? > > > If the set exists, its elements exist and can be counted. > > That depends on what one means by "counting". Outside of > Wolkenmuekenheim, there are sets which are not images of the naturals > under any function, and such sets are uncountable. No, then you would have proven ZFC consistent. Ross F.
From: Virgil on 1 Dec 2009 19:17 In article <5fa54469-e270-4070-b2bc-90f35bb8ce49(a)p8g2000yqb.googlegroups.com>, "Ross A. Finlayson" <ross.finlayson(a)gmail.com> wrote: > On Dec 1, 2:17�pm, Virgil <Vir...(a)home.esc> wrote: > > In article > > <f37ab501-a998-446b-8aaf-e88059d16...(a)z41g2000yqz.googlegroups.com>, > > > > �WM <mueck...(a)rz.fh-augsburg.de> wrote: > > > On 1 Dez., 20:46, Virgil <Vir...(a)home.esc> wrote: > > > > > > What makes you think that the elements of a limit set can be counted? > > > > > If the set exists, its elements exist and can be counted. > > > > That depends on what one means by "counting". Outside of > > Wolkenmuekenheim, there are sets which are not images of the naturals > > under any function, and such sets are uncountable. > > No, then you would have proven ZFC consistent. Not outside of Wolkenmuekenheim, I wouldn't. Ross seems to think that there are no uncountable sets in any set theory unless that set theory is embedded in ZFC.
From: Ross A. Finlayson on 1 Dec 2009 19:40 On Dec 1, 4:17 pm, Virgil <Vir...(a)home.esc> wrote: > In article > <5fa54469-e270-4070-b2bc-90f35bb8c...(a)p8g2000yqb.googlegroups.com>, > "Ross A. Finlayson" <ross.finlay...(a)gmail.com> wrote: > > > > > On Dec 1, 2:17 pm, Virgil <Vir...(a)home.esc> wrote: > > > In article > > > <f37ab501-a998-446b-8aaf-e88059d16...(a)z41g2000yqz.googlegroups.com>, > > > > WM <mueck...(a)rz.fh-augsburg.de> wrote: > > > > On 1 Dez., 20:46, Virgil <Vir...(a)home.esc> wrote: > > > > > > What makes you think that the elements of a limit set can be counted? > > > > > If the set exists, its elements exist and can be counted. > > > > That depends on what one means by "counting". Outside of > > > Wolkenmuekenheim, there are sets which are not images of the naturals > > > under any function, and such sets are uncountable. > > > No, then you would have proven ZFC consistent. > > Not outside of Wolkenmuekenheim, I wouldn't. > > Ross seems to think that there are no uncountable sets in any set theory > unless that set theory is embedded in ZFC. That's accurate. I just don't imagine you'd be using some other set theory (ZFC <=> NBG). It's also accurate that adherents of regular (well-founded) set theories get no absolutes, everything qualified by incompleteness. Then, where any of the other set theories so described contain the naturals thus encoding Peano arithmetic, purporting trans-finites, they can be lumped together with ZFC in never being provable, as a consequence of those incommensurable trans-finites, in the Goedelian incompleteness which is structurally defined regardless of whether, for example, there are sets too big for the theory that don't observe unmappable powersets (eg, NFU). Set theory: only sets. Class theory: non-sets theory. Large cardinals: non-trichotomous. Ross F.
From: Virgil on 1 Dec 2009 23:09
In article <05846d81-161e-4b2b-849b-7b5b24d9bb8a(a)s20g2000yqd.googlegroups.com>, "Ross A. Finlayson" <ross.finlayson(a)gmail.com> wrote: > On Dec 1, 4:17�pm, Virgil <Vir...(a)home.esc> wrote: > > In article > > <5fa54469-e270-4070-b2bc-90f35bb8c...(a)p8g2000yqb.googlegroups.com>, > > �"Ross A. Finlayson" <ross.finlay...(a)gmail.com> wrote: > > > > > > > > > On Dec 1, 2:17 pm, Virgil <Vir...(a)home.esc> wrote: > > > > In article > > > > <f37ab501-a998-446b-8aaf-e88059d16...(a)z41g2000yqz.googlegroups.com>, > > > > > > WM <mueck...(a)rz.fh-augsburg.de> wrote: > > > > > On 1 Dez., 20:46, Virgil <Vir...(a)home.esc> wrote: > > > > > > > > What makes you think that the elements of a limit set can be > > > > > > counted? > > > > > > > If the set exists, its elements exist and can be counted. > > > > > > That depends on what one means by "counting". Outside of > > > > Wolkenmuekenheim, there are sets which are not images of the naturals > > > > under any function, and such sets are uncountable. > > > > > No, then you would have proven ZFC consistent. > > > > Not outside of Wolkenmuekenheim, I wouldn't. > > > > Ross seems to think that there are no uncountable sets in any set theory > > unless that set theory is embedded in ZFC. > > That's accurate. I just don't imagine you'd be using some other set > theory (ZFC <=> NBG). There are classes in NBG which are not sets and which do not exist in ZFC, ergo, they are not "equivalent". Further garbage deleted. |