Another AC anomaly?
in [Logic]
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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.
From: Aatu Koskensilta on 1 Dec 2009 23:14 Virgil <Virgil(a)home.esc> writes: > There are classes in NBG which are not sets and which do not exist in > ZFC, ergo, they are not "equivalent". NBG is essentially just a notational variant of ZFC. The differences between these two theories are of technical logical interest only, and don't reflect anything in their mathematical content. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Dik T. Winter on 2 Dec 2009 08:14
In article <12b419b5-1a9f-42c1-b9bc-0e8a2cce2da8(a)z41g2000yqz.googlegroups.com> WM <mueckenh(a)rz.fh-augsburg.de> writes: > On 1 Dez., 14:57, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: .... > > > You may write this as often as you like, but you are wrong. If there > > > is a limit set then there is a limit cardinality, namely the number of > > > elements in that limit set. Everything else is nonsense. > > > > Can you prove the assertion that the limit of the cardinalities is the > > cardinality of the limit? > > With pleasure. My assertion is obvious if the limit set actually > exists. No it is not. > Limit cardinality = Cardinality of the limit-set. You are conflating within limit cardinality the limit of the cardinalities and the cardinality of the limit. You have to prove they are equal. > If it does not exist, set theory claiming the existence of actual > infinity is wrong. Both can exist but they are not necessarily equal, you have to *prove* that. > > I can prove that it can be false. As I wrote, > > given: > > S_n = {n, n+1} > > we have (by the definition of limit of sets: > > lim{n -> oo} S_n = {} > > This is wrong. You have never looked at the definition I think. Given a sequence of sets S_n then: lim sup{n -> oo} S_n contains those elements that occur in infinitely many S_n lim inf{n -> oo} S_n contains those elements that occur in all S_n from a certain S_n (which can be different for each element). lim{n -> oo} S_n exists whenever lim sup and lim inf are equal. With this definition lim{n -> oo} S_n exists and is equal to {}. > ong. You may find it helpful to see the approach where I > show a related example to my students. > > http://www.hs-augsburg.de/~mueckenh/GU/GU12.PPT#394,22,Folie 22 I see nothing related there. It just shows (I think) an open cube and a cylinder. > > and so > > 2 = lim{n -> oo} | S(n) | != | lim{n -> oo} S_n | = 0 > > or can you show what I wrote is wrong? > > Yes I can.* If actual infinity exists*, then the limit set exists. I have still no idea what the mathematical definition of "actual infinity" is, but given the definitions above, the limit of the sequence of sets exists and is the empty set. > Then the limit set has a cardinal number which can be determined > simply by counting its elements. > A simple example is > | lim[n --> oo] {1} | = | {1} | = lim[n --> oo] |{1}| = 1. Right. And as in the above example the limit is the empty set, we can count the elements and come at 0. > If you were right, that | lim[n --> oo] S_n | =/= lim[n --> oo] |S_n| > was possible, then the limit set would not actually exist (such that > its elements could be counted and a different limit could be shown > wrong). The limit set is empty. > > If so, what line is wrong? > > Wrong is your definition of limit set, or set theory claiming actual > infinity as substantially existing, or both. What is *your* definition of the limit of a sequence of sets? -- dik t. winter, cwi, science park 123, 1098 xg amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |