An uncountable countable set
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From: mueckenh on 28 Aug 2006 07:48 Virgil schrieb: > In article <1156501377.098380.50700(a)m79g2000cwm.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > Virgil schrieb: > > > > > > > > But as we just investigate consistency, you cannot presuppose it. With > > > > your attitude it is impossible to find any inconsistency even in an > > > > inconsistent theory. Deplorably you are too simple to recognize that. > > > > > > If "Mueckenh" can deduce from any axiom system both a statement within > > > the system and its negation, "Mueckenh" will have found his > > > inconsistency. > > > > Which part of my proof concerning the binary tree is not in accordance > > with the ZFC axioms in your opinion? > > The part that says a bijective image of the naturals bijects with a > bijective image of the power set of the naturals. I do not say that they biject. I only say that the bijective image of the naturals has more elements (or more precisely: not less elements) than the bijective image of the reals. But that is only the consequence which cannot be avoided if there are no other loopholes. Regards, WM
From: mueckenh on 28 Aug 2006 07:50 Virgil schrieb: > In article <1156501674.444594.242900(a)74g2000cwt.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > Dik T. Winter schrieb: > > > > > > > > > > > Mathematics is not about reality... > > > > It is. You have not yet recognized it. > > > Mathematics itself occurs only in the imagination, and is no more 'real' > than anything else that occurs there. Everything that occurs there has at least reality in the brain, in the form of electric currents. Therefore things which occur there are based upon the facilities offered there. But there is no facility whic enables one to distinguish 10^100 elements of a set. > > It is only in attempts to apply mathematics to reality that there is any > point of contact between the two. You are considering the results of mathematics. But you forget to consider its foundations. The roots of mathematics are necessarily made of reality. So the contact of mathematics and reality is twofold. (But whether the results describe reality too, is only a question of mathematics being useful or useless.) Regards, WM
From: mueckenh on 28 Aug 2006 07:51 Virgil schrieb: > In article <44eef04d(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > > > If the naturals are not a subset of the reals in ZFC and NBG, then those > > theories are even more screwed up than they already seemed. > > While there are sets of objects within the rationals and reals which > look very similar to the urset of naturals, they are not quite the same. > They are merely copies of the naturals within the larger systems. That is an opinion. I have the opposite opinion. An example supporting my opinion is the geometric interpretation. Another one is the mathematically correct equation 1 = 1.000... (in physics it would not hold). Regards, WM
From: mueckenh on 28 Aug 2006 07:53 Tony Orlow schrieb: > > But since no natural requires an infinite bit string, that is irrelevant. > > If no natural requires an infinite bit string, even the very largest, > and all bit positions are included in it, then no infinite set of bit > positions is required. Tony, please drop the "very largest". The rest is ok. If no natural requires an infinite set then all naturals together do not require an infinite set. If the contrary is asserted (and if infinity is a number larger than any finite number), then we may ask which natural is the first such that all of its predecessors require an infinite number of bit positions. Regards, WM
From: Virgil on 28 Aug 2006 15:02
In article <1156765413.469344.13770(a)75g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > In article <1156500736.055279.173430(a)p79g2000cwp.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > MoeBlee schrieb: > > > > > > > Albrecht wrote: > > > > > We must conclude that we can only say: we can find infinite many > > > > > points > > > > > on a line. This is a potential infinity. > > > > > > > > Do you have axioms for your mathematical theory of potential but not > > > > actual infinity? > > > > > > Five Peano axioms. > > > > Which version? In some versions, it is explicitly required that the set > > of all "naturals" exist. > > > But as it does not and cannot exist in any sense, I prefer the original > version (slimmed to 5 axioms). > "Mueckenh" keeps claiming that the set of all naturals cannot exist even though each natural can exist, but he never proves his claim, but just keeps arguing in circles about it. In order to prove that no such set of all naturals can exist, one must assume something which allows such a proof. No such assumption then no such proof! Neither "Mueckenh", nor anyone else, can prove anything without assuming something. |