An uncountable countable set
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From: Virgil on 26 Aug 2006 01:25 In article <44ef9c12(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > MoeBlee wrote: > > Tony Orlow wrote: > >> No, it all rests on the notions of identity and equality. As Leibniz > >> pointed out, when the properties of two objects are all exactly the > >> same, then they are the same object. So, when we say two numbers are > >> equal, that means all properties of the two are equal. > > > > Ha! The fallacy of reversing implication right there! An example of > > just about the most basic fallacy. > > When you say "a=b", you say "A a A b P(a)=P(b)". The two are equivlent > statements, and therefore imply each other. When one says of sets A and B that A = B, that is equivalent to For all x (x \in A <==> x \in B). At least in ZF, ZFC and NBG. > > > > > No, the indiscernibility of identicals does NOT imply the identity of > > indiscernibles. You need both implications; you can't derive one from > > the other. And, in first order logic, one direction can be posited only > > in the semantics not in the axioms. > > You prove two quantities equal by showing there is no difference, do you > not? When one says of sets A and B that A = B, that is equivalent to "For all x (x \in A <==> x \in B). At least in ZF, ZFC and NBG." http://en.wikipedia.org/wiki/Axiom_of_extension
From: Virgil on 26 Aug 2006 01:30 In article <44ef9c5a$1(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > MoeBlee wrote: > > Tony Orlow wrote: > >> Yes, and the universe is consistent by definition, so math should be > >> consistently overall as well. > > > > Unless the universe is a set of sentences, the notion of consistency > > does not even apply. > > The universe is governed by the properties of the elements within it, > which properties are statements true about those elements. But as those "properties" noumenon and we can only observe phenomema in the "real world". And they can never meet.
From: Virgil on 26 Aug 2006 01:37 In article <44ef9ecc(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > > There is NO system that will give you any kind of even minimal > > mathematics without adopting axioms that are not true in all domains of > > discourse. > > I am not convinced of that. TO is convinced only of the infallibility of his intuition. Others, having seen the quagmires into which where his intuition has led him, are even more convinced of its fallibility. > > > > >> If Ross and I each have our linearizations of the reals, then they > >> exist, independent of the axiom of choice (which might as well be called > >> the axiom of free will). > > > > We prove existence of objects in a theory from axioms. You posit the > > existence of objectw without any system of logic, primitives, or > > axioms. > > Logic is the primitive of argument. Logical truth is founded upon > quantity, and quantity upon geometry. Space is a priori. :) Only that empty one between TO's ears is a priori. > > Define 'finite'. In what context? That word is, like many, one whose meaning depends on the context in which it is used.
From: Virgil on 26 Aug 2006 01:40 In article <44ef9f9c(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > MoeBlee wrote: > > Tony Orlow wrote: > >> Given a set x, can we always determine card(x)? > > > > I'm guessing, but I'm pretty sure that there is not an algorithm that > > will tell you the ordinal index of the aleph that is the cardinality of > > any given set. > > Good guess, except I have no idea what your driving at. He is driving at the fact that for infinite ordinals there are infinite sets of different ones having the same cardinality.
From: Virgil on 26 Aug 2006 01:45
In article <44efa0e1(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > 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. > > But, you claim that a bijective image of the naturals bijects with a > bijective image of a set whose power set is the naturals That may be what TO claims, but not I. I have always denied that any bijection can exist between any set and its power set. And there is no set in ZF, ZFC or NBG whose power set is the set of naturals, so such a set could only exist in some odd place like TOmania. |