In article <email@example.com>, WM <firstname.lastname@example.org> wrote:
> On 11 Jun., 14:50, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote: > > In article > > <74f33c12-95fe-4b83-a7e3-941591def...@c9g2000yqm.googlegroups.com> WM > > <mueck...@rz.fh-augsburg.de> writes: > > > On 4 Jun., 04:16, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote: > > ... > > > > > And both say, in effect, about the existence of actual infinity, > > what > > > > > Kant said about the proof of the existence of God: These > > assumptions > > > > > (proof of God, axiom of infinity) are as ridiculous as a merchant > > who > > > > > would try to improve his balance by adding some zeros behind his > > > > > result. > > > > > > > > And you are deluded. An axiom is a statement of something that can > > not > > > > be proven, neither disproven using the remainder of the theory. > > > > > > The axiom can be contradicted. Simple example: The axiom could be: The > > > binary tree has uncountably many paths. > > > > Perhaps, although in ZF it is not an axiom. > > It is, because the paths of the tree are isomorphic with the real > numbers in [0, 1]
No they are not. Not even order isomprphic. In the maximal binary tree, with the usual lexicographic order, there are adjacent paths, but there are no adjacent reals in binary, or any other, notation. > > > > > I show that the end of each > > > path p of the set P can be mapped on a node, and that all paths p of P > > > cover all nodes of the tree. > > > > Ignoring that in ZF the paths do not have an end. > > The paths of the tree have no end. But it can be shown for every node > that it gets covered and that all nodes get covered by a countable set > of paqths.
But that is not at all the same as showing that all paths get "covered" but a countable set of nodes, with only one path per node. > > > > > Therefore, after having completed the > > > covering of the whole tree, there remains no node that could be used > > > to construct a path that does not belong to P.
But there remain uncountably many paths in the maximal tree unaccounted for by that asociation. > > > > This is the wrong way around. You assume that you can cover this way the > > whole tree (I think with this you mean each path in the tree). But that is > > what you have to prove. > > There is not much to prove. Append a tail of a path to every node. > Then every node is covered by at least one path, hence it does not > remain uncovered.
In fact every single node is covered by uncountably many paths this way. > > > > > This disproves the > > > mentioned axiom. > > > > Indeed, when you assume it is false, it is easy to prove it is false. > > I do not assume that the number of nodes is countable, but I count > them. Here:
You assume the number of paths is countable despite you inability to count (list) them.
> Like God and the set of natural numbers. He knows all of them, > according to Augustinus and Cantor. But I am sure he has no list of > all the reals.
Are you questioning the ability of God? Why not? You question the ability of everyone else, except yourself. > > > > > > So, if you are discussion Eucliedan > > > > geometry you should use the parallel axiom. Of course you can reject > > is > > > > but in that case you are not discussing Euclidean geometry but > > something > > > > else. > > > > > > That means, you are willing to believe in what the Vatican says? > > > > Well, no, because that "dogma" is not a valid "axiom". But can you tell > > me where that "dogma" actually is stated the way you say? > > Sorry, I only read it some time ago somwhere. But I think the set of > dogmas must be in the net for those who are interested. I am not.
Then why bring it up? You have much more serious problems to face.