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] > > > 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. > > > 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. > > 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. > > > 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: http://www.hs-augsburg.de/~mueckenh/GU/GU12.PPT#363,26,Folie 26 > > > > So comparing > > > the "axiom of infinity" with a "proof of God" is pretty stupid. In > > > mathematics a theory depends on the axioms used. There is nothing > > > sacred about the axioms, but as long as you are discussing a theory > > > you should use the axioms of that theory. > > > > That is same as with proofs of God. The Vatican published an axiom > > (they call it a dogma, but it is of the same meaning) according to > > which it is possible to prove the existence of God. > > I do not think such an axiom would be a valid axiom in mathematics. Axioms > do not state what is or what is not possible to prove. Axioms states > properties and existence of objects.
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. > > > > 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.