In article <firstname.lastname@example.org> WM <email@example.com> writes: > On 11 Jun., 14:50, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote: ... > > > > 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]
First off, the paths of the tree are not isomorphic with the real number. There is no 1-1 mapping that is order preserving.
Moreover, the statement "the binary tree has uncountably many paths" is a theorem, not an axiom. It can be proven.
In ZF there is no axiom that uses the word "uncountable". It is defined and all statements using it are theorems.
But apparently you still do not understand what an axiom is.
> > > 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.
So your statment "I show that the end of each path p of the set P can be mapped on a node" is blatant nonsense. Why do you utter such contradictionary statements?
> But it can be shown for every node > that it gets covered and that all nodes get covered by a countable set > of paqths.
You have not shown it. You only show it by assuming that there are countably many paths.
> > > 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.
Right. But do you use up *all* paths? *That* is what you have to prove. You are always going to it from the wrong way. It is right that all nodes can be covered by countably many paths. The set of paths where each path from some node always goes right is an example. It does indeed cover all nodes. But it is not the set of all paths, because there are paths that do not satisfy the condition that from some node onwards it always goes to the right.
Or can you find a node that is *not* covered by a path that after some node to the right? Can you find a node that is covered by a path that alternates going left and right that is not covered by a path that after some node always goes to the right?
> > > 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.
Darn. Misreeading *again*. The number of nodes *is* countable. You do assume the number of paths is countable. Why are you always misreading what people do write?
Sorry, I am not able to look at that, no powerpoint on this system.
> > > 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.
I am not interested in religious non-sequitors.
> > > 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.
So you just state something without being able to back it up. -- dik t. winter, cwi, science park 123, 1098 xg amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/