In article <firstname.lastname@example.org> WM <email@example.com> 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.
> 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.
> 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.
> This disproves the > mentioned axiom.
Indeed, when you assume it is false, it is easy to prove it is false.
> > 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.
> > 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?
> Probably they think that the Dutch better should have stayed within > the Spanish Empire.
As, currently of the people that say they belong to a church, nearly 50% is Roman Catholic there may be some validity. But that nearly 50% is actually about 17% of the total population... -- dik t. winter, cwi, science park 123, 1098 xg amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/