In article <firstname.lastname@example.org>, WM <email@example.com> wrote:
> On 4 Jun., 04:16, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote: > > In article > > <103a00da-91c5-47fc-b40e-79f330393...@e24g2000vbe.googlegroups.com> > > WM <mueck...@rz.fh-augsburg.de> writes: > On 3 Jun., 04:37, "Dik > > T. Winter" <Dik.Win...@cwi.nl> wrote: ... > > You apparently do > > not understand it at all. WM has posted this stuff for > > years > > in the German newsgroup on mathematics, in German. Moreover, he > > > > has two books about mathematics on his name, in German. > > 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. I show that the end of > each path p of the set P can be mapped on a node,
In a maximal infinite binary tree a path can have only one "end" and all paths have the same end: the root node.
But since one of WM's often hidden assumptions is that there re no infinite sets, in his world there is no such tree at all.
> and that all paths p of P cover all nodes of the tree. Therefore, > after having completed the covering of the whole tree
Which in WM's world is guaranteed to be finite anyway,
> there remains no node that could be used to construct a path that > does not belong to P.
That presumes that through each node passes only a less than uncountable infinity of paths, which is false in any system, such as ZF, in which a maximal infinite binary tree can exist.
> This disproves the mentioned axiom.
Not in ZF it doesn't.
Or, at least, not without sneaking in some of WM's hidden assumptions it doesn't. > > > 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. As long as you > are discussing catholicism, you should use it, i.e., believe in that > axiom.
Since mathematics is about what follows from a given system of axioms or assumptions without regard to whether those axioms or assumptions themselves are in any sense true, if one ACCEPTS, for purposes of argument, that Catholic axiom, one can see whether the proof their God is valid.
That is where WM fails to see what axiomatic mathematics is about. It does not CARE whether an axiom in some system is true, but only cares what follows from assuming the whole axiom system.
But the notion that WM has access to THE TRUTH, when he has such a profound lack of understanding of logic is ridiculous. > > > So, if you are discussion Eucliedan geometry you should use the > > parallel axiom. Of course you can reject it, 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?
Not at all, but the difference between what can be deduced from a set of axioms and whether the set of axioms represent what is actually true are quite different problems, neither of which WM is competent to discuss.
> Probably they think that the Dutch better should have stayed within > the Spanish Empire.
Possible for the Spanish, but almost certainly not for the Dutch.