On 14 Dez., 08:15, Virgil <vir...@ligriv.com> wrote:
> > Then take all paths that cover as many nodes as possible. > > Every path in an infinite tree "covers" infinitely many nodes.
Yes. > > > But they must be defined by more than the nodes, if there shall be > > more than conutably many. > > Nonsense! The countably infinite set of nodes necessarily has > uncountably many subsets, and uncountably many of those subsets will be > paths.
You are wrong. I cover all nodes with countably many paths such that there remains no subset that could represent a real number can be distinguished by uncovered nodes. > > Known infinite sequences may require definition, but there is nothing > that requires an infinite sequence to be known.
Everything that in principle can appear as the diagonal of a Cantor list must be known in principle. Otherwise it could never get known.
> Every one of the > uncountably many infinite ordered subsets of the ordered set N is an > infinite sequence and there are uncountably many of them, most of them > "unknown".
"There are infinitely many angels", (Cantor, Letter to Jeiler 1888). Most angels are unknown. But do they belong to mathematics?
Numbers that are knowable, are knowable by finite definitions. Others are neither knowable nor elements of mathematics.
> > You can also say to construct a collection of nodes, namely the > > complete Binary Tree, by countably many paths. > > While a set of countably many paths may cover all nodes, it necessarily > omits more paths than it can include.
There is no path omitted that can be defined by nodes. Recognize and agree or name the nodes of such a path the existence of which you are asserting. > > > > > > WM has been in here with this shit for over a decade. > > > Two wrong assertions in one simple sentence. > > Actually WM HAS been here with his shit for over a decade.
Neither is it shit, nor have I been in sci.logic or sci.math before 2005. But you will not recognize your errors. > > > Small wonder that you > > don't understand that Cantor has already been refuted. > > Cantor has not been refuted outside of Wolkenmuekenheim.
But you will not recognize your errors.
> > > > I construct the complete infinite Binary Tree by means of countably > > many paths. > > You often have claimed so, but since there are many proofs, quite valid > everywhere here outside of your Wolkenmuekenheim, that you are wrong, > your "proofs" do not hold up here!
But you will not recognize your errors.
> > > that you ar won If there were more paths definable by nodes, then you > > should be able to say which one. > > If not, then you should be able to > > see that only finite definitions define paths like that of 1/3. Alas, > > there are only countaby many finite definitions. Is it really that > > hard to understand? > > Where is it written that one must be able to name every member of a set > in order to count its members?
Without being able to name the elements, one cannot distinguish them. But a set is undefined unless all elements are distinct.
Extensionality: Sets containing the same elements are equal. If elements are not nameable, one cannot prove whether two sets are equal or not.
> The set of reals is known to have more > members than can be named
The set of matheologians is known to have more members than can think. Correction: to have members that cannot think.
> > I construct the complete infinite Binary Tree by means of countably > > many paths. > > Since it is well known that, at last outside Wolkenmuekenheim, the set > of all possible such paths trivially bijects with the uncountable set of > all subsets of N, no countable set of paths can include every path. > My set includes every path that can be identified by nodes.
> > > If there were more paths definable by nodes, then you > > should be able to say which one. > > We can find lots of them that you have missed as soon as you tell us > which ones you have included
I include all finite paths. The infinite endings are nonsense and not definable by nodes. I use them only in order to show you that they are nonsense by proving that you cannot distinguish them by nodes. > > > If not, then you should be able to > > see that only finite definitions define paths like that of 1/3. > > But the Cantor diagonal process, which proves any listing that you > present must be incomplete,is totally independent of any need for or > reliance on finite definitions.
Wrong. No Cantor list has ever been completely defined and supplied a complete diagonal number other than by a finite definition. > > > Alas, > > there are only countaby many finite definitions. Is it really that > > hard to understand? > > There are perfectly legitimate ways to prove that there have to be more > paths, and f functions, and subset of N, than your limited > definitionism will cover.
There are perfectly legitmate ways to prove that there are infinitely many angels. But all that is not part of mathematics.