On 4 Jun., 04:04, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote:
> > That is the logic of potential infinity, i.e., of incomplete sets. > > Eh? As far as I know logic is *not* about sets. It is about axioms, > statements and inference rules.
Logic has been obtained from the bahviour of parts of reality that can be called sets. > > > In Cantor's diagonal argument you can use the same logic : There is no > > last line, therefore there is always a line beyond the checked lines. > > But there you don't. > > Because you do not check the lines in order. It is always your basic > assumption that you first check the first line and after that the next line. > That is wrong.
That is necessary because you cannot find the n-th line unless you know the line number n - 1 or some equivalent mark.
> You simply give a definition of a new number so it is clear > that it will be different from each of the omega lines, without checking > directly any of them.
And in my binary tree I give a definition of an end of a path that contains aleph_0 nodes. Every end of a path contains aleph_0 nodes. These nodes can be mapped on that path. Every node will eventually be mapped on one path. Therefore all nodes are used up for constructing a countable set of paths.
> > When it is derived from logic of finite sets, then it is not the > > reverse of the logic of finite sets. But that is claimed in ZF. > > It is *not* the reverse of the logic of finite sets. For finite sets it is > identical.
For infinite sets it is not identical. And the reverse of being identical is being not identical.
> Ok. The logic of finite complete sums of rational elements gives that the > sum is rational. Using your logic we get that the complete (i.e. in the > limit) sum of rational elements is also rational. And so by that logic, e, > pi and whatever are rational, and all numbers we do use are rational.
In fact there are no binary expansions of irrational numbers. > > > Either those sets obey that logic or they do not > > exist in a science that is subject to the application of logic. > > Ah, so 'e', 'pi', 'sqrt(2)' do not exist. Still I think you use at least > one of them on occasion.
They exist as ideas but not as sums of series. > > > In fact thoses sets contradict their own existence under the > > government of logic. > > So 'sqrt(2)' contradicts itself under the government of logic. I think you > are a few thousand years behind.
I am only a few years ahead.
> > Small wonder. There cannot be a complete list, because the existence > > of a complete infinite linear set like N contradicts logic. > > Apparently your logic. What are the rules of inference in your logic?
The union of a complete set of linear sets is one of the sets.
> And, also apparently, in your logic the length of the diagonal of a square > with sides with size 1 does not exist.
It exists, but not as binary or decimal expansion. . > > > > That is a blatant lie. > > Eh? What is the lie? I just state that in ZF there are no non-static sets.
Then every element should be accessible. Why do you start always with a diminishingly small one? > > > Every static linear set has a last element. > > As that is false in ZF...
This shows that ZF deals with non-static sets if necessary. > > > If you say you cannot choose the last elemement because there is none, > > then you apply potentially infinite sets. > > You have still not provided a definition of "potentially infinite sets" in > ZF, so I cannot comment on this.
A potentially infinite set is a set that has no last element. > > > Then for every element that > > you choose there is a larger one. If it has been there all time, why > > the hell did you not start with this one ? > > Because in ZF a set does not need to have a last element. So whatever > element you chose, there is *always* a larger one.