Date: Mar 23, 2013 10:01 PM
Subject: Re: Matheology � 224
WM <email@example.com> wrote:
> On 23 Mrz., 21:26, William Hughes <wpihug...@gmail.com> wrote:
> > > > > I say that there is no finite line that changes the union.
> > > > Correct
> > > > > So the union would be the same if there was no finite line.
> > > > Nope, does not follow.
> In set theory we can construct the set of all elements that have a
> certain property. Does that mean that the property vanishes if too
> many elements belong to the set?
It only vanishes from among non-members of that set.
> The set of all negative numbers cannot be subtracted from the real
> line without subtracting also a positive number ?
One may subtract one real number from another, but one normally deletes
a member of a set like the real line from that set.
And deleting one member does not usually require deletion of any other.
> The set of all red cars cannot be formed without containing a green
> car too?
Sounds line Wolkenmuekenheim again!
> The set of all lines without any relevance for the union cannot be
> subtracted without changing the union?
If WM is talking about his lines/FISONs of naturals again and the union
of all such FISONs, he should note that for a natural to be in the union
of all FISONs or in the union of any non-empty set of FISONs, it must be
in at least on FISON of the set of FISONs being unioned.
> You claim that no finite line of the set changes the union.
Appending one FISON to an infinite set of FISONs does not enlarge its
union and removing one FISDON from an infinite set of FISONs does not
diminish its union. In fact any two infinite sets of FISONs have the
same union, namely |N, and only finite sets of FISONs can have unions
which are proper subsets of |N.
at lesat outsice Wolkenmuekenheim.
> You claim that when every finite line which does not change the union,
> is deleted, then the union is changed.
That statement is far too abmigious to be meaningful.
Removing any one FISON from the set of all FISONs does not change the
But that does not mean that removing all FISONs from the set of all
FISONs does not leave an empty set whose union is also the empty set.
The critical point is that the union of any infinite set of FISONs
results in the infinite set of all naturals,|N, and the union of any
non-empty finite set of FISONs results in its maximal member.
So the union of any non-empty but finite set of of FISONs is always a
FISON, but the union of any infinite set of FISONs is always |N, which
is not a FISON.
At least everywhere outside Wolkenmuekenheim.