Re: infinity
- From: "zuhair" <zaljohar@xxxxxxxxx>
- Date: 27 Oct 2005 10:33:39 -0700
Matt Gutting wrote:
> zuhair wrote:
> > Matt wrote:
> > As Randy said, "arrangements" of a set do not have cardinality: they
> > have
> > ordinality. The set itself has one cardinality, independent of any
> > arrangements;
> > but the arrangements do not have cardinality.
> > ----------------------------------------------------
> >
> > The relation between cardinality and ordinality is a deep subject more
> > deeper than
> > you illustrated, in reality many times we cannot reach a specific
> > cardinality without ordinality
> > for example Aleph-1 is only reached through the Omegas with fields
> > having a cardinality of Aleph-0
> > So in reality it is the ordinal number Omega-1 is the one which gave
> > birth to Aleph-1 and not the reverse. I am doing something of that kind
> > in my question , but neither you nor Randy here figured what I am
> > trying to show. You will not solve matters by giving different
> > terminology for things , a thing which has ordinal number has
> > cardinality also.
>
> Not quite. The *set* has a cardinal number. The *order* in which the set is
> arranged has an ordinal number.
As in my reply to Randy what you call order is involved in series and
its called
the serial relation ( which must be ordered ie aliorelative,transitive
and connected) . but the field of a serial relation (serial relation is
what you like to call as "order" ) is a set.
The field of a serial relation ( or series) is the terms that have that
relation.
So series S=1,2,3,4,5 is not a set of coarse. but I can form a set P
having members that are defined by being identical to terms in the
series above.
so P= { 1,2,3,5,4 }
Their is no order involved in the definition of P. and in reality P is
called
the field of the serial relation in S.
Now since P is a set, then it has Cardinality.
This explains what I meant by saying things which has an ordinal
number(ie series)
has Cardinality as well ( of the fields of that series )
Review Introduction to mathematical philosophy. Bertrand Russell.
Zuhair
>
> >
> > But if you insist I will use your language , if set S can be equally
> > defined in a way that do not involve order and that can be rearranged
> > in the arrangment I pointed and can be bijected to Peano's set of
> > natural numbers starting from zero(which is impossible)
> > then and only then I will say the cardinality of set S is Aleph-0.
> >
>
> I thought Randy already came up with a bijection between the Peano-defined
> natural numbers and your example set. Given a set, do you deny that it can be
> arranged in any arbitrary way (including your arrangement)? If so, on what
> basis? Do you deny that Randy's function is a bijection? If so, can you prove
> it? If you do not deny either assertion, then you must (by your own admission)
> say that the cardinality of your example set is Aleph-0.
>
> Just to clarify: Your example set, as I recall, was S =
> {(0,0),(0,1),(1,0),(1,1),...}; and the ordering you desired was
> (0,0),(1,0),(2,0),...,(0,1),(1,1),(2,1),.... (Note the lack of set brackets
> here.) Randy's function - again from my recollection - was
>
> f:N -> S : f(n) = (floor(n/2),n mod 2)
>
> (Its inverse would be g:S -> N : g(a,b) = 2a + b.)
Both are not the set I defined , the set I defined is the field of the
series I defined and it is not bijected by the above function.
>
> Given the bijection between S and N, one can then order S in the desired
> arrangement.
You should first have the bijection which you didn't.
>
> > Cardinality is not that seperate from ordinality , they are different
> > of coarse but still one depends on the other , the generation of the
> > (Omega-x)s in Cantorian analysis and the Cardinals that comes from them
> > the (Aleph-[x+1})s is a good example of what I am saying
> > However I will wright a post on that matter to the group soon.
> >
> > And generally speaking I think I figured out the difference in views
> > between TOmatics and mine and Ross on one side and the standard views
> > adopted by mathematicians on the other side, I think it is
> > philosophical and I will wrigth something on that soon.
> >
> > Zuhair
> >
.
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