Re: infinity


zuhair wrote:
> Randy Poe wrote:
> > zuhair wrote:
> > > Randy Poe wrote:
> > > > zuhair wrote:
> > > > > Randy Poe wrote:
> > > > > > zuhair wrote:
> > > > > > > Randy wrote:
> > > > > > >
> > > > > > > Oh yes? Name one.
> > > > > > > ---------------------------------
> > > > > > >
> > > > > > > Before I relpy to that point let me be sure of your interpretation of
> > > > > > > bijection
> >
> > You said you were goint to answer this question. You never did.
>
> I am going to do that , but wait a little. The picture will be clear to
> you.
>
> of coarse You know very well I don't regard the bijection you are doing
>
> a real bijection , it is a pseudo-bijection.

I presume by now you know the definition of a bijection from
set A to set B.

1. For every element of B, there is some element of A mapped
to that element of B (surjection).

2. No two elements of A are mapped to the same element of B
(injection).

Since it's not a real bijection, which of these two properties
that completely characterize a "real bijection" is not
satisfied?

> But I want to clarify that to you , so please try to be patient.

> > > w= ||||||......
> >
> > OK, I need to modify the language again. w has a stick
> > at positions x=0, 1, 2, ...
> > >
> > > w'= |0|0|0......
> >
> > w' has a stick at positions 0, 2, 4, ...
>
> Randy If you read my articles especially the discussions between me
> and Loius H. Kauffman you will see that we are modelling in the way
> you are doing.Good start Randy.
> >
> > > Now Randy the number of sticks in both w and w' is infinite without
> > > doubt
> >
> > More specifically, the number of sticks is the same, and equal
> > to the cardinality of N.
>
> Lets take an intermediate position before we decide they are the same
> or not.

You understand that since I know there is a bijection between
these two sets, that I know how to reach that conclusion from
simple axioms, that no amount of axiom-free insistence from
you is going to convince me otherwise. "Convincing" in this
forum requires logical deduction, and it is precisely the
process of logical deduction which you reject.

> > > but does w contain the same number of sticks as w' does.
> >
> > Yes.
>
> wait please.
> >
> > I really can't see a better way than matching sticks to see
> > which has more (the now familiar "bijection").
>
> right!!!

OK. But we know the bijection exists. End of story.

> > > My answer is that w' contains half as many sticks in w.
> > >
> > > What is yours?
> >
> > See above.
> >
> > You will argue that w' is created by removing every second
> > stick from w. But w', the very same collection, can be
> > created in the ways I said without removing a single stick.
> >
> Now Randy suppose that those sticks cannot be moved from their places.
> the zeros in |0|0|0.... are placeholding zeros , and you can imagine
> them
> as a barrier that forbids the sticks from moving from their positions.

In effect, what you are arguing over and over is that because
a set has the same cardinality as a proper subset of w, it
can not have the same cardinality as w. This is the same
as your argument with the set of ordered pairs {(k,0),(k,1),k=0,1,...}

Yet this is an axiom introduced by you, something you want to
be true, which I am not going to accept. You see contradictions
because our bijections violate this axiom of yours. But it
doesn't violate any axiom of ours. The contradiction is introduced
by you, by insisting that something can't have the same cardinality
as both a set and a proper subset of that set.

It doesn't matter to me if I can biject 100 copies of N to a
set S (in fact, I can easily biject 100 copies of N to N).
That doesn't make S 100 times the size of N. I can choose
to "prove" that S is k times the size of N for any integer value of
k you choose > 0. I can prove that your set of ordered pairs
is also 4 times the size of N, or 8, or 17. So which of those
is the "real" size?

The only consistent measure I know is in terms of the MINIMUM
value of k, which is 1.

I know that I can make the set of evens from the set of naturals
by removing every other element. But I also know I can construct
it by doubling every element without adding or subtracting any.
That second operation clearly doesn't change the number of
elements in the set. Since the resulting set is EXACTLY THE SAME
SET as the removal operation, I don't think it makes sense
to say the two sets have different sizes.

It doesn't matter if you tell me the sticks are tied down and
unmovable. I know that w' is equivalent to a set created with
untied sticks. It is the very same set. Should two identical
objects have two different sizes?

You have asserted several times that the content is not the
same. You're wrong, and you know you're wrong which is why
you won't justify that statement.

- Randy

.

Relevant Pages

  • Re: A simple question?
    ... Your answer is that there cannot exist a bijection between a well ... ordered set and a not well ordered set. ... Since according to what you say, without the axiom of choice ... accordingly cardinality of Pas defined by Von Neumann, ...
    (sci.math)
  • Re: Set Theory-Axiom of Congruence.
    ... The cardinality of any set other than the empty set and the set of all sets is always greater than the cardinality of any proper subset of it. ... This axiom is meaningless until you tell us that. ... cardinality which becomes equal when their is a bijection: ... system inconsistent. ...
    (sci.math)
  • Re: Set Theory-Axiom of Congruence.
    ... The cardinality of any set other than the empty set and the set of all sets is always greater than the cardinality of any proper subset of it. ... This axiom is meaningless until you tell us that. ... cardinality which becomes equal when their is a bijection: ... So Card A = Card B iff A can be bijected to B. I am not referring to any different concept of cardinality. ...
    (sci.math)
  • Re: infinity
    ... > Randy wrote: ... > Is this a bijection from N to N'? ... > No of coarse not! ... > It is a surjection of N to N'. ...
    (sci.math)
  • Re: infinity
    ... Randy wrote: ... and at the same time claim that a bijection exists between them? ... Zuhair ... Prev by Date: ...
    (sci.math)