Re: Infinity......


zuhair wrote:
William Hughes wrote:

No. I am saying that either you must abandon one of

i: if you rename all of the elements of A then A does
not change size

(from which it follows trivially that
ia: if the elements of A are replacable without repetition
by the elements of B then A and B have the same size)

or

ii: A and B do not have the same size.

If you adopt any definition of size such that A and B do not
have the same size then you must abandon i.

- William Hughes

I don't know what all of the above has got to do with what I am
professing here.

Very little. What you are professing has little to
do with the point I am trying to make. You are
giving details and justifications for your definition
of size. However, I wish to
point out that *any* definition of size is
subject to the following..

The two statements

a: You can use the names of the elements of
B to uniquely rename the elements of A and visa versa.

and

b: There is a bijection between A and B


are equivalent.

Now before you go on. Do you
agree that the two staments are equivalent?

Therefore. If you adopt the principal

Changing the name of every element of A does not change
the size of A

then

If there is a bijection between A and B then A and B have
the same size.


The above will apply to *any* defintion of size, whether based on
bijection, injection, surjection, simplest injection, prettiest
injection ...
You cannot adopt "changing names does not change the size"
if you have a definition of size where there can be a bijection
between sets A and B of different sizes.

Do you agree with this?

Now, you can give as many reasons as you want why one should
pick a definition of size such that there can be a bijection
between sets A and B of different sizes. The fact remains
that under such a definition there exists a set A
such that changing the names of every element in A changes
the size of A.

You are putting the cart before the horse.

Decide what you want from your defintion of size

Find out what restrictions this puts on your definition of size.

Then, and only then, attempt to define size.

- William Hughes

.

Relevant Pages

  • Re: Infinity......
    ... If you adopt any definition of size such that A and B do not ... have the same size then you must abandon i. ... B to uniquely rename the elements of A and visa versa. ... If there is a bijection between A and B then A and B have ...
    (sci.math)
  • Re: Infinity......
    ... I cannot rename all of A elements by replacing ... the same absolute size, then the absolute size of A ... Yes, there is a way of renaming the elements of B, ... if there is a bijection between sets A and B, ...
    (sci.math)
  • Re: Infinity......
    ... I cannot rename all of A elements by replacing ... the same absolute size, then the absolute size of A ... Yes, there is a way of renaming the elements of B, ... if there is a bijection between sets A and B, ...
    (sci.math)
  • Re: Infinity......
    ... I cannot rename all of A elements by replacing ... the same absolute size, then the absolute size of A ... Yes, there is a way of renaming the elements of B, ... if there is a bijection between sets A and B, ...
    (sci.math)
  • Re: Infinity......
    ... If there is a bijection between A and B then A and B have ... bijection, injection, surjection, simplest injection, prettiest ... Now we desire to know the absolute set size comparison not these ... We should have a fare criterion that could be f or g or h, ...
    (sci.math)