Re: Infinity......


William Hughes wrote:
zuhair wrote:
William Hughes wrote:
zuhair wrote:
William Hughes wrote:
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.

Yes, that is clear man, who is dicussing this. But I have something to
add here,
which is the following:

"if there is a bijection between A and B , then we can change the name
of every element of A by an element of B without changing the size of
A, As far as size A and the replacements are defined after this
bijective function" I will elaborate on this below.


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

Ok, I got your point.
But only to express my new definition of size

I don't think the point has sunk in. As soon as you
adopt a "new definition of size" you have to accept the
fact that there will be a set A such you can change
the size of A by renaming each of the elements.

Please answer the following question.

Are you willing to adopt a definition of size under which
there exists a set A such that you can change the size
of A by renaming each of its elements?

- William Hughes

Though I said I will not post a message to this forum again, yet since
you asked then there is a moral obligation on my side to answer.

The answer to your question is no. we cannot change the size of any set
by merelly renaming each of it's elements.


In that case the two sets

A={0,1,2,3,...}
and
B={0,2,4,6,...}


have the same size because we can use the elements of B to
rename the elements of A.

Yes, the have the same size as determined by f when f is bijective.
but when f is not bijective then they don't have the same size_f.

see what I wrote. you are not differentiating between reltive and
absolute size.
that's your problem

Note that the *only* property of size that was used here was
the fact that renaming the elements of a set does not change its size.
Thus, under *any* definition of size you come up with
A and B will have the same size.

What kind of a size, the relative or the absolute size.


- William Hughes

My ideas are different from the standard, they might look so complex.
but they are nearer to the truth.

Cantor's system is inconsistent.because it hold together two arguments
that cannot be held together and these are

1) f:A->B, f is injective and serjective <-> size A = size B
2) f:A->B, f is injective and not serjective <-> size A <= size B.

These two arguments cannot be held together.

because 1)-> f:A->B, f is injective and not serjective <->( size A <
size B )

But with 2) then this leads to

size A < size B <-> size A <= size B, which is contradictive.

Therefore Cantorian system of defining cardinality is inconsistent.

In reality it is even not precise. since the correct statements are

1) f:A->B, f is injective and serjective <-> ( size A = size B )_f
2) f:A->B, if is injective and not serjective <-> ( size A < size B
)_f.

The cantorian system is neither precise nor consistent. I showed that,
and you are still stubborn holding to it true, and due to prejedous
nothing else.

When this fog will be washed out of your minds, I don't know.

Zuhair

.

Relevant Pages

  • Re: Infinity......
    ... William Hughes wrote: ... If you adopt any definition of size such that A and B do not ... If there is a bijection between A and B then A and B have ... Now which one is the function reflecting the absolute size comparison ...
    (sci.math)
  • Re: Infinity......
    ... William Hughes wrote: ... Note that there is a bijection from A to B. ... Thus there is a way to rename the elements of B to the elements ... nothing of my dichotomy between Absolute and relative size., ...
    (sci.math)
  • Re: Infinity......
    ... William Hughes wrote: ... Note that there is a bijection from A to B. ... Thus there is a way to rename the elements of B to the elements ... nothing of my dichotomy between Absolute and relative size., ...
    (sci.math)
  • Re: Infinity......
    ... If you adopt any definition of size such that A and B do not ... 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 ... the same absolute size, then the absolute size of A ...
    (sci.math)
  • Re: Infinity......
    ... William Hughes wrote: ... Note that there is a bijection from A to B. ... Thus there is a way to rename the elements of B to the elements ... nothing of my dichotomy between Absolute and relative size., ...
    (sci.math)
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