# Re: Bijections (was OM = SIGMA1)

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Le 14-nov.-07, à 17:23, Torgny Tholerus a écrit :```
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>
> Bruno Marchal skrev:
>> 0) Bijections
>>
>> Definition: A and B have same cardinality (size, number of elements)
>> when there is a bijection from A to B.
>>
>> Now, at first sight, we could think that all *infinite* sets have the
>> same cardinality, indeed the "cardinality" of the infinite set N. By
>> N,
>> I mean of course the set {0, 1, 2,  3,  4,  ...}
>>
> What do you mean by "..."?

Are you asking this as a student who does not understand the math, or
as a philospher who, like an ultrafinist, does not believe in the
potential infinite (accepted by mechanist, finistist, intuitionist,
etc.).

I have already explained that the meaning of "...'" in {I, II, III,
IIII, IIIII, IIIIII, IIIIIII, IIIIIIII, IIIIIIIII, ...}  is *the*
mystery.

A beautiful thing, which is premature at this stage of the thread, is
that accepting the usual meaning of "..." , then we can mathematically
explained why the meaning of "..." has to be a mystery.

>> By E, I mean the set of even number {0, 2, 4, 6, 8, ...}
>>
>> Galileo is the first, to my knowledge to realize that N and E have the
>> "same number of elements", in Cantor's sense. By this I mean that
>> Galileo realized that there is a bijection between N and E. For
>> example, the function which sends x on 2*x, for each x in N is such a
>> bijection.
>>
> What do you mean by "each x" here?

I mean "for each natural number".

>
> How do you prove that each x in N has a corresponding number 2*x in E?
> If m is the biggest number in N,

There is no biggest number in N. By definition of N we accept that if x
is in N, then x+1 is also in N, and is different from x.

> then there will be no corresponding
> number 2*m in E, because 2*m is not a number.

Of course, but you are not using the usual notion of numbers. If you
believe that the usual notion of numbers is wrong, I am sorry I cannot

Bruno

>> Now, instead of taking this at face value like Cantor, Galileo will
>> instead take this as a warning against the use of the infinite in math
>> or calculus.
>>
> --
> Torgny Tholerus
>
> >
>
http://iridia.ulb.ac.be/~marchal/

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