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begin quote ita"
CHAPTER I [:] Notions from Set Theory [...] 4. FINITE AND INFINITE SETS. "end quote
These pages assume that the reader has the book and is reading along.    
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Please read these notes from the beginning to here and then paragraph 2 and 3 of 4.

DEFINITIONS FOR FINITE AND INFINITE SETS

begin quote ita-1-4-2-pg11 "
Let us therefore assume knowledge of the set   {1,2,3,...}.   A set   X   is called   finite   if it is empty or there is a positive integer   n   such that   X   can be put into one-one correspondence with the set   {1,2,3,...,n},   that is there is a one-one function from   {1,2,3,...,n}   onto   X.
"end quote

begin quote ita-1-4-3-pg11 "
A set is   infinite   if it is not finite.
"end quote

I would like to do in a slightly different form what Rosenlicht did in the paragraph cited above which defines the infinite set. Let   X   be any infinite set, and recall the general definition of a function. We know from the definitions above that   X   is not empty. Therefore, let   x   be any element of   X   and define a function

f       =       (    ( {1}, X   ),   { (1, x) }    ).

The function   f   is obviously one-one because the first set   {1}   has only one element.   Recall that one-one means different elements of the first set correspond under the function to different elements of the second set, which is   X   here.   However, the function is not onto, because then it would be finite by the definitions above.   Therefore, there is an element   x   in   X   which is different from   x,   and we can define a function

f       =       (    ( {1,2}, X   ),   { (1, x), (2, x) }    ).

Again, this function is obviously one-one, but it is not onto because that would make   X   finite.   Therefore let   x   be an element of   X   which is different from   x   and   x,   and define a function

f       =       (    ( {1,2,3}, X   ),   { (1, x), (2, x), (3, x) }    ).

As before, the function is one-one but not onto, and we can continue this process as far as we like: Let   n   be any positive integer. We will have a function

f       =       (    ( {1,2,3,...,n}, X   ),   { (1, x), (2, x), (3, x), ..., (n, x) }    ).

which is one-one but not onto. Then:

begin quote ita-1-4-3-pg11 "
[...] there exist distinct elements   x, x, x, ... in X.
"end quote

Rosenlicht's ... means we can go on without end, but that is also what we have above if   n   can be any positive integer.

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