Re: Cantor Confusion

On May 9, 3:52 am, WM <mueck...@xxxxxxxxxxxxxxxxx> wrote:
On 8 Mai, 20:32, Virgil <vir...@xxxxxxxxxxx> wrote:
In article <1178636838.282834.114...@xxxxxxxxxxxxxxxxxxxxxxxxxxx>,

WM <mueck...@xxxxxxxxxxxxxxxxx> wrote:
On 7 Mai, 23:50, Virgil <vir...@xxxxxxxxxxx> wrote:

http://books.google.com/books?id=Er1r0n7VoSEC&pg=PA23&ots=1afi1e6mts&;...
k++Jech+set+theory+function&sig=Zx5hPqZZ2icNy3mkguHi9kyrFVA#PPA24,M1
/Introduction to Set Theory/, by Karel Hrbacek, Thomas J. Jech
(1999) [pages 23-4]
:
: * 3. Functions *
:
: Function, as understood in mathematics, is a procedure, a
: rule, assigning to any object /a/ from the domain of the
: function a unique object /b/, the value of the function
: at /a/. A function, therefore, represents a special type
: of relation, a relation where every object /a/ from the
: domain is related to precisely one object in the range,
: namely, to the value of the function at /a/.
:
: * 3.1 Definition * A binary relation /F/ is called a
: /function/ (or /mapping/, /correspondence/) if /aFb_1/
: and /aFb_2/ imply /b_1 = b_2/ for any /a/, /b_1/, and
: /b_2/. In other words, a binary relation /F/ is a function
: if and only if for every /a/ from dom /F/ there is exactly
: one /b/ such that /aFb/. This unique /b/ is called
: /the value of F at a/ and is denoted /F(a)/ or /F_a/.
: [F(a) is not defined if /a [not in] dom F/.] If /F/ is
: a function with /dom F = A/ and /ran /F/ [subset] B/,
: it is customary to use the notations /F: A -> B/,
: /<F(a)| a [in] A>/, /<F_a>_a[in]A/ for the function /F/.
: The range of the function /F/ can then be denoted
: /{F(a)| a [in] A}/ or /{F_a}_a[in]A/.
:

As usual, WM includes only the irrelevant bits

I only wanted to avoid typing infinite definitions.

and excludes the part
that gives the formal definition, and, incidentally, proves him wrong

The second paragraph proves the first one wrong, in your opinion?

A formal definition always REPLACES any informal ones, and governs the
meaning and usage of the thing defined.

But the formal definition does not specify how a and b are related
other than by mentioning F. This F however is undefined unless you
know from the first paragraph that it is a procedure or rule.

WRONG. F is a SET.

Further definition 3.1 contains: It is customary to use the notations /
F: A -> B/. Why do you think A and B were not two sets and F was not a
formula, as I said (and as anybody says who ever studied some
mathematics)?

What are you talking about? F, A, and B are all SETS. In Z set
theories, every object is a set (and in ordinary class theory, every
object is either a set or a proper class).

'F' is NOT a variable of the meta-lanaguage ranging over formulas. 'F'
is a variable of the OBJECT language, thus F is a set (or more
precisely, upon an appropriate definition of 'set' (such as 'x is a
set <-> Ey xey) we have, in Z set theories, the theorem 'Ax x is a
set', thus we have, as a derivable line in a proof,. the formula 'F is
a set'.

And the formal definition requires a function to be a set of ordered
pairs (a relation), so that it is first of all a SET, which is quite
different from being merely a rule.-

Of course it is not merely a rule. It is a rule and two sets, which
are connected by this rule.

No, here it is:

3.1 Definition * A binary relation F is called a function (or mapping,
correspondence) if aFb_1 and aFb_2 imply b_1 = b_2 for any a, b_1 and
b_2.

There it is in EXPLICIT mathematical English.

F is a relation. And a relation is a set (it is a set of ordered
pairs). F is not required to be a formula.

No matter that is has been EXPLICITLY shown to you that you are
incorrect, you still persist to confuse the informal notion (and you
even get THAT wrong) with the rigorous definition.

MoeBlee

.

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