Re: Cantor Confusion

In article <1170413330.341310.269660@xxxxxxxxxxxxxxxxxxxxxxxxxxxx> mueckenh@xxxxxxxxxxxxxxxxx writes:
On 2 Feb., 02:19, "*** T. Winter" <***.Win...@xxxxxx> wrote:
In article <1170348341.624257.130...@xxxxxxxxxxxxxxxxxxxxxxxxxxxx> mueck.=
....
If 4 is the set of all existing (whatever that does mean) sets with 4
elements and 5 is the set of all existing sets with 5 elements, we find
immediately that 4 is *not* a subset of 5. With this definition of
numbers as sets subsetting does not give what you wish.

4 is the set of all sets with 4 elements. 4 is also every set with 4
elements.

That can not be true. 4 can not be all kinds of different things at once,
at least not in mathematics. With this kind of definition 4 contains 4.

> Therefore we have to investigate whether such a representation exists.
> Note that I use the definition of a number by Peano in order to look
> for the existence of that number.

What does the latter sentence *mean*? But indeed, for most numbers some
representations do not exist, while other representations do exist. The
existence of a number is independent of the existence of a representation.

This point of view has lead to the present mess-math.

For instance, for most rational numbers a decimal representation does not
exist.

Correct, for instance for 1/7.

And for computable numbers some representation does exist.

> The definition of a number is given as non-circular as possible: p. 3:
> 1) 1 ist eine nat=FCrliche Zahl.
> 2) Jede Zahl a in N hat einen bestimmten Nachfolger a' in N.

in N is wrong here; N is not defined.

N is already in (1), because (1) is identical to "1 is in N".

N is not *defined*. Whether you use it in (1) is irrelevant, what is
relevant is that it is not defined.

A more proper version is:
2) Jede natürliche Zahl a had einen bestimmten Nachfolger a', und
dass ist auch ein natürliche Zahl.

What is every natural number if N is not defined?

This is a recursive definition of natural numbers. By (1) we have one
natural number, by (2), from that single natural number we get a lot of
other natural numbers.

> 3) Es gibt keine Zahl mit dem Nachfolger 1.
> 4) Aus a' = b' folgt a = b.
> 5) Jede Menge M von nat=FCrlichen Zahlen, welche die Zahl 1 und zu jed=
er
> Zahl a in M auch den Nachfolger a' enth=E4lt, enth=E4lt alle nat=FC=
rlichen
> Zahlen.

Yes, I know Peano pretty well. But this is *not* a circular definition. =
I
wonder why you think it is circular.

I did not say it is circular. I said "as non-circular as possible". If
it is not circular in your opinion, then be happy. (In fact every
definition is circular, because every language is. You cannot explain
anything without already using some unexplained wording. If you want
to explaine the unexplained by the words already explained, you get
circular. That is unavoidable. Cp. N being undefined but appearing in
the Peano axioms. But that is not the point here.)

This is getting phylosofical. In mathematics a definition is circular if
in the definition one of the deciding features is the term you want to
define. So a definition as you gave:
3 is the set of all sets of 3 elements
is a perfect example of a circular definition. The reason is that it
states precisely nothing about what 3 is. It is not better than the
definition:
3 is 3.
In order to know whether a particular set fits in 3 you have to know
what 3 is, and to know that you have to know whether that particular
set does fit.

> p. 130:
>
> 1) 1 in M.
> 2) If n in M then n + 1 in M.
I prefer the successor of n, rather than n + 1 here.

I think that everybody able to read and understand these lines will
know what "+ 1" means while the successor is not immediately clear.
(The successor of n coul be n+2 or 2*n or 10*n or ....)

Yes, indeed, and that is the crux. Abstraction. When you use the
'+ 1' notation you are already assuming the existence of addition.
When you do not, you can properly define addition (and all other
operations) using the Peano axioms. You are *defining* the natural
numbers, presumably without any knowledge about what natural numbers
even are. And I may note that with '2*n' you can get a set that is
isomorphic (with respect to all operations) to the natural numbers,
only the naming is different. Consider the set of powers of 2,
(your 2*n case) define:
a '+' b = 2^[ log_2(a) + log_2(b) + 1 ]
a '*' b = 2^[ log_2(a) * log_2(b) + log_2(a) + log_2(b) ]
Call this set K. You may verify that the rings:
R(N, +, *) and R(K, '+', '*')
are isomorphic.

And *in this context* the definition of 3 is:
succ(succ(1))
no set at all. And the existence of 3 follows from the first set of
axioms you gave.

The existence does not follow from axioms. Axioms state something in
an arbitrary way. They can even state the famous pink elephant.

The non-existence of a pink elephant in the realm of the axiom that
states that there is a pink elephant is not mathematics but philosophy.
Axioms state what things exist (or do not exist) in their realm.
In geometry there is the parallel axiom (that you also use as example
for the axiom of choice in your book). In two forms of geometry there
exists a line through a point parallel to a given line not through that
point. In one form of geometry such a line does not exist. (Exist
meaning exist mathematically here.) The axiom states existence or
non-existence, and that is all.

Whether it exists remains to be investigated.

Your existence is not a mathematical existence.

Your definition is circular, this one is not.

I do not give a definition but I look for the existence of the number
already defined.

So when I ask you for a definition you do not give a definition? Yes,
I have been thinking that all along.
--
*** t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~***/
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