# Re: The seven step-Mathematical preliminaries

```
Torgny,```
```
I agree with Quentin.
You are just showing that the naive notion of set is inconsistent.
Cantor already knew that, and this is exactly what forced people to
develop axiomatic theories. So depending on which theory of set you
will use, you can or cannot have an universal set (a set of all sets).
In typical theories, like ZF and VBG (von Neuman Bernay GĂ¶del) the
collection of all sets is not a set. In NF, some have developed
structure with universal sets, and thus universe containing
themselves. Abram is interested in such universal sets. And, you can
interpret the UD, or the Mandelbrot set as (simple) model for such
type of structure.

Your argument did not show at all that the set of natural numbers
leads to any trouble. Indeed, finitism can be seen as a move toward
that set, viewed as an everything, potentially infinite frame (for
math, or beyond math, like it happens with comp).

The problem of naming (or given a mathematical status) to "all sets"
is akin to the problem of giving a name to God. As Cantor was
completely aware of. We are confused on this since we exist. But the
natural numbers, have never leads to any confusion, despite we cannot
define them.

You argument against the infinity of natural numbers is not valid. You
cannot throw out this "little infinite" by pointing on the problem
that some "terribly big infinite", like the "set" of all sets,  leads
to trouble. That would be like saying that we have to abandon all
drugs because the heroin is very dangerous.
It is just non valid.

Normally, later  I will show a series of argument very close to
Russell paradoxes, and which will yield, in the comp frame,
interesting constraints on what computations are and are not.

Bruno

On 13 Jun 2009, at 13:26, Torgny Tholerus wrote:

>
> Quentin Anciaux skrev:
>> 2009/6/13 Torgny Tholerus <tor...@dsv.su.se>:
>>
>>> What do you think about the following deduction?  Is it legal or
>>> illegal?
>>> -------------------
>>> Define the set A of all sets as:
>>>
>>> For all x holds that x belongs to A if and only if x is a set.
>>>
>>> This is an general rule saying that for some particular symbol-
>>> string x
>>> you can always tell if x belongs to A or not.  Most humans who think
>>> about mathematics can understand this rule-based definition.  This
>>> rule
>>> holds for all and every object, without exceptions.
>>>
>>> So this rule also holds for A itself.  We can always substitute A
>>> for
>>> x.  Then we will get:
>>>
>>> A belongs to A if and only if A is a set.
>>>
>>> And we know that A is a set.  So from this we can deduce:
>>>
>>> A beongs to A.
>>> -------------------
>>> Quentin, what do you think?  Is this deduction legal or illegal?
>>>
>>
>> It depends if you allow a set to be part of itselft or not.
>>
>> If you accept, that a set can be part of itself, it makes your
>> deduction legal regarding the rules.
>
> OK, if we accept that a set can be part of itself, what do you think
> about the following deduction? Is it legal or illegal?
>
> -------------------
> Define the set B of all sets that do not belong to itself as:
>
> For all x holds that x belongs to B if and only if x does not belong
> to x.
>
> This is an general rule saying that for some particular symbol-
> string x
> you can always tell if x belongs to B or not.  Most humans who think
> about mathematics can understand this rule-based definition.  This
> rule
> holds for all and every object, without exceptions.
>
> So this rule also holds for B itself.  We can always substitute B for
> x.  Then we will get:
>
> B belongs to B if and only if B does not belong to B.
> -------------------
> Quentin, what do you think?  Is this deduction legal or illegal?
>
>
> --
> Torgny Tholerus
>
> >

http://iridia.ulb.ac.be/~marchal/

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