# Re: The seven step-Mathematical preliminaries

```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

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