# Re: The seven step-Mathematical preliminaries

```Quentin Anciaux skrev:
> Well it is illegal regarding the rules meaning with these rules set B
> does not exist as defined.
>   ```
```
What is it that makes set A to exist, and set B not to exist?  What is
the (important) differences between the definition of set A and the
definition of set B?  In both cases you are defining a set by giving a
property that all members of the set must fulfill.

Why is the deduction legal for set A, but illegal for set B?  There is
the same type of deduction in both places, you are just making a
substitution for the all quantificator in both cases.

--
Torgny Tholerus

>
> 2009/6/13 Torgny Tholerus <tor...@dsv.su.se>:
>
>> 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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