Well it is illegal regarding the rules meaning with these rules set B
does not exist as defined.
2009/6/13 Torgny Tholerus <torgny.domain.name.hidden>:
>
> Quentin Anciaux skrev:
>> 2009/6/13 Torgny Tholerus <torgny.domain.name.hidden>:
>>
>>> 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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Received on Sat Jun 13 2009 - 13:30:57 PDT