# Re: The seven step-Mathematical preliminaries

```2009/6/14 Torgny Tholerus <tor...@dsv.su.se>:
>
> 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.```
```
Yes and one fulfil it according to the given rules the other not.

are.

> 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.

That's all the point of puting rules and checking that something is
correct or not according to it. 1+1=3 is false according to PA... that
doesn't mean you couldn't find a rule or mapping that would render
this statement true ***regarding the chosen rules***/

Regards,
Quentin

> --
> 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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```