2009/6/17 Torgny Tholerus <torgny.domain.name.hidden>:
>
> Bruno Marchal skrev:
>> 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 Gödel) the
>> collection of all sets is not a set.
>
> It is not the naive notion of set that is inconsistent. It is the naive
> *handling* of sets that is inconsistent.
>
> This problem has two possible solutions. One possible solution is to
> deny that it is possible to create the set of all sets. This solution
> is chosen by ZF and VBG.
>
> The second possible solution is to be very careful of the domain of the
> All quantificator. You are not allowed to substitute an object that is
> not included in the domain of the quantificator. It is this second
> solution that I have chosen.
>
> What is illegal in the two deductions below, is the substitutions.
> Because the sets A and B do not belong to the domain of the All
> quantificator.
>
> You can define "existence" by saying that only that which is incuded in
> the domain of the All quantificator exists. In that case it is correct
> to say that the sets A and B do not exist, because they are not included
> in the domain. But I think this is a too restrictive definition of
> existence. It is fully possible to talk about the set of all sets. But
> you must then be *very* careful with what you do with that set. That
> set is a set, but it does not belong to the set of all sets, it does not
> belong to itself. It is also a matter of definition; if you define
> "set" as the same as "belonging to the set of all sets", then the set of
> all sets is not a set. This is a matter of taste. You can choose
> whatever you like, but you must be aware of your choice. But if you
> restrict yourself too much, then your life will be poorer...
>
>> 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.
>>
>
> The "proof" that there is no biggest natural number is illegal, because
> you are there doing an illegal deduction, you are there doing an illegal
> substitution, just the same as in the deductions below with the sets A
> and B. You are there substituting an object that is not part of the
> domain of the All quatificator.
No the proof is based on PA and in PA you do not have an axiom
restricting the successor function and as such it is defined in the
axiom that you don't have an upper bound limit. The proof is *valid*
against the axioms. *You* are doing an illegal deduction by not taking
into accound the rules with wich you work.
Regards,
Quentin
> --
> Torgny Tholerus
>
>> 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 <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?
>>>
>
>
> >
>
--
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Received on Wed Jun 17 2009 - 17:03:46 PDT