# Re: The seven step-Mathematical preliminaries

```2009/6/13 Torgny Tholerus <tor...@dsv.su.se>:
>
> Jesse Mazer skrev:
>>
>> > Date: Fri, 12 Jun 2009 18:40:14 +0200
>> > From: tor...@dsv.su.se
>> > Subject: Re: The seven step-Mathematical preliminaries
>> >
>> > It is, as I said above, for me and all other humans to understand what
>> > you are talking about. It is also for to be able to decide what
>> > deductions or conclusions or proofs that are legal or illegal.
>>
>> Well, most humans who think about mathematics can understand
>> rule-based definitions like "0 is a whole number, and N is a whole
>> number if it's equal to some other whole number plus one"--you seem to
>> be the lone exception.
>>
>> As for being "able to decide what deductions or conclusions or proofs
>> that are legal or illegal", how exactly would writing out all the
>> members of the "universe" solve that? For example, I actually write
>> out all the numbers from 0 to 1,038,712 and say that they are members
>> of the "universe" I want to talk about. But if I write out some axioms
>> used to prove various propositions about these numbers, they are still
>> going to be in the form of general *rules* with abstract variables
>> like x and y (where these variables stand for arbitrary numbers in the
>> set), no? Or do you also insist that instead of writing axioms and
>> making deductions, we also spell out in advance every proposition that
>> shall be deemed true? In that case there is no room at all for
>> mathematicians to make "deductions" or write "proofs", all of math
>> would just consist of looking at the pre-established list of true
>> propositions and checking if the proposition in question is on there.
>
> 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. If you don't then the statement
is illegal regarding the rules (it violates the rule saying that a set
can't contains itself, which means that A in this system is not a set
thus all the reasoning in *that system* is false.

Choosing one rule or the other tells nothing about the rule itself
unless you can find a contradiction by choosing one or the other.

Regards,
Quentin

But I can't see why a set as I understand it cannot be part of
itself... {1,2,3} is included in {1,2,3} is true, what is the exact
problem with that statement ? (written differently all elements of the
set A are elements of the set B ===> A is included in B, here as A and
B are the same A is included in A.

> --
> Torgny Tholerus
>
> >
>

--
All those moments will be lost in time, like tears in rain.

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