# Re: The seven step-Mathematical preliminaries

```
Torgny Tholerus wrote:
> Brian Tenneson skrev:
>
>> On Thu, Jun 4, 2009 at 8:27 AM, Torgny Tholerus <tor...@dsv.su.se
>> <mailto:tor...@dsv.su.se>> wrote:
>>
>>
>>     Brian Tenneson skrev:
>>     >
>>     >
>>     > Torgny Tholerus wrote:
>>     >> It is impossible to create a set where the successor of every
>>     element is
>>     >> inside the set, there must always be an element where the
>>     successor of
>>     >> that element is outside the set.
>>     >>
>>     > I disagree.  Can you prove this?
>>     > Once again, I think the debate ultimately is about whether or not to
>>     > adopt the axiom of infinity.
>>     > I think everyone can agree without that axiom, you cannot "build" or
>>     > "construct" an infinite set.
>>     > There's nothing right or wrong with adopting any axioms.  What
>>     results
>>     > is either interesting or not, relevant or not.
>>
>>     How do you handle the Russell paradox with the set of all sets
>>     that does
>>     not contain itself?  Does that set contain itself or not?
>>
>>
>> If we're talking about ZFC set theory, then the axiom of foundation
>> prohibits sets from being elements of themselves.
>> I think we agree that in ZFC, there is no set of all sets.
>>
>
> But there is a set of all sets.  You can construct it by taking all
> sets, and from them doing a new set, the set of all sets.  But note,
> this set will not contain itself, because that set did not exist before.
>
If that set does not contain itself then it is not a set of all sets.```
```
>
>>
>>
>>
>>
>>     My answer is that that set does not contain itself, because no set can
>>     contain itself.  So the set of all sets that does not contain
>>     itself, is
>>     the same as the set of all sets.  And that set does not contain
>>     itself.
>>     This set is a set, but it does not contain itself.  It is exactly the
>>     same with the natural numbers, *BIGGEST+1 is a natural number, but it
>>     does not belong to the set of all natural numbers.  *The set of
>>     all sets
>>     is a set, but it does not belong to the set of all sets.
>>
>> How can BIGGEST+1 be a natural number but not belong to the set of all
>> natural numbers?
>>
>
> One way to represent natural number as sets is:
>
> 0 = {}
> 1 = {0} = {{}}
> 2 = {0, 1} = 1 union {1} = {{}, {{}}}
> 3 = {0, 1, 2} = 2 union {2} = ...
> . . .
> n+1 = {0, 1, 2, ..., n} = n union {n}
> . . .
>
> Here you can then define that a is less then b if and only if a belongs
> to b.
>
> With this notation you get the set N of all natural numbers as {0, 1, 2,
> ...}.  But the remarkable thing is that N is exactly the same as
> BIGGEST+1.  BIGGEST+1 is a set with the same structure as all the other
> natural numbers, so it is then a natural number.  But BIGGEST+1 is not a
> member of N, the set of all natural numbers.  BIGGEST+1 is bigger than
> all natural numbers, because all natural numbers belongs to BIGGEST+1.
>
Right, so n+1 is a natural number whenever n is.
>
>>
>>
>>
>>     >
>>     >> What the largest number is depends on how you define "natural
>>     number".
>>     >> One possible definition is that N contains all explicit numbers
>>     >> expressed by a human being, or will be expressed by a human
>>     being in the
>>     >> future.  Amongst all those explicit numbers there will be one
>>     that is
>>     >> the largest.  But this "largest number" is not an explicit number.
>>     >>
>>     >>
>>     > This raises a deeper question which is this: is mathematics
>>     dependent
>>     > on humanity or is mathematics independent of humanity?
>>     > I wonder what would happen to that human being who finally expresses
>>     > the largest number in the future.  What happens to him when he wakes
>>     > up the next day and considers adding one to yesterday's number?
>>
>>     This is no problem.  If he adds one to the explicit number he
>>     expressed
>>     yesterday, then this new number is an explicit number, and the number
>>     expressed yesterday was not the largest number.  Both 17 and 17+1 are
>>     explicit numbers.
>>
>> This goes back to my earlier comment that it's hard for me to believe
>> that the following statement is false:
>> every natural number has a natural number successor
>> We -must- be talking about different things, then, when we use the
>> phrase natural number.
>> I can't say your definition of natural numbers is right and mine is
>> wrong, or vice versa.  I do wonder what advantages there are to the
>> ultrafinitist approach compared to the math I'm familiar with.
>>
>
> The biggest advantage is that everything is finite, and you can then
> really know that the mathematical theory you get is consistent, it does
>
>
From what you said earlier, BIGGEST={0,1,...,BIGGEST-1}.  Then
BIGGEST+1={0,1,...,BIGGEST-1} union {BIGGEST} = {0,1,...,BIGGEST}.
Why would {0,1,...BIGGEST} not be a natural number while
{0,1,...,BIGGEST-1} is?

--~--~---------~--~----~------------~-------~--~----~
You received this message because you are subscribed to the Google Groups
"Everything List" group.
To post to this group, send email to everything-list@googlegroups.com
To unsubscribe from this group, send email to