Brian Tenneson skrev:
> On Thu, Jun 4, 2009 at 8:27 AM, Torgny Tholerus < 
> <>> 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.

>     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.

>     >
>     >> 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 
not contain any contradictions.

Torgny Tholerus

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