On Thu, Jun 4, 2009 at 8:27 AM, Torgny Tholerus <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.

> 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?

> >
> >> 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.

> --
> Torgny Tholerus
> >

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