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 > > > > --~--~---------~--~----~------------~-------~--~----~ 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 everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~----------~----~----~----~------~----~------~--~---