Torngy, How many numbers do you think exist between 0 and 1? Certainly not only the ones we define, for then there would be a different quantity of numbers between 1 and 2, or 2 and 3.

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Jason On Thu, Jun 4, 2009 at 10:27 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? > > 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. > >> >>> 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. > > -- > 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 -~----------~----~----~----~------~----~------~--~---