On 2/23/2014 1:01 PM, Virgil wrote: > In article <firstname.lastname@example.org>, > "Julio Di Egidio" <email@example.com> wrote: > >> Indeed. To reiterate, mine was not an objection to the notion of 1-to-1 >> mappings, e.g. I have no qualms with the idea that there are as many even >> natural numbers as there are natural numbers: both collections are simply >> *endless*. It is the idea that the natural numbers form a set, i.e. a >> complete totality that is, as I am contending, untenable. > > If you can speak of the natural numbers collectively, as you just did, > then in your mind you can distinguish them for everything else, which is > just what claiming a set(collection) of them is doing. > > So you are, in effect, saying that you cannot do what you have just done. >
Julio, di Egidio, I wouldn't let that the numbers weren't a collection, just because we can collect them.
Because most assuredly you don't intend to give up other features of the numbers as they already are.
Then, Hancher, neither, these sets as regular in all the regular spaces they comprise, is not that as well-founded then as the continuum of numbers, that they aren't so. The numbers are regular for the continuum of the naturals or reals, as of the naturals. Yet, it is not as well-founded, that the "infinity" of the collection of the numbers, would preclude they were regular (here well-ordered in their usual sense, infinite numbers for example from zero).
The natural integers are infinite: there is no greatest, this is via induction. Then, Goedel proves the consistent arithmetic incomplete, there are true facts about its objects, only integers, and consistent thus with all other true facts, that are not theorems already of "there are only finite(-ly) many integers that are infinite". Then, there are infinitely many integers that are infinite, this with that there are still only integers.
Then, for what infinity is or would, it is symmetric to zero that all its predecessors reflect and go to zero as all zero's successors go to it.