
Re: The set of natural numbers
Posted:
Aug 30, 2017 6:39 AM


"Julio P. Di Egidio" <julio@diegidio.name> writes:
> On 29/08/2017 05:35, George Greene wrote: >> On Monday, August 28, 2017 at 6:47:45 PM UTC4, Julio Di Egidio wrote: >>> Theorem: N = {1,2,3,...;w} (where w denotes a simple infinity). >> >> You are FUCKING STUPID, speaking of Pavlovian anticrankery. N >> *HAS*A*DEFINITION*!!! Nobody can prove that ANYthing is something >> other than what it IS DEFINED to be!
> You (and the others) are just mistaken on that: the set of natural > numbers is *the minimal set that satisfies the axiom of infinity* and > my theorem is about which set that is. So, at least the problem > statement is surely legitimate.
A basic property of sets (not involving undefinable elements, or uncountability) is that a definable collection of elements from a given set is also a set.
If you accept the existence of X = {1,2,3,...;w} as a set, then by the property above there is also a set
Y = { x in X : x =/= w }.
That is a smaller set, and its existence is enough to ensure that there is an infinite set (even if "infinite" is taken as "potentially infinite").
Do you reject the idea that a definable collection from a given set is also a set?
> > Julio
 Alan Smaill

