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Topic: The set of natural numbers
Replies: 29   Last Post: Oct 1, 2017 5:33 PM

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Alan Smaill

Posts: 1,052
Registered: 1/29/05
Re: The set of natural numbers
Posted: Aug 30, 2017 6:39 AM
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"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 UTC-4, Julio Di Egidio wrote:
>>> Theorem: |N = {1,2,3,...;w} (where w denotes a simple infinity).
>>
>> You are FUCKING STUPID, speaking of Pavlovian anti-crankery. 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



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