On 10/15/2024 11:29 AM, Ross Finlayson wrote:
On 10/15/2024 08:51 AM, joes wrote:
Am Mon, 14 Oct 2024 18:37:31 +0200 schrieb WM:
On 14.10.2024 18:04, joes wrote:
Am Mon, 14 Oct 2024 15:40:01 +0200 schrieb WM:
On 14.10.2024 14:15, joes wrote:
>
No, we are taking the complete, actually infinite set which reaches
to "before" w.
and fills the space between 0 and ω evenly. Same happens with the
doubled set between 0 and ω2.
No, there is no consequent infinity. The even numbers do not go 0, 2,
4, ..., w, w+2, w+4, ..., w*2
Either the doubled numbers are natural, then half of them have not been
among the original set, or all natural numbers have been doubled, then
the result contains infinite numbers.
Which "half"?
>
That doubling _all_ natural numbers only yields _all_ natural numbers is
impossible.
Of course. But all doubled naturals are themselves natural.
>
>
Aliquot parts, i.e. density where the "half of the integers are even",
and the equivalency of the countably infinite, where "integers are
1-1 squares" were around centuries and THOUSANDS of years before
Grosseteste and Galileo, and are such OLD HAT that this "old wrapped
as new" is long, long, long past its freshness dating.
>
>
You know, sets model some numbers,
and, numbers model some sets.
This is called "descriptive" set theory
or usually enough "arithmetization".
Anyways it's totally usual that there's
a model of integers that is each of the
bounded above and below fragments
of integers, where, the operations
aren't closed, except, to enough small
numbers close to zero, they are.
I.e., it's exactly the same thing to have
a bunch of bounded models at least
one of which suffices, as to have
no least which doesn't suffice.
Or, this way then it's totally obvious
that a fair sampling, for example the
fair coin toss or Bernoulli trial, of integers,
results P(x is even) = 1/2, "in the limit",
i.e. what's called "asymptotic density".
That's its _name_ and that's what it _is_,
and that the "each least bound" and
"no least bound" models of integers
have both "asymptotic density" and
"countable cardinality" is so dirt-obvious
that it should go without saying, except
some people claim there's only the one
way and that what's a direct result in the
one is so in-direct in the other makes it
not a thing, well, that's a falsehood.
So, integers have _both_ these
"each least bound"
and
"no least bound"
what are totally inverted their
existential and universal quantifiers
and brought together in a fuller dialectic,
there is and are both, there are and is.
Opinionated closed-minded fools, thinking
that the only one that's closed is that
one that isn't each closure.