On 4/27/2024 10:46 AM, WM wrote:
Le 26/04/2024 à 20:53, Jim Burns a écrit :
On 4/26/2024 10:37 AM, WM wrote:
That is true for definable numbers
but not for the last numbers before ω.
>
If any number below n canNOT be counted to from 0
then n itself canNOT be counted to from 0
>
Thus,
each number which CAN be counted to from 0
is not above
any number which canNOT be counted to from 0
>
By definition,
ω is between
numbers which CAN be counted to from 0 and
numbers which canNOT be counted to from 0
>
Imagine being someone who denies that definition of ω
>
Because the following isn't a claim about ω
you (the denier) should still admit:
if n can be counted to from 0
then n⋅2 can be counted to from n
then n⋅2 can be counted to from 0 (through n)
>
If ω exists as defined,
then doubling never crosses ω
(from CAN to canNOT)
>
Hence
not all natural numbers exist as defined
for visible numbers.
1.
We describe some things.
For our convenience,
we refer to what we describe by some label.
Whatever label we use, "natural number" or
"flying rainbow sparkle pony",
using that label uses _that description_
2.
We describe some things as
things which can be counted to from 0
Everything which is
a thing which can be counted to from 0
is
a thing which can be counted to from 0
Nothing which is not
a thing which can be counted to from 0
is
a thing which can be counted to from 0
For our convenience,
I refer to
things which can be counted to from 0
as
flying rainbow sparkle ponies.
3.
I describe a thing as
the first upper bound of
flying rainbow sparkle ponies.
For our convenience,
I refer to that thing as ω
ω is the first upper bound
of flying rainbow sparkle ponies
(of things which can be counted to from 0).
4.
If n can be counted to from 0
then n⋅2 can be counted to _from n_
then n⋅2 can be counted to from 0 _through n_
If n is a flying rainbow sparkle pony
then n⋅2 is a flying rainbow sparkle pony.
If n is before ω
then n⋅2 before ω
5.
For each flying rainbow sparkle pony m,
it is not always true that,
if n is before m
then n⋅2 before m
However,
it is always true that,
if n is before ω
then n⋅2 before ω
The explanation for that is that ω
least upper bound of flying rainbow sparkle ponies
is not
a flying rainbow sparkle pony.
6.
My guess is that you (WM) feel that
you (WM) should be able to _define_ ω into being
a flying rainbow sparkle pony.
Definitions do not have that power.
The most that you (WM) can do in that regard
is to define ω into "being" non.existent, since
a flying.rainbow.sparkle.pony least.upper.bound of
flying.rainbow.sparkle.ponies must not.exist.
7.
A claim that a non.existent thing exists
has consequences.
It has all the consequences you can wish for
and all the consequences you can wish against.
It has both, for and against.
For that reason, it's useless as an argument.
Imagine that the existence of darkᵂᴹ numbers
is a consequence of a flying.rainbow.pony
first.upper.bound of flying.rainbow.ponies.
That's useless as an argument because
the non.existence of darkᵂᴹ numbers follows
equally well.
V
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