Re: Incompleteness of Cantor's enumeration of the rational numbers (infinite-middle, completions, poor rhetoric, lacking criticism)

On 11/26/2024 03:19 AM, FromTheRafters wrote:

WM laid this down on his screen :

On 25.11.2024 19:08, Jim Burns wrote:

On 11/25/2024 7:11 AM, WM wrote:

On 24.11.2024 19:16, Jim Burns wrote:

On 11/24/2024 6:06 AM, WM wrote:

On 24.11.2024 06:37, Jim Burns wrote:

>

Consider the set of natural numbers for which
there are NOT enough hats.

>
It is dark.

>
It is not dark what we mean by 'natural number'.
A natural number is countable.to from.0

>
That is a definable number.

>
Thus
it is not dark that
what we mean by 'natural number' is
countable.to from.0  and is
what you call a definable number.

>
Consider the black hats at every 10 n and white hats at all other
numbers n. It is possible to shift the black hats such that every
interval (0, n] is completely covered by black hats. There is no first
n discernible that cannot be covered by  black hat. But the origin of
each used black hat larger than n is now covered by a white hat.
Without deleting all white hats it is not possible to cover all n by
black hats. But deleting white hats is prohibited by logic. Exchanging
can never delete one of the exchanged elements. Therefore we have
here, like in all Cantor-pairings, the same impediment and further
disussion is futile: You must deny logic. I do not.

>
How much paint would be needed to paint the entire cartesian plane? Too
much huh? then just paint the first quadrant with one quarter the amount
of paint. Your 'logic' is too easily denied.

How much paint thinner when it's already painted?
See, now WM has framed it not simply "countable sets are countable"
and any old bijection from integers makes something countable and
composition of functions and inverses makes it simple for that
lazy and forgetful mathematicians can effect conceptually
something "done", instead, he's given a slightly less "practical"
problem with regards to an infinite population, and, overall,
the length of each member from the head of a linear queue they
all share.
Then, framing something as "not so", that countable sets are
countable, because something related is "not so", makes I would
say that it makes the interlocutors not even figuring that out,
when it is yet so that absent creation and destruction it is
not so that an ordering containing two distinct populations
eliminates either from re-ordering, first of all starts with
a bad question, then arrives at an embarrassing lack of taste.