Re: Does the number of nines increase?

On 7/6/2024 1:25 AM, Ross Finlayson wrote:

On 07/05/2024 07:12 PM, Moebius wrote:

[...]

>
I'm reminded many years ago,
when studying size relations in sets,
that one rule that arrived was that
a proper subset, had a size relation,

? smaller than the superset,

and was told that
it was not so,
while, still it was written how
it was so.
>
Then, Fred Katz pointed me to
his Ph.D. from M.I.T. and OUTPACING,
showing that it was a formal result that
it was so.
>
So,
the "conclusion", seems to be, "not a conclusion",
for all the "considerations",
their conclusions, together.

No.
Each of the conclusions (no scare quotes)
is a conclusion
in a different discussion.
A human body has height, weight, age,
body.fat.index, and on, and on.
It is not an error
that there are these different numbers,
and that they are different.
The various human sizes are
somewhat.predictive of each other, but
where they aren't predictive, they aren't,
and I don't know of anyone who thinks
human sizes being not.always.predictive
is a mistake of some kind.
When we discuss the various set sizes,
a mistake is seen by some people,
because the set sizes are _too good_
⎛ This is my personal theory of
⎝ the psychological reasons for these objections.
For _all these sets_ (the everyday sets)
inclusion.size predicts matching.size.
But, look! Over there are sets for which
inclusion.size doesn't predict matching.size.
The sets are wrong!
Inclusion.size is wrong!
Matching.size is wrong!
Something must be wrong!
For some people,
being predictive for _all these sets_
but NOT for _these other sets_ (the infinites)
is just a bridge too far.
Situations in which,
for Albert and Betty or for Cathy and Daniel,
one is taller and the other is heavier
are not encountered rarely enough
to be seen as an error of some kind.

Then another one was asymptotic density and
the size relation of sets not just being ordered
but also having a rational value,
this was the "half of the integers are even".
>
It involves a bit of book-keeping, yet,
it is possible to keep these various notions,
while still there's cardinality sort of in the middle,
where of course on the other side of
these refinements of the notion of
the relation of size in infinite sets of
numbers their spaces their elements then
there's an entire absolute of "ubiquitous ordinals",
that have the infinite sets as of a "size".
>
So, when you mean cardinal, say cardinal.

A better rule is:
When you mean "cardinal", be sure
"cardinal" is understood.
Which is what we have been doing.
How is "ubiquitous ordinal" to be understood?

There are other notions of
"size", and "measure", and, "number".

The advantage the word "size" has over
the word "cardinal" is that
"size" is an everyday.word,  and,
as such, "size" travels with a cloud of
intuitions buzzing around it.
That advantage is a disadvantage
if our intuitions lead us astray.
If they do, "cardinal" might serve better.