Re: Replacement of Cardinality

On 8/28/2024 10:36 PM, Jeff Barnett wrote:

On 8/28/2024 2:10 PM, Jim Burns wrote:

[...]

>
"Trichotomous" generates the right mental images,
I must admit.
But I'm not sure how one could show that
a relation R on S, where R is Trichotomous,
implies that
I can't 1-1 map a strict subset of R to R.

I agree that one can't prove that.
I am doing something else here.
My apology for having stripped so much context.
I get complaints about the length of my posts.
And, as much as stripping context is not to my taste,
if my chief correspondent won't read my brilliant text,
Something Must Be Done.
The Something which Must Be Done is that
my side of this multi.year correspondence has become
a search for perfect zingers by which to transmit
principles of predicate logic, finiteness, infiniteness,
sets, natural numbers, real numbers, and other topics,
as appropriate.
The search for zingers has kept me tolerably amused,
so far, enough to keep me plugging away, at least.
The downside is that a random passerby, such as yourself,
might (understandably) find whatever.we.have.here opaque.
Again, my apology.
Context:
Wolfgang Mückenheim has been rejecting
Dedekind.infinite sets, sets R such that
I can 1-1 map a strict subset of R to R.
WM calls them "potentially infinite", by which
WM means that these sets change, which means that
WM is not talking about _those sets_
which do not change.
A typical WM.argument sets up some sequence and
asserts by mathematics, by logic
(in reality, by "common sense", by "obviousness")
that the sequence has two ends.
Since the sequence has only one _visible_ end,
WM considers that proof of a second end which is _dark_
It is an argument which focuses on sequences.
There is another definition of finiteness/infiniteness
which doesn't immediately call for Dedekind.infinite sets
and which does focus on sequences.
⎛ A finite set has an order which is well.ordered
⎜ in both directions.
⎝ -- Paul Stäckel, 1862...1919
I have been re.deriving familiar (to us) results
for Stäckel.finite and Stäckel.infinite sets, instead of
for Dedekind.finite and Dedekind.infinite sets.
'Stäckel.finite' is messier.
I frame it in terms of Stäckel.finite orders and
Stäckel.infinite orders, but what do we say about
sets which have both?
My answer to that is what you, Random Passerby,
have before you, here.
Lemma.
⎛ A set cannot have both a Stäckel.finite order and
⎝ a Stäckel.infinite order.
That's what I prove, which allows me to me to offer
a clearer (or at least a different) view of
finiteness and infiniteness.

But I'm not sure how one could show that
a relation R on S, where R is Trichotomous,
implies that
I can't 1-1 map a strict subset of R to R.

And I don't show that.
I show that,
if one order of B is Stäckel.finite
and a second order is trichotomous,
then the second order is also Stäckel.finite.
Thus,
because ℕ has a standard order with one end,
there is no Stäckel.finite order of ℕ.
There is no superset of ℕ with a Stäckel.finite order,
either.