Re: Replacement of Cardinality (infinite middle)

On 08/19/2024 02:43 PM, Jim Burns wrote:

On 8/19/2024 3:27 PM, Ross Finlayson wrote:

On 08/18/2024 09:56 PM, Jim Burns wrote:

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[...]

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I mean it's a great definition that finite has that
there exists a normal ordering that's a well-ordering

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...in both directions...
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and that all the orderings of the set are well-orderings.

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...in both directions...
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That's a great definition of finite and now it stands
for itself in enduring mathematical definition in defense.

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...for comfortably more than a century.
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Why is it you think that Stackel's definition of finite
and "not Dedekind's definition of countably infinite"
don't agree?

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They mostly agree.
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Given the Axiom of Choice
(let us say, if an inaccessible cardinal exists),
they completely agree.
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My impression from somewhere is that,
if they disagree,
they disagree on some very weird sets.
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https://en.wikipedia.org/wiki/Finite_set
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[...] is another little fact of mathematics
missing from your neat little hedgerow.

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I mark my neat little hedgerow, and
I describe what's true everywhere inside the hedgerow.
That allows me to learn about
what's inside the hedgerow,
even though it's infinite and I am finite.
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I like doing that.
It isn't wrong for me to do that.
I will continue doing that.
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I try to avoid using the word "weird",
it's associated with too much jingoism of
the knee-jerk variety, i.e. Pavlovian or
operant-conditioning the conditioned-responses,
"odd", for example, and for some time if you
recall at some point I declared that the use of
the word "but" was modally unhygienic and that
that is only "yet", so, if yet oddly, there are sets
that are totally regular as being divvied up as
by parts from part theory, or for example for
Brentano for boundaries, totally regular,
yet that for the usual regularity of a set
theory like ZF, "I am least deep", is the opposite.
It is _not_ yet oddly that that is so, that the
same underlying universe of substrate of elements
in relation has to model sets and parts, and that
the rulialities, the regularities, than in the universals
fall out not-ultimately-untrue thus having been at some
point not-first-false, eg where your limit ordinal comes
from as a matter of definition or axiomn if necessary,
yet for those for whom deductive inference is available
not necessarily so contrived, has that now that you've
definitely declared that your hedgerow has an inside
and an outside, that there is an outside, and,
it's its own inside, about complementary duals,
like parts and sets, where the comprehension either
divides or increments, like sets and and classes where
the comprehension either collects or elaborates, and
like numbering and counting where the comprehension
either enumerates as arithmetically or algebraically,
that each one of these pairs shows itself what must
be an entire meta-theory all outside your theory,
then whether you ever enjoy the light of day, as it were.
Each of these insides is finite to itself yet they
are infinite outsides to each other.
Give yourself some credit for having a perfect twin
who simply was inculcated with axioms starting from
the opposite side.
Or, you know, whether or not "complementary duals"
results "complimentary duels".