Re: Replacement of Cardinality (infinite middle)

On 08/29/2024 04:32 PM, Jim Burns wrote:

On 8/29/2024 6:46 PM, Ross Finlayson wrote:

On 08/19/2024 12:27 PM, Ross Finlayson wrote:

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

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Definition.
⎛ An order ⟨B,<⟩ of B is finiteᵖᵍˢˢ  iff
⎜ each non.empty subset S ⊆ B holds
⎝ both min[<].S and max[<].S

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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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I don't think they disagree, normally.
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Note: If you mean Dedekind's definition of infinite,
it isn't limited to countably.infinite.
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The entire idea here that there's a particular _regularity_
due dispersion and modularity only courtesy division down
from a fixed-point, that "Peano's axioms" don't give integers,
they only give increments, i.e. not necessarily constant increments,
that there's more than one _regularity_, REQUIRED, is another
little fact of mathematics missing from your neat little hedgerow.

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..., REQUIRED, ....

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Things missing from my neat little hedgerow are
missing because I intend for them to be missing.
My neat little hedgerow has no weeds.
It has not had and will not have weeds.
And weeds would not be an improvement.
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My neat little hedgerow is well.ordered;
each non.empty subset holds a minimum.
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In my neat little hedgerow,
each Little Bunny Foo Foo has a successor,
scooping up the field mice and bopping them on the head,
and is a successor, except the first, named 0.
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Successors are non.0 non.doppelgänger non.final.
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You are welcome to talk about something else, Ross.
Note, though, that,
if you are talking about something else,
then you are talking about something else.
Non.triangles are not counter.examples to triangles.
Non.Bunny.Foo.Foos are not counter.examples to Bunny.Foo.Foos.
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Have a nice day.
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The other day I read or leafed through and enjoyed
this pretty good little book called "Us & Them: The
Science of Identity", by a D. Berreby. Now, I don't
necessarily adhere to any same opinions, yet it's
rather didactic and establishes a sort of discourse
about what is so and considered so and what's not
and considered not.
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Then, the idea that that sort of reflexivity is or isn't
symmetrical, about the usual notions of conservation
and symmetry in this sort of world, is explored as
for matters of Berreby's opinion and lens about
how science that isn't physics or "mathematical",
i.e. that it's "non-logical", at all, isn't science.
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So, for nominalist fictionalists of the formalist
sort, while there may be strong mathematical
platonists who are also formalist constructivists,
it's suggested that a reading of Berreby might
result them being non-logical and fundamentally
as of matters of mere opinion and not of relevance,
here as with respect to the Relephant, since at least
times when flying rainbow sparkle ponies were
putative models of continuous domains or "sets
of reals", and various ones at that.
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Huntington's postulates are mentioned again,
quite all about universals. (A president of the MAA.)
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Peter of Spain's appositve and suppositive and
about use/mention distinction making it so that
"terms" in some "universal particulars" are
REQUIRED their context, helps explain why
theories like universal ordinals for any model
of an integer continuum and the duBois-Reymond
long-line of all real expressions, which has a larger
cardinal than c and is on the same line already,
all make one milieu, and it's logical.
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No-one's trying to take away your triangles,
nor anything else that's mathematical for
that matter, it is though pointed out that
this wider world of a strong mathematical
platonist's universal criteria _always exists_,
basically pointing out that you can't wish that away.
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Often this is mentioned, "that is like the pot,
one of the implements in the fire along with
the kettle, who are both blackened by the fire,
that is like the pot, calling the kettle black,
when indeed the pot and the kettle are both
quite black", yet it's not relevant here, because,
the issue is that for all your reasonable and correct
criticisms of perceived and demonstrated formal
incorrectness according to formal constructions,
then you claim ignorance of "theories with universes",
for example, without which there isn't one, or,
this simple "only diagonal" after you've spent
an entire course establishing why the non-constructive
"anti-diagonal" makes your system of inequalities
giving measure after least-upper-bound (axiomatized)
and measure 1.0 (axiomatized), why the one is so yet
the other with pretty much the exact same form
is not: it demonstrates that a hedgerow without
it would be a mathematical absurdity, and thus
not mathematical.
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Or, you know, trivial, which is acceptable for itself,
a "fragment", of a, "the mathematics", this though
is about a "the mathematics" for _all_ the objects
of the universe of mathematical objects, including
itself.
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For example, at one point it was brought out that
in the theories about relations of triangles, that
sine and cosine and the Pythagorean has another
way to make it, where the Pythagoarean triples
are basically the end result or the completions
instead of the other way around, demonstrating
that lines don't make points and points don't make
lines, according to induction, yet they do, according
to deduction, hence/whence/thence they do.
Then for example the Phythian, more or less
does the same for uniqueness of Fourier-style series,
liberating what are some "uniqueness" results to
"distinctness" results, and making more "repleteness"
of this "completeness", "re-pletion".
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2500 years later, ....
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So anyways, there's basically the "only diagonal" bit
that sets up there's a non-Cartesian function so that
there's a model of a countable continuous domain,
with least-upper-bound and measure according to
there being sigma algebras established, then you
get both or none.
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Of course you needn't _apply_ such definitions,
that for whatever reasons you don't use, there's
though for the mathematically conscientious,
that one established is an _enduring effect_.
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