On 10/19/2024 12:40 PM, Ross Finlayson wrote:
On 10/19/2024 11:04 AM, Ross Finlayson wrote:
On 10/19/2024 09:04 AM, Jim Burns wrote:
On 10/19/2024 4:16 AM, WM wrote:
On 18.10.2024 00:34, Jim Burns wrote:
On 10/1v7/2024 2:22 PM, WM wrote:
On 17.10.2024 00:39, Jim Burns wrote:
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The only set of natural numbers with no first
is the empty set..
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No, the set of dark numbers is
another set without smallest element.
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A nonempty set without a first element
is not a set of only finite ordinals.
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The set of dark numbers contains
only natural numbers.
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There is a general rule not open to further discussion:
Things which aren't natural numbers
shouldn't be called natural numbers.
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What you call a "set of finite ordinals" is
not a set
but a potentially infinite collection.
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There is a general rule not open to further discussion:
Finite sets aren't potentially infinite collections.
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----
Consider nonempty S of only finite ordinals:
only ordinals with only finitely.many priors.
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k ∈ S is a finite ordinal
Its set ⦃j∈𝕆:j<k⦄ of priors is finite.
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⦃j∈𝕆:j<k⦄∩S ⊆ ⦃j∈𝕆:j<k⦄
⦃j∈𝕆:j<k⦄∩S is a finite set
⦃j∈𝕆:j<k⦄∩S holds its first or is empty.
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⎛ If Priors.in.S ⦃j∈𝕆:j<k⦄∩S is empty
⎝then k is first.in.S
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⎛ If Priors.in.S ⦃j∈𝕆:j<k⦄∩S is not empty
⎜ then i is first.in.⦃j∈𝕆:j<k⦄∩S
⎜
⎜⎛ For i and m ∈ S, i≠m,
⎜⎜ consider set {i,m} of finite ordinals
⎜⎜ {i,m} holds first.in.{i,m}
⎜⎜ i<m ∨ m<i
⎜⎜
⎜⎜ i<m
⎜⎜⎛ Otherwise, m<i and
⎜⎜⎜ m ∈ ⦃j∈𝕆:j<k⦄∩S and
⎜⎝⎝ i isn't first.in.⦃j∈𝕆:j<k⦄∩S
⎜
⎜ for i and m ∈ S, i≤m
⎝ i is first.in.S
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Nonempty S of only finite ordinals
holds first.in.S
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No, the set of dark numbers is
another set without smallest element.
>
A nonempty set without a first element
is not a set of only finite ordinals.
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The set of dark numbers contains
only natural numbers.
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If dark numbers 𝔻 doesn't hold first.in.𝔻
then
either 𝔻 is empty
or 𝔻 isn't only finite ordinals.
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Proof:
If you double all your finite ordinals
you obtain only finite ordinals again,
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Yes.
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although the covered interval is
twice as large as the original interval
covered by "all" your finite ordinals.
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No.
The least.upper.bound of finites is ω
The least.upper.bound of doubled finites is ω
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The washing of dishes is one of those things
where the basic idea is, when it's deemed
necessary to wash a dish, and for some it's
right away and that's a good way of doing things,
that the idea is that once it's put away,
then you don't go hauling it out and washing it
again just for fun.
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What I'm saying is that WM never introduces
anything new so there's no reason to reply,
because, the readership here is already having
the benefit of any needful knowledge about it
otherwise.
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Then though besides where it's like neither of
"countable cardinality" nor "asymptotic density"
need attack nor defense, each being a thing,
then the only amusement is that AP is an abstract
thinker with a langauge like Leonardo in the mirror
though it's broken, so a generous reading has to
be particularly generous and even a contrived sort
of way - then that what possible meaning the
infinite numbers or "the high side" of the integers,
may have, they're not "dark numbers" they're infinite
numbers, then there are simple theories where it's
so that "half the naturals are infinitely-grand each"
or "one of the naturals is infinitely-grand" or
"none of the naturals are infinitely-grand" then
usual Archimedean aspect, and usual enough non-Archimedean.
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I have a job washing dishes one summer when what it
is: is that when one turns 16, then they could get a job,
and it was expected, because it was, so anyways I washed
dishes for a couple months, and got pretty good at it,
I'm a pro. Then I got some computer work, yet, that's
because most anybody should know how to do usual menial
things with acceptable quality like manual/manuel labor.
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There was this one song in the '80's called "On the Dark Side",
it got very heavy radio rotation for sure, one-hit wonder
of a sort.
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The only Google hit for "non-Archimedean integer" is this
Bottazi et alia about Robinsin's useless hyper-reals, ...,
mostly seeming to shill "Easwaran and Towsner, ET", ...
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"Earlier in their text, ET do admit an uncountable number
system for reasons of elegance, so as to be able to defend
the use of R as the basic number system. But their insistence
on trimming the language to countable size does not deliver
the desired disqualification of non-Archimedean systems,
since such countable systems can be constructed that admit
no automorphisms, i.e., are rigid ...."
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So, the "non-standard countable" is a usual thing.
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The only search hit was a cross-mention of Skolem
about non-Archimedean integer, yet apparently
the search results these days are much reduced
and should not be said to reflect more than the cursory,
and indeed reflect a recent dis-accreditation of the corpus.
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>
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Anyways the non-standard countable, like Paris and Kirby,
or Boucher, or various sorts of things that have an
infinitely-grand member naturally in the extra-ordinary
naturals, it's a usual concept like Skolem and Mirimanoff
give, in case you wonder how such concepts of course
have plentiful bibliographic after formal reference.
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The other usual case than:
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half the numbers are infinite
one of the numbers are infinite
none of the numbers are infinite
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is
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most of the numbers are infinite.
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(Almost all of the numbers are infinite.)
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Towsner has this
"That is, the nonstandard analyist [sic] and the
conventional mathematician outright entirely
agree on the definition of continuity, they merely
express the notion in different notation.
(Sanders expands on this point in [Sanders, 2016].)"
That is to say, "Robinson is use-less and since
Dedekind is only one notion of continuity",
that since there are others and at least _three_
definitions of continuity (Aristotle, Eudoxus, Nyquist/Fourier),
thusly talk about anything involved in the
non-Archimedean or non-standard countable
(not the non-standard uncountable or "Robinson's
use-less hyperintegers that actually belong to Skolem"),
that those who claim to be talking about foundations
of mathematics yet only have one definition of
continuity, are simply not actually talking about foundations,
only a closed little fragment where their law of large
numbers is the usual law of small numbers (that what's
so for a few small numbers isn't so for large numbers).
Towsner (or Easwaran, "Realism in Mathematics:
The case of the hyper-reals"):
"Here we agree with Bishop, who remarked that
Keisler’s calculus book using the hyperreals
“offers no evidence that the hyperreal numbers
are anything except a device for proving theorems
about the real numbers” [Bishop, 1977]. Connes
makes a similar point. But unlike Bishop and Connes,
we do not see this as a criticism — when one’s goal
is to prove theorems, devices for doing so are
all to the good. Indeed, this is precisely the
strength of the hyperreals: they are a linguistic
tool for keeping track of “epsilonics”, and quantifiers
more generally, in a cognitively easier way without
fundamentally changing our understanding
of the reals."
See, useless.
Then that there are more REPLETE definitions
of continuity, of course may be of interest to
mathematicians and others interested in
mathematical theory like physicists, and here
particularly of the strongly platonist, strongly
constructivist, strongly logical positivist, and
strongly realist sort, particularly.
How many different unit fractions are lessorequal than all unit fractions?
Re: How many different unit fractions are lessorequal than all unit fractions?