Re: How many different unit fractions are lessorequal than all unit fractions? (repleteness)

On 09/25/2024 02:00 PM, Jim Burns wrote:

On 9/25/2024 2:44 PM, Ross Finlayson wrote:

On 09/25/2024 10:11 AM, Jim Burns wrote:

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[...]

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How would you define "atom"
the otherwise "infinitely-divisible"?

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I would proceed by defining
a non.existent definiendum.
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It's not a problem to define
a definiendum which doesn't exist.
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It is a problem to interpret a definition as
a claim that its definiendum exists.
I strongly recommend against doing that.
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In a similar vein, define d to be
a positive lower.bound of finite.unit.fractions.
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The interpretation of the definition of d as
a claim that d exists
makes impossibilities necessary.
Necessary impossibilities are a problem.
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But we can avoid that problem by
_not_ doing that, by
concluding that d does not exist.
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That is most of the argument that,
in the Dedekind.complete line,
there are no infinitesimals
(AKA points between 0 and unit.fractions).
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Defining non.existent objects can be
even better than non.disastrous.
It can be downright useful to do so.
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It's seems quite Aristotlean to be against atomism,
yet, at the same time
it's a very useful theory,
for example, with Democritan chemistry, atomic chemistry,
and stoichiometry.
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This is foundations under consideration here,
not merely "pre-calc".

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I think we don't choose foundations which
choose for us what is to be built on them.
(I think they shouldn't, so, Yay!)
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ℝ is anti-atomic, that is, without infinitesimals.
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And yet, ℝ is very useful for describing solutions to
the hydrogen.atom Hamiltonian ̂H = ̂p²/2m - e²/̂r
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The periodic table and the complete ordered field
seem to connect differently from
the way i which you (RF) think they connect.
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It reminds me of a story, once upon a time about
twenty years ago, me and a friend were hiking into
this wilderness area about fifteen miles off the
trailhead, and as we were hiking in with the provisions
and comestibles got to talking about set theory,
infinitesimals, and measurable character, and he
mentioned "you know I wrote a paper about the
infinitesimals with measurable character in Nelson's
Internal Set Theory, IST, that IST, was co-consistent,
with ZFC, Zermelo-Fraenkel set theory". So, that was
very enjoyable, as it helped reflect that modernly,
there already of course are developments like Nelson's
what have measurable infinitesimals, and that in the
consideration that they're co-consistent with ZFC.
Then, as with regards to those being in one theory,
of course gets involved the extra-ordinary for set theory,
then also this account of the equivalency function, as
with regards to it being central, primary, fundamental,
and elementary.
Then he said "in this next section through this bug-kill,
you'll be happy I brought this DEET".
About foundations, then it seems that reflects on
what Leibniz and others call, "the fundamental
question of meta-physics", then why there's
any one theory at all. (Truth, ....)
Now if you'll excuse me there's a flu coming on.