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

On 10/02/2024 11:57 AM, Jim Burns wrote:

On 10/1/2024 6:13 PM, Ross Finlayson wrote:

On 10/01/2024 01:13 PM, Jim Burns wrote:

On 10/1/2024 2:02 PM, Ross Finlayson wrote:

>

Here it's that "Eudoxus/Dedekind/Cauchy is
_insufficient_ to represent the character
of the real numbers".
>
Then, that there are line-reals and signal-reals
besides field-reals, has that of course there are
also models of line-reals and signal-reals in the
mathematics today, like Jordan measure and the ultrafilter,
and many extant examples where a simple deliberation
of continuity according to the definitions of
line-reals or signal-reals, results any contradictions
you might otherwise see as arriving their existence.
>
Then, besides noting how it's broken, then also
there's given a reasoning how it's repaired,
resulting "less insufficient", or at least making
it so that often found approaches in the applied,
and their success, make the standard linear curriculum,
unsuited.
>
Then, I think it's quite standard how I put it,
really very quite standard.

>
I hope this will help me understand you better.
Please accept or reject each claim and
-- this is important --
replace rejected claims with
what you _would_ accept.
>
⎛ ℝ, the complete ordered field, is
⎝ the consensus theory in 2024 of the continuum.
>
⎛ ℝ contains ℚ the rationals and
⎜ the least upper bound of
⎝ each bounded nonempty subset of ℚ and of ℝ
>
( The greatest lower bound of ⅟ℕ unit fractions is 0
>
⎛ A unit fraction is reciprocal to a natural>0
⎜
⎜ A set≠{} ⊆ ℕ naturals holds a minimum
⎜ A natural≠0 has a predecessor.natural.
⎜ A natural has a successor.natural.
⎜
⎜ The sum of two naturals is a natural
⎝ the product of two naturals is a natural.
>
⎛ There are no points in ℝ
⎜ between 0 and all the unit fractions
⎝ (which is what I mean here by 'infinitesimal').
>
Thank you in advance.

>
Well, first of all there's a quibble that
R is not usually said to contain Q as much as that
there's that in real-values that
there's a copy of Q embedded in R.

>
I take your lack of an explicit rejection of
the Dedekind.complete continuum.consensus
to be an implicit acceptance of
the Dedekind.complete continuum.consensus.
>
A quibble for your quibble:
A set isomorphic to ℚ is usually said to _be_ ℚ
Each model of ℚ is ℚ
>
A model of complete.ordered.field ℝ supersets
a model of rational.ordered.field ℚ,
which is to say, ℝ contains ℚ
>
I think that you (RF) are pointing to this:
>
⎛ Consider a model ℚ₀ of the rationals
⎜ which has only urelements.
⎜
⎜ Using ℚ₀  construct
⎜ a model of ℝ complete ordered field
⎜ in any of several known ways:
⎜ a partition of Cauchy sequences of rationals,
⎜ open.foresplits of rationals,
⎜ or something else.
⎜
⎜ The set ℝₛ of open.foresplits of ℚ₀
⎜ { S⊆ℚ₀: {}≠Sᵉᵃᶜʰ<ₑₓᵢₛₜₛSᵉᵃᶜʰ<ᵉᵃᶜʰℚ₀\S≠{} }
⎜ is a model of ℝ
⎜ It holds an open foresplit for each irrational point,
⎜ and an open foresplit for each rational point
⎜
⎜ The set ℚₛ of open.foresplits for rationals
⎜ { {q′∈ℚ₀:q′<q}: q∈ℚ₀ }
⎝ is not ℚ₀
>
Yes,
ℚₛ ≠ ℚ₀
However,
both ℚₛ and ℚ₀ are models of ℚ,
we say both ℚₛ and ℚ₀ we are ℚ,
Each theorem we prove for ℚ,
for example, that no element of ℚ is √2,
is true of both ℚₛ and ℚ₀,
and that's enough for (consensus) us.
>

The, "1/N unit fractions", what is that,
that does not have a definition.

>
Read a bit more and you'll see a definition.
>

Is that some WM-speak?
I suppose that
if it means the set 1/n for n in N
then the g-l-b is zero.

>
Thank you.
My motivation has been to find out if you accept that.
The rest is to make sure we're talking about
the same things.
>
Because
g.l.b of ⅟ℕ (⅟n for n in ℕ)  is 0
there is no positive lower bound of ⅟ℕ
>
A point between 0 and ⅟ℕ would be
a positive lower bound of ⅟ℕ
Such a point doesn't exist.
>
When I say infinitesimals don't exist,
I mean points between 0 and set ⅟ℕ
in the complete.ordered.field
don't exist.
>
When I say that, and then you name.check
various other systems which have infinitesimals,
it _sounds to me_ as though
you object to my claim.
All of this has been my attempt to sort out
_what you're saying_
>

Then otherwise what you have there appear facts
about N and R.

>
They're facts which identify ℕ and ℝ from among
a host of other possible things.called ℕ or ℝ
I take your lack of an explicit rejection
to be an implicit acceptance, and
I take you and I to be talking about
the same ℕ and the same ℝ
>

Then,
where there exists a well-ordering of R,
then to take the well-ordering it results that
first there's a well-ordering of [0,1]
for both simplicity and necessity,

>
Yes.
Note that a well.order of ℝ
is not the usual order of ℝ
>
However,
well.orders being well.orders,
a well.order for ℝ is
a well.order for each subset of ℝ,
including [0,1]
and is also not the usual order of [0,1]
>

and it's as the range of the function n/d
with 0 <= n < d and as d -> oo,
i.e., only in the infinite limit,
that the properties of the range of naturals,
apply to the properties of the range of [0,1].

>
No.
>
lim.infᵈᐧᐧᐧ⟨0/d,1/d,...,d/d⟩  ⊆
limᵈᐧᐧᐧ⟨0/d,1/d,...,d/d⟩  ⊆
lim.supᵈᐧᐧᐧ⟨0/d,1/d,...,d/d⟩
>
⋃ᵈᐧᐧᐧ⋂ᵈᑉᵐᐧᐧᐧ⟨0/m,1/m,...,m/m⟩  ⊆
limᵈᐧᐧᐧ⟨0/d,1/d,...,d/d⟩  ⊆
⋂ᵈᐧᐧᐧ⋃ᵈᑉᵐᐧᐧᐧ⟨0/m,1/m,...,m/m⟩
>
⋃ᵈᐧᐧᐧ{0,1}  ⊆
limᵈᐧᐧᐧ⟨0/d,1/d,...,d/d⟩  ⊆
⋂ᵈᐧᐧᐧ[0,1]∩ℚ
>
{0,1}  ⊆
limᵈᐧᐧᐧ⟨0/d,1/d,...,d/d⟩  ⊆
[0,1]∩ℚ  ⊉  [0,1]
>
limᵈᐧᐧᐧ⟨0/d,1/d,...,d/d⟩  ≠  [0,1]
>
----
I have been less bewildered by your (RF's) responses
since I have started to think of your posts as
brainstorming exercises with the prompt being
the previous post,
especially its last dozen lines.
https://en.wikipedia.org/wiki/Brainstorming
>
I can't guess if you'd consider that good or bad.
I thought you deserved to know how
what you're sending out
is being received.
Received by me, anyway.
>
>

Such casual extensionality as "these Q's are those Q's"
is certainly usual, it's considered a lax sort of
accommodation that thereafter that "all proofs
reflecting on a model of Q will be as entirely
agnostic to the actual set modeling Q as any
other may do, only fulfilling the role of the
model of Q of representing structurally all the
relations of all the elements of Q with all the
elements of all the elements of other structures
so related, model theory".
Then about how you find properties of the function
_in_ the limit _as_ a limit, it's that: those
properties don't exist at all, that function
doesn't exist at all except _in the limit_. Now,
you're talking about a family of functions that
model this not-a-real-function, in the limit,
yet, they are not it, in the limit.
This way the extent, density, completeness,
and measure, aren't from being the union
of ranges of functions that model it,
they're the infinite integers in it.