On 2/24/2023 3:44 AM, WM wrote:
> Jim Burns schrieb am Donnerstag,
> 23. Februar 2023 um 19:54:18 UTC+1:
>> On 2/23/2023 10:16 AM, WM wrote:
>>> Jim Burns schrieb am Mittwoch,
>>> 22. Februar 2023 um 18:12:55 UTC+1:

>>>> Mathematically,
>>>> no matter which neighborhood (-x,x) of 0
>>>> is "near 0"
>>>> almost all unit fractions are in (-x,x)
>>>
>>> Alas
>>> you cannot remove any unit fraction which
>>> has ℵo unit fractions on the right-hand side.
>>> Why?
>>
>> Each unit fraction ends a sequence from 1/1
>> such that, for each split between 1/1 and it,
>> some 1/i,1/i⁺⁺ is last-before,first-after.
>
> That is potential infinity.

That is
not-possibly-false-about a unit fraction.

Augment that with claims which are
visibly not-first-possibly-false
and about a unit fraction,
and you will have more claims which are
not-possibly-false-about a unit fraction.

>> Your description exists, but it describes
>> something which does not exist.
>
> So you deny the existence of
> ℵo real points on the real line.

I deny the existence of unit fractions
which are not unit fractions.

Look!
"unit fractions which are not unit fractions"
I typed it!
That doesn't mean such things exist.

> If they existed at the RHS of 0,
> then they would cover a finite distance,
> part of which would cover
> less than ℵo real points on the real line
> together with their internal distances.

Each distance has the same number (cardinality)
as any other distance. Geometry.

Each unit fraction 1/n has
its own unit fraction (1/mₓ)(1/n) in (-x,x)
mₓ = ⌈1/x⌉

"Each does" means
"It does not" is a description of
something which does not exist.

Jim Burns schrieb am Freitag, 24. Februar 2023 um 14:49:07 UTC+1:
> On 2/24/2023 3:44 AM, WM wrote:

> > So you deny the existence of
> > ℵo real points on the real line.
> I deny the existence of unit fractions
> which are not unit fractions.

They are existing, according to ZFC, at the left-hand side of all definable unit fractions. Hence they cannot be definable unit fractions.

> > If they existed at the RHS of 0,
> > then they would cover a finite distance,
> > part of which would cover
> > less than ℵo real points on the real line
> > together with their internal distances.
> Each distance has the same number (cardinality)
> as any other distance. Geometry.

Not the same length however. An infinite sum of finite dístances has a finite lenth. This length, next to zero, contains unit fractions which have less than this length at their left-hand side.

Regards, WM

On 2/24/2023 5:16 PM, WM wrote:
> Jim Burns schrieb am Freitag,
> 24. Februar 2023 um 14:49:07 UTC+1:
>> On 2/24/2023 3:44 AM, WM wrote:

>>> So you deny the existence of
>>> ℵo real points on the real line.
>>
>> I deny the existence of unit fractions
>> which are not unit fractions.
>
> They are existing, according to ZFC,
> at the left-hand side of
> all definable unit fractions.

No.
You are wrong about what ZFC says.

Define
0 = ∅
n⁺⁺ = n ∪ {n}
S is inductive ⟺
0 ∈ S ∧ ∀n ∈ S: n⁺⁺ ∈ S

ZFC says that
an inductive set exists.

Define
ℕ to be the unique inductive set which is
subset of each inductive set

ZFC says that
ℕ exists.

For ⟨j,k⟩,⟨m,n⟩ ∈ ℕ×ℕ
define
⟨j,k⟩ ≈⁻ ⟨m,n⟩ ⟺
k-j = n-m ⟺
k+m = n+j

Define
ℤ = ℕ×ℕ\≈⁻
the partition of ℕ×ℕ by '≈⁻'

ZFC says that
ℤ exists.

Define
0 = [⟨0,0⟩] ∈ ℤ

For ⟨p,q⟩,⟨r,s⟩ ∈ ℤ×(ℤ\{0})
define
⟨p,q⟩ ≈ᐟ ⟨r,s⟩ ⟺
p/q = r/s ⟺
p*s = r*q

Define
ℚ = ℤ×(ℤ\{0})\=\
the partition of ℤ×(ℤ\{0}) by '≈ᐟ'

ZFC says that
ℚ exists.

Define
B is an open split of ℚ ⟺
B ⊆ ℚ
B ≠ ∅ ≠ ℚ\B
∀p ∈ B, ∀r ∈ ℚ\B: p < r
¬∃q ∈ B, ∀r ∈ Q: q < r ⇒ r ∈ Q\B

Define
ℝ = {B ⊆ ℚ: B is an open split of ℚ}

ZFC says that
ℝ exists.

>>> So you deny the existence of
>>> ℵo real points on the real line.
>>
>> I deny the existence of unit fractions
>> which are not unit fractions.
>
> They are existing, according to ZFC,
> at the left-hand side of
> all definable unit fractions.

No.
ZFC says that
ℝ has the least upper bound property.

ZFC says that
if
any positive point exists at the left hand side
of all definable unit fractions,
then
the set of all such points has a
least upper bound
and
twice that least-upper-bound
is both in and not-in the set of
all such points,
which is a contradiction.

Therefore,
no,
ZFC says that
a positive point ∈ ℝ is to the left of
all definable unit fractions ∈ ℝ
does not exist.

> Hence they cannot be definable unit fractions.

A description can exist of
something which does not exist.

You have described things which are not
definable unit fractions.

Your apparent feeling that
| your description is
| an argument that
|| things which aren't definable unit fractions
|| exist
indicates that
| you (WM) don't understand how any of this works.

A description can exist of
something which does not exist.

Jim Burns schrieb am Samstag, 25. Februar 2023 um 01:44:57 UTC+1:
> On 2/24/2023 5:16 PM, WM wrote:

> > They are existing, according to ZFC,
> > at the left-hand side of
> > all definable unit fractions.
> No.
> You are wrong about what ZFC says.

No.

> ZFC says that
> an inductive set exists.

By induction we prove
∀n ∈ ℕ_ind: |ℕ \ {1, 2, 3, ..., n}| = ℵo.

Hence for every 1/n_ind there are ℵo unit fractions at its left-hand side.
>
> Define
> ℕ to be the unique inductive set which is
> subset of each inductive set
>
> ZFC says that
> ℕ exists.

And it proves that ∀n ∈ ℕ_ind: |ℕ \ {1, 2, 3, ..., n}| = ℵo.
But |ℕ \ {1, 2, 3, ...}| = 0
This proves the existence of dark natnumbers.

> ZFC says that
> ℚ exists

and that by moving X in

XOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO...
XOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO...
XOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO...
XOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO...
XOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO...
XOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO...
XOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO...
XOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO...
XOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO...
XOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO...
XOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO...
XOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO...
XOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO...
XOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO...
XOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO...
XOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO...
XOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO...
XOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO...
XOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO...
XOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO...
XOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO...
XOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO...
XOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO...
XOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO...
XOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO...
XOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO...
XOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO...
XOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO...
XOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO...
XOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO...
XOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO...
XOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO...
XOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO...
XOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO...
XOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO...
XOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO...
XOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO...
XOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO...
XOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO...
XOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO...
XOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO...
XOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO...
XOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO...
XOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO...
XOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO...
XOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO...
XOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO...
XOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO...
XOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO...
XOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO...
....................

the whole matrix can be covered.

> ZFC says that
> ℝ exists.

But that is a diversion. What you denied above is that every descending sequence from omega to zero is finite. There are less natural numbers in a finite sequence than in an infinite sequence. Since all ℵo natnumbers can appear in an ascending sequence, not all natnumbers can appear in a descending sequence.

> ZFC says that
> if
> any positive point exists at the left hand side
> of all definable unit fractions,
> then
> the set of all such points has a
> least upper bound
> and
> twice that least-upper-bound
> is both in and not-in the set of
> all such points,
> which is a contradiction.

It is.
>
> Therefore,
> no,
> ZFC says that
> a positive point ∈ ℝ is to the left of
> all definable unit fractions ∈ ℝ
> does not exist.

No. For all *definable* unit fractions it proves by induction via |ℕ \ {1, 2, 3, ..., n}| = ℵo that ℵo unit fractions are between it and zero.

> > Hence they cannot be definable unit fractions.
> A description can exist of
> something which does not exist.
> You have described things which are not
> definable unit fractions.

But which must exist in actual infinity.
>
> Your apparent feeling that
> | your description is
> | an argument that
> || things which aren't definable unit fractions
> || exist
> indicates that
> | you (WM) don't understand how any of this works.

My proofs are not a feeling. Your disappearing Bob is contradicting mathematics.

Regards, WM

On 2/25/2023 5:28 AM, WM wrote:
> Jim Burns schrieb am Samstag,
> 25. Februar 2023 um 01:44:57 UTC+1:
>> On 2/24/2023 5:16 PM, WM wrote:

>>> They are existing, according to ZFC,
>>> at the left-hand side of
>>> all definable unit fractions.
>>
>> No.
>> You are wrong about what ZFC says.
>
> No.
>
>> ZFC says that
>> an inductive set exists.
>
> By induction we prove
> ∀n ∈ ℕ_ind: |ℕ \ {1, 2, 3, ..., n}| = ℵo.

ZFC says that
ℕ is the unique inductive set which is
subset of each inductive set.
ℕ_ind is inductive. ℕ ⊆ ℕ_ind

Also,
∀n ∈ ℕ⁺: |ℕ⁺\{ 1,2,3,...,n }| = |ℕ⁺| = ℵ₀

because
∀n ∈ ℕ⁺:
∀i,j ∈ ℕ⁺:
i*n⁺⁺ ∈ ℕ⁺\{ 1,2,3,...,n } ∧
j*n⁺⁺ ∈ ℕ⁺\{ 1,2,3,...,n } ∧
i ≠ j ⇒ i*n⁺⁺ ≠ j*n⁺⁺

> Hence for every 1/n_ind
> there are ℵo unit fractions at
> its left-hand side.

>>>>> you cannot remove any unit fraction which
>>>>> has ℵo unit fractions on the right-hand side.

>>> They are existing, according to ZFC,
>>> at the left-hand side of
>>> all definable unit fractions.

No.
ZFC says that
they are in a bounded subset of ℝ
with no least upper bound.

ZFC says that
there are no bounded _non-empty_ subsets of R
with no least upper bound.

ZFC says that
they are in the _empty_ subset of ℝ

Jim Burns schrieb am Samstag, 25. Februar 2023 um 19:06:03 UTC+1:
> On 2/25/2023 5:28 AM, WM wrote:

> ∀n ∈ ℕ⁺: |ℕ⁺\{ 1,2,3,...,n }| = |ℕ⁺| = ℵ₀

Alldefinable natural numbers have ℵo undefined natural successors, ℵo of which will forever remain undefined.

> > Hence for every 1/n_ind
> > there are ℵo unit fractions at
> > its left-hand side.

> No.

Yes. Proof by induction. Since the yall are existing as real points on the real axis, there are some which have same number of unit fractions at their LHS and their RHS. But they are undefinable.

> ZFC says that
> there are no bounded _non-empty_ subsets of R
> with no least upper bound.
>
> ZFC says that
> they are in the _empty_ subset of ℝ

ZFC says a lot of self-contradictory nonsense. But it cannot be defended by simply adhering to this nonsense. Your arguing is titally in vain as long as you claim that Bob could vanish.

Regards, WM