Very nice typography. Doesn't take this ages to make
such a post? (More time than compute Peano 100!)
Jim Burns schrieb am Dienstag, 17. Januar 2023 um 20:31:32 UTC+1:
> On 1/17/2023 3:54 AM, WM wrote:
> > Therefore an infinite sequence of
> > infinite and inclusion-monotonic [sets]
> > cannot have an empty intersection.
> I changed "elements" to "sets".
> What's the reason for that?
>
> ----
> Assume that 𝓐 = ⟨ 𝐴₁ 𝐴₂ 𝐴₃ ... ⟩
> is an inclusion-monotonic
> infinite sequence of sets
> ∀𝐴ₖ ∈ 𝓐: 𝐴ₖ₊₁ ∈ 𝓐
> 𝐴ₖ ⊇ 𝐴ₖ₊₁ ∧ 𝐴ₖ ≠ 𝐴ₖ₊₁
>
> Define 𝓓 = ⟨ 𝐷₁ 𝐷₂ 𝐷₃ ... ⟩
> to be the infinite sequence of
> set differences of inclusion-monotonic 𝓐
> ∀𝐷ₖ ∈ 𝓓: 𝐷ₖ₊₁ ∈ 𝓓
> 𝐷ₖ = 𝐴ₖ\𝐴ₖ₊₁
>
> By an all-not-first-possibly-false
> finite sequence of claims (not shown here),
> 𝓓 is an each-non-empty, disjoint
> infinite sequence of sets.
> 𝐷ₖ ≠ ∅
> j < k ⟹ 𝐷ⱼ ∩ 𝐷ₖ = ∅
>
> Define 𝓔 = ⟨ 𝐸₁ 𝐸₂ 𝐸₃ ... ⟩
> to be the infinite sequence of
> end unions of set differences 𝓓
> ∀𝐸ᵢ ∈ 𝓔: 𝐸ᵢ₊₁ ∈ 𝓔
> 𝐸ᵢ = ⋃[i ≤ k] 𝐷ₖ
>
> By an all-not-first-possibly-false
> finite sequence of claims (not shown here),
> 𝓔 is an each-infinite, inclusion-monotonic
> infinite sequence of sets.
> |𝐸ᵢ| ≥ |ℕ|
> 𝐸ᵢ ⊇ 𝐸ᵢ₊₁ ∧ 𝐸ᵢ ≠ 𝐸ᵢ₊₁
>
> _The intersection ⋂𝓔 of 𝓔 is empty_
> ⋂𝓔 = ∅
>
> | Assume otherwise.
> | Assume x is in ⋂𝓔
> |
> | x is in each 𝐸ᵢ in 𝓔
> |
> | However,
> | x is in some set difference 𝐷ⱼ
> ||
> || Otherwise
> || x is in no end union of differences.
> || and not in any 𝐸ᵢ not in ⋂𝓔
> |
> | The 𝐷ₖ are disjoint
> | x is in 𝐷ⱼ
> | x is not in any 𝐷ₖ j < k
> | x is not in ⋃[j < k] 𝐷ₖ = 𝐸ⱼ₊₁
> | x is not in each 𝐸ᵢ in 𝓔
> | Contradiction.
>
> Therefore,
> the intersection ⋂𝓔 of 𝓔 is empty.
> > Therefore an infinite sequence of
> > infinite and inclusion-monotonic [sets]
> > cannot have an empty intersection.
> No.
> 𝓔 = ⟨ 𝐸₁ 𝐸₂ 𝐸₃ ... ⟩
> is an infinite inclusion-monotonic sequence
> of infinite sets which has an empty intersection.
> ⋂𝓔 = ∅
>
> ∀𝐸ᵢ ∈ 𝓔: 𝐸ᵢ₊₁ ∈ 𝓔
> 𝐸ᵢ = ⋃[i ≤ k] 𝐷ₖ
> |𝐸ᵢ| ≥ |ℕ|
> 𝐸ᵢ ⊇ 𝐸ᵢ₊₁ ∧ 𝐸ᵢ ≠ 𝐸ᵢ₊₁
>
> ∀𝐷ₖ ∈ 𝓓: 𝐷ₖ₊₁ ∈ 𝓓
> 𝐷ₖ = 𝐴ₖ\𝐴ₖ₊₁
> 𝐷ₖ ≠ ∅
> j < k ⟹ 𝐷ⱼ ∩ 𝐷ₖ = ∅
>
> ∀𝐴ₖ ∈ 𝓐: 𝐴ₖ₊₁ ∈ 𝓐
> 𝐴ₖ ⊇ 𝐴ₖ₊₁ ∧ 𝐴ₖ ≠ 𝐴ₖ₊₁

On 1/17/2023 4:03 PM, Mostowski Collapse wrote:

> Very nice typography.

Thank you.

> Doesn't take this ages to make
> such a post? (More time than compute Peano 100!)

Probably the total time spent is somewhere in
WM's dark numbers.
But I cut and paste _a lot_ from previous posts.
I fuss once, and then re-use many.

Given how repetitive these threads are,
it's not too bad, on average.

I once was very determined to use only ASCII
characters to "imitate" math text.
I wanted to be accessible as universally as possible.

But some of the "imitations" were very obscure.
I dislike giving my reader extra work trying to
guess what I'm saying. I want them to spend their
energy on more important things like splits and
last-befores and first-afters.

I saw other people using special characters here
and there, like 𝝎 or ℵ₀ and the sky didn't fall.

It turns out that Wikipedia has pages and pages
of Unicode characters. Lovely characters!
Multiple fonts! Amharic!!! I fell in love.

Currently, I still want posts to be readable as
universally as possible, but I have re-defined
"as universally as possible" from "in ASCII"
to "in Unicode".

> Jim Burns schrieb am Dienstag,
> 17. Januar 2023 um 20:31:32 UTC+1:
>> On 1/17/2023 3:54 AM, WM wrote:

>>> Therefore an infinite sequence of
>>> infinite and inclusion-monotonic [sets]
>>> cannot have an empty intersection.
>>
>> No.
>> 𝓔 = ⟨ 𝐸₁ 𝐸₂ 𝐸₃ ... ⟩
>> is an infinite inclusion-monotonic sequence
>> of infinite sets which has an empty intersection.
>> ⋂𝓔 = ∅
>>
>> ∀𝐸ᵢ ∈ 𝓔: 𝐸ᵢ₊₁ ∈ 𝓔
>> 𝐸ᵢ = ⋃[i ≤ k] 𝐷ₖ
>> |𝐸ᵢ| ≥ |ℕ|
>> 𝐸ᵢ ⊇ 𝐸ᵢ₊₁ ∧ 𝐸ᵢ ≠ 𝐸ᵢ₊₁
>>
>> ∀𝐷ₖ ∈ 𝓓: 𝐷ₖ₊₁ ∈ 𝓓
>> 𝐷ₖ = 𝐴ₖ\𝐴ₖ₊₁
>> 𝐷ₖ ≠ ∅
>> j < k ⟹ 𝐷ⱼ ∩ 𝐷ₖ = ∅
>>
>> ∀𝐴ₖ ∈ 𝓐: 𝐴ₖ₊₁ ∈ 𝓐
>> 𝐴ₖ ⊇ 𝐴ₖ₊₁ ∧ 𝐴ₖ ≠ 𝐴ₖ₊₁

Jim Burns explained on 1/17/2023 :
> On 1/17/2023 4:03 PM, Mostowski Collapse wrote:
>
>> Very nice typography.
>
> Thank you.
>
>> Doesn't take this ages to make
>> such a post? (More time than compute Peano 100!)
>
> Probably the total time spent is somewhere in
> WM's dark numbers.
> But I cut and paste _a lot_ from previous posts.
> I fuss once, and then re-use many.
>
> Given how repetitive these threads are,
> it's not too bad, on average.
>
> I once was very determined to use only ASCII
> characters to "imitate" math text.
> I wanted to be accessible as universally as possible.
>
> But some of the "imitations" were very obscure.
> I dislike giving my reader extra work trying to
> guess what I'm saying. I want them to spend their
> energy on more important things like splits and
> last-befores and first-afters.
>
> I saw other people using special characters here
> and there, like ? or ℵ₀ and the sky didn't fall.
>
> It turns out that Wikipedia has pages and pages
> of Unicode characters. Lovely characters!
> Multiple fonts! Amharic!!! I fell in love.
>
> Currently, I still want posts to be readable as
> universally as possible, but I have re-defined
> "as universally as possible" from "in ASCII"
> to "in Unicode".
>
>> Jim Burns schrieb am Dienstag,
>> 17. Januar 2023 um 20:31:32 UTC+1:
>>> On 1/17/2023 3:54 AM, WM wrote:
>
>>>> Therefore an infinite sequence of
>>>> infinite and inclusion-monotonic [sets]
>>>> cannot have an empty intersection.
>>>
>>> No.
>>> ? = ⟨ ?₁ ?₂ ?₃ ... ⟩
>>> is an infinite inclusion-monotonic sequence
>>> of infinite sets which has an empty intersection.
>>> ⋂? = ∅
>>>
>>> ∀?ᵢ ∈ ?: ?ᵢ₊₁ ∈ ?
>>> ?ᵢ = ⋃[i ≤ k] ?ₖ
>>> |?ᵢ| ≥ |ℕ|
>>> ?ᵢ ⊇ ?ᵢ₊₁ ∧ ?ᵢ ≠ ?ᵢ₊₁
>>>
>>> ∀?ₖ ∈ ?: ?ₖ₊₁ ∈ ?
>>> ?ₖ = ?ₖ\?ₖ₊₁
>>> ?ₖ ≠ ∅
>>> j < k ⟹ ?ⱼ ∩ ?ₖ = ∅
>>>
>>> ∀?ₖ ∈ ?: ?ₖ₊₁ ∈ ?
>>> ?ₖ ⊇ ?ₖ₊₁ ∧ ?ₖ ≠ ?ₖ₊₁

Here's an example of how it looks to me.

https://ibb.co/QPL7Bdv

William schrieb am Dienstag, 17. Januar 2023 um 17:08:13 UTC+1:

> A Peano set has cardinality ℵo.

No.

> A potentially infinite collection has cardinality less than ℵo. A Peano set is not a potentially infinite collections.

The set of unit fractions 1/n has cardinality ℵo.
No further unit mfractions fits between all unit fractions and zero.

There are less Peano unit fractions because for every Peano unit fraction ℵo smaller unit fractions fit between it and zero.

Regards, WM

Jim Burns schrieb am Dienstag, 17. Januar 2023 um 19:02:24 UTC+1:
> On 1/17/2023 3:54 AM, WM wrote:
>
> > I assert that mathematics and logic are true.
> If 𝓅 and 𝓅⇒𝓆 precede 𝓆,
> then 𝓆 is not first possibly-false.
>
> If each claim in a sequence is
> not first possibly-false,
> then each claim in that sequence is
> not possibly-false.
>
> When I say
> | Mathematics and logic are true
> I mean that claims like those are true.
>
> I also assert that
> mathematics and logic are true
> Do you disagree with what I mean?

Your chain of claims does not reach until infinity. But Cantors sets do.
The set of unit fractions 1/n has cardinality ℵo.
No further unit fraction fits between all unit fractions and zero.
That means: reaching until zero. Therefore the natnumbers reach until infinity.

There are less unit fractions covered by n ==> n+1 because for every such unit fraction ℵo smaller unit fractions fit between itself and zero.

Therefore your If 𝓅 and 𝓅⇒𝓆 approach is not always true. It is true for every number having a FISON.

Note: "classical logic was abstracted from the mathematics of finite sets and their subsets. (The word finite is here to be taken in the precise sense that the members of such set are explicitly exhibited one by one.) Forgetful of this limited origin, one afterwards mistook that logic for something above and prior to all mathematics, and finally applied it, without justification, to the mathematics of infinite sets. This is the Fall and original sin of set-theory, for which it is justly punished by the antinomies. Not that such contradictions showed up is surprising, but that they showed up at such a late stage of the game! [H. Weyl: "Levels of infinity: Selected writings on mathematics and philosophy", Peter Pesic (ed.), Dover Publications (2012) p. 140f]

Regards, WM

FromTheRafters schrieb am Dienstag, 17. Januar 2023 um 20:29:17 UTC+1:
> WM formulated the question :
> > FromTheRafters schrieb am Dienstag, 17. Januar 2023 um 11:45:08 UTC+1:
> >> WM was thinking very hard :
> >
> >>> I know that Peano-numbers have FISONs
> >> How do you know, did you check them step-by-step?
> >
> > By definition 1 is a Peano number. Its successor is a Peano number. Its
> > successor is a Peano number. Its successor is a Peano number. And so on..
> > Therefore they all have FISONs.
> Oh, so you accept some things as infinite readily, and others
> reluctantly if at all in a step-by-step manner.

In a step-by-step manner we cannot reach until infinity. But Cantors sets do.
The set of unit fractions 1/n has cardinality ℵo.
No further unit fraction fits between all unit fractions and zero.
That means: reaching until zero. Therefore the natnumbers reach until infinity.

There are less unit fractions covered by n ==> n+1 because for every such unit fraction ℵo smaller unit fractions fit between itself and zero.

Note: "classical logic was abstracted from the mathematics of finite sets and their subsets. (The word finite is here to be taken in the precise sense that the members of such set are explicitly exhibited one by one.) Forgetful of this limited origin, one afterwards mistook that logic for something above and prior to all mathematics, and finally applied it, without justification, to the mathematics of infinite sets. This is the Fall and original sin of set-theory, for which it is justly punished by the antinomies. Not that such contradictions showed up is surprising, but that they showed up at such a late stage of the game! [H. Weyl: "Levels of infinity: Selected writings on mathematics and philosophy", Peter Pesic (ed.), Dover Publications (2012) p. 140f]

Regards, WM

Jim Burns schrieb am Dienstag, 17. Januar 2023 um 20:31:32 UTC+1:
> On 1/17/2023 3:54 AM, WM wrote:
> > Therefore an infinite sequence of
> > infinite and inclusion-monotonic [sets]
> > cannot have an empty intersection.
> I changed "elements" to "sets".
> What's the reason for that?

The reason is that all elements in an endsegment are there from the beginning. As long as there remains one element, the intersection of this endsegment and all its predecessors is not empty.

Regards, WM

WM formulated the question :
> William schrieb am Dienstag, 17. Januar 2023 um 17:08:13 UTC+1:
>
>> A Peano set has cardinality ℵo.
>
> No.
>
>> A potentially infinite collection has cardinality less than ℵo. A Peano set
>> is not a potentially infinite collections.
>
> The set of unit fractions 1/n has cardinality ℵo.
> No further unit mfractions fits between all unit fractions and zero.
>
> There are less Peano unit fractions because for every Peano unit fraction ℵo
> smaller unit fractions fit between it and zero.

But always only countably more.

On 1/18/2023 4:23 AM, WM wrote:
> Jim Burns schrieb am Dienstag, 17. Januar 2023 um 20:31:32 UTC+1:
>> On 1/17/2023 3:54 AM, WM wrote:
>>> Therefore an infinite sequence of
>>> infinite and inclusion-monotonic [sets]
>>> cannot have an empty intersection.
>> I changed "elements" to "sets".
>> What's the reason for that?
>
> The reason is that all elements in an endsegment are there from the beginning.
>
> Regards, WM
>

sets do not change.

On Wednesday, January 18, 2023 at 6:07:17 AM UTC-4, WM wrote:
> William schrieb am Dienstag, 17. Januar 2023 um 17:08:13 UTC+1:
>
> > A Peano set has cardinality ℵo.
> No.

A Peano set does not have finite cardinality. A Peano set does not change so it cannot be finite but bounded. A Peano has cardinality ℵo.

On Tuesday, January 17, 2023 at 3:21:40 PM UTC-5, Fritz Feldhase wrote:
> On Tuesday, January 17, 2023 at 4:27:26 PM UTC+1, timba...@gmail.com wrote:
> .
> > Claiming that large values are readily workable in radix-one is problematic.
> > Nobody here will even touch the value 4000.
> Oh, really?!
>
> ssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssss
> ssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssss
> ssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssss
> ssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssss
> ssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssss
> ssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssss
> ssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssss
> ssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssss
> ssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssss
> ssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssss
> ssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssss
> ssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssss
> ssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssss
> ssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssss
> ssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssss
> ssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssss
> ssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssss
> ssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssss
> ssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssss
> ssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssss
> ssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssss
> ssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssss
> ssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssss
> ssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssss
> ssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssss
> ssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssss
> ssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssss
> ssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssss
> ssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssss
> ssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssss
> ssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssss
> ssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssss
> ssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssss
> ssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssss
> ssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssss
> ssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssss
> ssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssss
> ssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssss
> ssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssss
> ssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssss0
>
> Actually, not that hard.

Actually you've just verified an interesting side effect of working modulo one.There is a need for numerical punctuation.
If a teacher has asked you to show your work, so as to make their life easier on this problem there would be even more notation.
The futility of working large numbers through natural analysis I hope is being felt here.
That some argument should hinge on this as the proper medium to approaching infinity is as clumsy as it gets.
Ahh, possibly there lays a thought: the number as a medium rather than as a material. Quite so.

On 1/18/2023 3:18 AM, FromTheRafters wrote:
> Jim Burns explained on 1/17/2023 :
>> On 1/17/2023 4:03 PM, Mostowski Collapse wrote:

>>> Very nice typography.

>>>> ? = ⟨ ?₁ ?₂ ?₃ ... ⟩
>>>> is an infinite inclusion-monotonic sequence
>>>> of infinite sets which has an empty intersection.
>>>> ⋂? = ∅
>>>>
>>>> ∀?ᵢ ∈ ?: ?ᵢ₊₁ ∈ ?
>>>> ?ᵢ = ⋃[i ≤ k] ?ₖ
>>>> |?ᵢ| ≥ |ℕ|
>>>> ?ᵢ ⊇ ?ᵢ₊₁ ∧ ?ᵢ ≠ ?ᵢ₊₁
>>>>
>>>> ∀?ₖ ∈ ?: ?ₖ₊₁ ∈ ?
>>>> ?ₖ = ?ₖ\?ₖ₊₁
>>>> ?ₖ ≠ ∅
>>>> j < k ⟹ ?ⱼ ∩ ?ₖ = ∅
>>>>
>>>> ∀?ₖ ∈ ?: ?ₖ₊₁ ∈ ?
>>>> ?ₖ ⊇ ?ₖ₊₁ ∧ ?ₖ ≠ ?ₖ₊₁
>
> Here's an example of how it looks to me.
>
> https://ibb.co/QPL7Bdv

That's terrible.
It is exactly what I want to avoid.

I've replaced all the '?' with ASCII below,
and it doesn't look _too_ bad, but
I really liked signaling elements, sets,
sets of sets with different fonts.

To say nothing of function names,
propositional variables, transfinite ordinals,
matrices, the occasional dark number, and
formulas with less-common characters, like
P+q = ⌈(((8n+1)¹ᐟ²+1)/2)⌉
Does that get the '?' treatment?

Superscript-slash is in the Unified Canadian
Aboriginal Syllabics block.
| Anything worth doing is worth overdoing.
-- me

I suppose I could forego describing the location
of Bob, the O from 1/2, in Amharic, but, oh!
It is so adorably alien-looking and yet
translatable in Google.

With "my" fonts,
I can show that a FISON F is in
the set of all FISONs F without it looking
as though F and F are the same thing.
In a more reasonable exchange, that
would matter less. In this one, it's huge.

I would really like you, Mr Rafters, to at least
read my posts. I mentally place you firmly in
the Sanity Caucus, which grows smaller with
each passing year.

I don't want to go all the way back to all-ASCII
and it looks as though you can see many non-ASCII
characters, so that would serve no one, anyway.

It doesn't seem practical for me to check
other people's software for '?' in its rendering.

Could you please quote back to me '?'ified
posts. I will do my best to clarify, on a
case-by-case basis.

Perhaps, at some future point in time, all this
will be unnecessary.
| Technology is stuff that doesn't work _yet_
-- multiple sources (emphasis added)

----
EE = ⟨ E₁ E₂ E₃ ... ⟩
is an infinite inclusion-monotonic sequence
of infinite sets which has an empty intersection.
⋂EE = ∅

∀Eᵢ ∈ EE: Eᵢ₊₁ ∈ EE
Eᵢ = ⋃[i ≤ k] Dₖ
|Eᵢ| ≥ |ℕ|
Eᵢ ⊇ Eᵢ₊₁ ∧ Eᵢ ≠ Eᵢ₊₁

DD = ⟨ D₁ D₂ D₃ ... ⟩
∀Dₖ ∈ DD: Dₖ₊₁ ∈ DD
Dₖ = Aₖ\Aₖ₊₁
Dₖ ≠ ∅
j < k ⟹ Dⱼ ∩ Dₖ = ∅

AA = ⟨ A₁ A₂ A₃ ... ⟩
∀Aₖ ∈ AA: Aₖ₊₁ ∈ AA
Aₖ ⊇ Aₖ₊₁ ∧ Aₖ ≠ Aₖ₊₁

On 1/18/2023 5:16 AM, WM wrote:
> Jim Burns schrieb am Dienstag,
> 17. Januar 2023 um 19:02:24 UTC+1:
>> On 1/17/2023 3:54 AM, WM wrote:

>>> I assert that
>>> mathematics and logic are true.
>>
>> If 𝓅 and 𝓅⇒𝓆 precede 𝓆,
>> then 𝓆 is not first possibly-false.
>>
>> If each claim in a sequence is
>> not first possibly-false,
>> then each claim in that sequence is
>> not possibly-false.
>>
>> When I say
>> | Mathematics and logic are true
>> I mean that claims like those are true.
>>
>> I also assert that
>> mathematics and logic are true
>> Do you disagree with what I mean?
>
> Your chain of claims does not reach
> until infinity. But Cantors sets do.

You are saying that you disagree with
mathematics and logic
without saying that you disagree with
mathematics and logic.

> Your chain of claims does not reach
> until infinity.

No, that's wrong.
The opposite of that
is a short summary of several years of
my posts to you.

In natural language,
we can make claims which we know are true,
even while we don't know which one of the
possible referents is actually referred to.

| A duck-billed platypus has webbed feet.
| I know it's true.
_Which_ duck-billed platypus?
Which doesn't matter, I know it's true.

| A natural-number successor
| is not 0
| is not the successor of anything else
| and has a successor.
| I know it's true.
_Which_ natural-number successor?
Which doesn't matter, I know it's true.

The first claim reaches all duck-billed
platypuses. Correctly or not, I am claiming
that all duck-billed platypuses have webbed feet.

The second claim reaches all natural-number
successors. Correctly or not, I am claiming
that all natural-number successors are not 0,
are not a successor of anything else, and
has a successor.

> Your chain of claims does not reach
> until infinity.

No,
elsewhere, we see that there are infinitely-many
natural-number successors. There is no natural-
-number successor which my claim does not reach.

There is "all" and there is "more than all".
You (WM) tend to read claims as though they are
about more than all.
I have no responsibility for more-than-all
readings of a claim of mine.

I am not claiming that natural-number successors
have webbed feet.
I am not claiming that _anything other than_
a natural-number successor is not 0, is not
a successor of anything else, and has a successor.

Suppose that you (WM) prove that there are
natural-number successors without webbed feet.
You (WM) have not contradicted _my_ claim about
duck-billed platypuses, only _your own_
more-than-all straw-man version of my claim.

> Therefore your If p and p⇒q approach
> is not always true.

If p and p⇒q precede q
then
p is possibly-false xor necessarily-true
p⇒q is possibly-false xor necessarily-true
q is possibly-false xor necessarily-true

Is that not always true?

If p and p⇒q precede q
and one of p or p⇒q is possibly-false,
then
q is not first possibly-false.

Is that not always true?

If both of p and q are necessarily-true,
then,
because
p q p⇒q
1 1 1 ⟵
0 1 1
1 0 0
0 0 1
q is necessarily-true

If q is first possibly-false
then
q is possibly-false and not necessarily-true.

Is that not always true?

If p and p⇒q precede q
then
q is not first possibly-false.

Is that not always true?

If some claim in a sequence is possibly-false
then
some claim is first-possibly-flase.

Is that not always true?

> It is true for every number having a FISON.

When only numbers having a FISON
*might be a referent* of the claims on the table,
it is always true *for those claims*

Other claims about more-than-all are
not my claims, and not my problem.

> Note:
> "classical logic was abstracted from the
> mathematics of finite sets and their subsets.
> ...
> [H. Weyl: ... ]

In a _finite_ sequence of claims in which
each claim
either is
true of each one of infinitely-many real numbers
and thus not possibly-false about real numbers
or is
preceded by claims whereby we can _see_
that it is a not-first-possibly-false claim,

we can _see_ that
the claims in that sequence are
all-not-possibly-false, are all-necessarily-true
about infinitely-many real numbers.

That requires that
a finite sequence of claims be finite
and that
each real number not be something other than
_what we mean_ by "real number",
in other words, that there isn't "more than all".

Is that not always true?

Jim Burns <james.g.burns@att.net> writes:

> On 1/17/2023 4:03 PM, Mostowski Collapse wrote:
>
>> Very nice typography.
>
> Thank you.
>
>> Doesn't take this ages to make
>> such a post? (More time than compute Peano 100!)
>
> Probably the total time spent is somewhere in
> WM's dark numbers.
> But I cut and paste _a lot_ from previous posts.

I can't tell if you use Windows or something else, but I use X11's
compose key function. I make up my own memorable compose sequences for
the symbols I tend to use. ∀ is <compose> <f> <a> for example.

> I fuss once, and then re-use many.

My fussing is updating my .XCompose file. I've just added a few from
your post since I can see myself using them in my replies.

Of course, cutting and pasting whole texts is still faster for these
threads since the same things need to be said ℵ₀ (<compose> <a> <0>)
times. (I give this example because it shows that the result can be
multiple gylphs.)

My main usability issue with all this is that my newsreader seems to
want extra line spacing for the symbols that come from other fonts. The
∀ is not any taller than A, but the font used to realise it obviously
has different line-height spacing rules.

BTW, all your posts display perfectly well in Gnus (the Emacs-based
reader I use).

--
Ben.

On 1/18/2023 5:16 AM, WM wrote:

> Note:
> "classical logic was abstracted from the
> mathematics of finite sets and their subsets.
> (The word finite is here to be taken in the
> precise sense that the members of such set
> are explicitly exhibited one by one.)
> Forgetful of this limited origin, one afterwards
> mistook that logic for something above and
> prior to all mathematics, and finally applied it,
> without justification, to the mathematics of
> infinite sets. This is the Fall and original sin
> of set-theory, for which it is justly punished
> by the antinomies. Not that such contradictions
> showed up is surprising, but that they showed up
> at such a late stage of the game!
> [H. Weyl: "Levels of infinity: Selected writings
> on mathematics and philosophy", Peter Pesic (ed.),
> Dover Publications (2012) p. 140f]

By "classical logic", I mean the assembling of
sequences of claims, each claim of which can be
seen to be not the first of any possibly-false
claims. I hope I'm not too far from what
Hermann Weyl means.

The claims in these sequences can be about
either the finite or the infinite. Sinfully!

However,
prior to reasoning about the infinite,
we meta-reason about claims about the infinite.
We first treat claims themselves as
referents of meta-claims.

For example,
_why_ do we assemble sequences of claims,
no first claim of which is possibly false?

Because, in a totally-ordered finite set
-- of anything -- if any element has a property
-- any property -- then some element is first
to have that property.

_Why_ is claim q preceded by claims p and p⇒q
not the first of those possibly-false (if any)?

List out all the cases, if you like.
There aren't many.
In each case, q is either not first or
not possibly-false.

Whatever subtleties are involved elsewhere,
the meta-reasoning for our classical logic
is decidedly unsubtle.
Our meta-reasoning treats claims the way that
our reasoning treats sheep and pebbles.
Our logic has not seen the Fall.

onsdag 18 januari 2023 kl. 11:19:35 UTC+1 skrev WM:
> FromTheRafters schrieb am Dienstag, 17. Januar 2023 um 20:29:17 UTC+1:
> > WM formulated the question :
> > > FromTheRafters schrieb am Dienstag, 17. Januar 2023 um 11:45:08 UTC+1:
> > >> WM was thinking very hard :
> > >
> > >>> I know that Peano-numbers have FISONs
> > >> How do you know, did you check them step-by-step?
> > >
> > > By definition 1 is a Peano number. Its successor is a Peano number. Its
> > > successor is a Peano number. Its successor is a Peano number. And so on.
> > > Therefore they all have FISONs.
> > Oh, so you accept some things as infinite readily, and others
> > reluctantly if at all in a step-by-step manner.
> In a step-by-step manner we cannot reach until infinity. But Cantors sets do.
> The set of unit fractions 1/n has cardinality ℵo.
> No further unit fraction fits between all unit fractions and zero.
> That means: reaching until zero. Therefore the natnumbers reach until infinity.
>
> There are less unit fractions covered by n ==> n+1 because for every such unit fraction ℵo smaller unit fractions fit between itself and zero.
> Note: "classical logic was abstracted from the mathematics of finite sets and their subsets. (The word finite is here to be taken in the precise sense that the members of such set are explicitly exhibited one by one.) Forgetful of this limited origin, one afterwards mistook that logic for something above and prior to all mathematics, and finally applied it, without justification, to the mathematics of infinite sets. This is the Fall and original sin of set-theory, for which it is justly punished by the antinomies. Not that such contradictions showed up is surprising, but that they showed up at such a late stage of the game! [H. Weyl: "Levels of infinity: Selected writings on mathematics and philosophy", Peter Pesic (ed.), Dover Publications (2012) p. 140f]
>
> Regards, WM
do you understand anything?

Jim Burns formulated the question :
> On 1/18/2023 3:18 AM, FromTheRafters wrote:
>> Jim Burns explained on 1/17/2023 :
>>> On 1/17/2023 4:03 PM, Mostowski Collapse wrote:
>
>>>> Very nice typography.
>
>>>>> ? = ⟨ ?₁ ?₂ ?₃ ... ⟩
>>>>> is an infinite inclusion-monotonic sequence
>>>>> of infinite sets which has an empty intersection.
>>>>> ⋂? = ∅
>>>>>
>>>>> ∀?ᵢ ∈ ?: ?ᵢ₊₁ ∈ ?
>>>>> ?ᵢ = ⋃[i ≤ k] ?ₖ
>>>>> |?ᵢ| ≥ |ℕ|
>>>>> ?ᵢ ⊇ ?ᵢ₊₁ ∧ ?ᵢ ≠ ?ᵢ₊₁
>>>>>
>>>>> ∀?ₖ ∈ ?: ?ₖ₊₁ ∈ ?
>>>>> ?ₖ = ?ₖ\?ₖ₊₁
>>>>> ?ₖ ≠ ∅
>>>>> j < k ⟹ ?ⱼ ∩ ?ₖ = ∅
>>>>>
>>>>> ∀?ₖ ∈ ?: ?ₖ₊₁ ∈ ?
>>>>> ?ₖ ⊇ ?ₖ₊₁ ∧ ?ₖ ≠ ?ₖ₊₁
>>
>> Here's an example of how it looks to me.
>>
>> https://ibb.co/QPL7Bdv
>
> That's terrible.
> It is exactly what I want to avoid.
>
> I've replaced all the '?' with ASCII below,
> and it doesn't look _too_ bad, but
> I really liked signaling elements, sets,
> sets of sets with different fonts.
>
> To say nothing of function names,
> propositional variables, transfinite ordinals,
> matrices, the occasional dark number, and
> formulas with less-common characters, like
> P+q = ⌈(((8n+1)¹ᐟ²+1)/2)⌉
> Does that get the '?' treatment?
>
> Superscript-slash is in the Unified Canadian
> Aboriginal Syllabics block.
> | Anything worth doing is worth overdoing.
> -- me
>
> I suppose I could forego describing the location
> of Bob, the O from 1/2, in Amharic, but, oh!
> It is so adorably alien-looking and yet
> translatable in Google.
>
> With "my" fonts,
> I can show that a FISON F is in
> the set of all FISONs F without it looking
> as though F and F are the same thing.
> In a more reasonable exchange, that
> would matter less. In this one, it's huge.
>
>
> I would really like you, Mr Rafters, to at least
> read my posts. I mentally place you firmly in
> the Sanity Caucus, which grows smaller with
> each passing year.
>
> I don't want to go all the way back to all-ASCII
> and it looks as though you can see many non-ASCII
> characters, so that would serve no one, anyway.
>
> It doesn't seem practical for me to check
> other people's software for '?' in its rendering.
>
> Could you please quote back to me '?'ified
> posts. I will do my best to clarify, on a
> case-by-case basis.
>
> Perhaps, at some future point in time, all this
> will be unnecessary.
> | Technology is stuff that doesn't work _yet_
> -- multiple sources (emphasis added)
>
> ----
> EE = ⟨ E₁ E₂ E₃ ... ⟩
> is an infinite inclusion-monotonic sequence
> of infinite sets which has an empty intersection.
> ⋂EE = ∅
>
> ∀Eᵢ ∈ EE: Eᵢ₊₁ ∈ EE
> Eᵢ = ⋃[i ≤ k] Dₖ
> |Eᵢ| ≥ |ℕ|
> Eᵢ ⊇ Eᵢ₊₁ ∧ Eᵢ ≠ Eᵢ₊₁
>
> DD = ⟨ D₁ D₂ D₃ ... ⟩
> ∀Dₖ ∈ DD: Dₖ₊₁ ∈ DD
> Dₖ = Aₖ\Aₖ₊₁
> Dₖ ≠ ∅
> j < k ⟹ Dⱼ ∩ Dₖ = ∅
>
> AA = ⟨ A₁ A₂ A₃ ... ⟩
> ∀Aₖ ∈ AA: Aₖ₊₁ ∈ AA
> Aₖ ⊇ Aₖ₊₁ ∧ Aₖ ≠ Aₖ₊₁

Mostly, I understand what you are trying to show, so I can guess pretty
well what some of the mangled symbols are supposed to represent. My
view of the post sometimes has little squares or question marks, and
gets even worse when I attempt to reply because the editor pane uses a
different character set with a fixed width font and further mangles the
symbols.

My reader, MesNews, is currently using utf-8 and your headers indicate
that you are also. This might be a fault in my reader, but I like it
anyway.

William schrieb am Mittwoch, 18. Januar 2023 um 15:53:39 UTC+1:
> On Wednesday, January 18, 2023 at 6:07:17 AM UTC-4, WM wrote:
> > William schrieb am Dienstag, 17. Januar 2023 um 17:08:13 UTC+1:
> >
> > > A Peano set has cardinality ℵo.
> > No.
> A Peano set does not have finite cardinality. A Peano set does not change so it cannot be finite but bounded.

A Peano set is not a set (in the meaning of ZFC).

Please try to comprehend the following text, perhaps in a moment where your matheologial defence meachanism is sleeping.

The set of unit fractions 1/n has cardinality ℵo.
No further unit fraction fits between all unit fractions and zero.

There are less Peano unit fractions because for every Peano unit fraction ℵo smaller unit fractions fit between it and zero.

Regards, WM

Jim Burns schrieb am Mittwoch, 18. Januar 2023 um 23:37:28 UTC+1:
> On 1/18/2023 5:16 AM, WM wrote:
>
> > Your chain of claims does not reach
> > until infinity. But Cantors sets do.
> You are saying that you disagree with
> mathematics and logic
> without saying that you disagree with
> mathematics and logic.

I adhere to logic. You adhere to nonsense called mathematical logic. "'Mathematical logic' has completely deformed the thinking of mathematicians and of philosophers" [L. Wittgenstein: "Remarks on the foundations of mathematics", Wiley-Blackwell (1991)]

> > Your chain of claims does not reach
> > until infinity.
> No, that's wrong.

You confuse the infinities.

The set of unit fractions 1/n has cardinality ℵo.
No further unit fraction fits between all unit fractions and zero.

There are less Peano unit fractions because for every Peano unit fraction ℵo smaller unit fractions fit between it and zero.

That is a difference, a huge difference, an infinite difference.

> In natural language,
> we can make claims which we know are true,
> even while we don't know which one of the
> possible referents is actually referred to.

But every referent that you can choose belongs to Peano infinity.

> | A natural-number successor
> | is not 0
> | is not the successor of anything else
> | and has a successor.
> |
> I know it's true.
> _Which_ natural-number successor?
> Which doesn't matter, I know it's true.

And yiu forget that each one has more successors than predecessors. But you claim that your proof covers all. That is your big mistake.
>
> There is "all" and there is "more than all".
> You (WM) tend to read claims as though they are
> about more than all.

More than all that are covered by your induction.

> I have no responsibility for more-than-all
> readings of a claim of mine.

You claim that all endegments are infinite. That is true as far as induction reaches. But it is not true that their intersection is empty. You conclude that only from the fact that you cannot see the dark numbers. Thus you are trapped in the contradiction that infinite inclusion-monotonic sets could have an empty intersection. Blatant nonsense.
>
> in other words, that there isn't "more than all".
>
> Is that not always true?

See the set of unit fractions above. Try to understand it.

Regards, WM

Jim Burns schrieb am Donnerstag, 19. Januar 2023 um 02:59:41 UTC+1:

> Our meta-reasoning treats claims the way that
> our reasoning treats sheep and pebbles.
> Our logic has not seen the Fall.

Can it distinguish this?
No further unit fraction fits between all unit fractions and zero.
For every Peano unit fraction ℵo smaller unit fractions fit between it and zero.

Regards, WM

After serious thinking WM wrote :
> Jim Burns schrieb am Mittwoch, 18. Januar 2023 um 23:37:28 UTC+1:
>> On 1/18/2023 5:16 AM, WM wrote:
>>
>>> Your chain of claims does not reach
>>> until infinity. But Cantors sets do.
>> You are saying that you disagree with
>> mathematics and logic
>> without saying that you disagree with
>> mathematics and logic.
>
> I adhere to logic. You adhere to nonsense called mathematical logic.
> "'Mathematical logic' has completely deformed the thinking of mathematicians
> and of philosophers" [L. Wittgenstein: "Remarks on the foundations of
> mathematics", Wiley-Blackwell (1991)]
>
>>> Your chain of claims does not reach
>>> until infinity.
>> No, that's wrong.
>
> You confuse the infinities.
>
> The set of unit fractions 1/n has cardinality ℵo.

Sure, inversion is monotonic.

Explain how 'fitting between it and zero' works when there is no
guarantee of convergence.

On Thursday, January 19, 2023 at 6:22:58 AM UTC-4, WM wrote:
> William schrieb am Mittwoch, 18. Januar 2023 um 15:53:39 UTC+1:
> > On Wednesday, January 18, 2023 at 6:07:17 AM UTC-4, WM wrote:
> > > William schrieb am Dienstag, 17. Januar 2023 um 17:08:13 UTC+1:
> > >
> > > > A Peano set has cardinality ℵo.
> > > No.
> > A Peano set does not have finite cardinality. A Peano set does not change so it cannot be finite but bounded.
> A Peano set is not a set (in the meaning of ZFC).

Piffle. Of course it is.

On 1/19/2023 4:22 AM, WM wrote:
> William schrieb am Mittwoch, 18. Januar 2023 um 15:53:39 UTC+1:
>> On Wednesday, January 18, 2023 at 6:07:17 AM UTC-4, WM wrote:
>>> William schrieb am Dienstag, 17. Januar 2023 um 17:08:13 UTC+1:
>>>
>>>> A Peano set has cardinality ℵo.
>>> No.
>> A Peano set does not have finite cardinality. A Peano set does not change so it cannot be finite but bounded.
>
> A Peano set is not a set (in the meaning of ZFC).
>
> Please try to comprehend the following text, perhaps in a moment where your matheologial defence meachanism is sleeping.
>
> The set of unit fractions 1/n has cardinality ℵo.
> No further unit fraction fits between all unit fractions and zero.
>
> There are less Peano unit fractions because for every Peano unit fraction ℵo smaller unit fractions fit between it and zero.
>
> Regards, WM
>

You are just making stuff up, incoherent.

You are proved wrong, however you must disagree to protect your ego.

Your responses are those of a pretender, not a professor.

On 1/19/2023 4:37 AM, WM wrote:
> Jim Burns schrieb am Mittwoch, 18. Januar 2023 um 23:37:28 UTC+1:
>> On 1/18/2023 5:16 AM, WM wrote:
>
>
>> I have no responsibility for more-than-all
>> readings of a claim of mine.
>
> You claim that all endegments are infinite.

they are defined that way, got that ?

> That is true as far as induction reaches.

wrong. they are defined that way, got that ?

> But it is not true that their intersection is empty.

Wrong, there intersection is not an endsegment, Miss Director.

> You conclude that only from the fact that you cannot see the dark numbers.

now you state you know what others think.
there are no dark numbers - you have never proved they exist.
your dark numbers are not associated with endsegments, nor intersections.

> Thus you are trapped in the contradiction that infinite inclusion-monotonic sets could have an empty intersection. Blatant nonsense.

Liar.

>>

>
> Regards, WM

On 1/19/2023 4:45 AM, WM wrote:
> Jim Burns schrieb am Donnerstag, 19. Januar 2023 um 02:59:41 UTC+1:
>
>> Our meta-reasoning treats claims the way that
>> our reasoning treats sheep and pebbles.
>> Our logic has not seen the Fall.
>
> Can it distinguish this?

> No further unit fraction fits between all unit fractions and zero.

that is a self conflicting statement, it is moot.

and the discussion was on natural numbers, not your change over to real numbers.

> For every Peano unit fraction ℵo smaller unit fractions fit between it and zero.

Wrong, Peano is not involved.

your unit fractions is too restrictive.

between any two real numbers there are an infinity of numbers.

>
> Regards, WM

you have a lot to learn. When did you stop ?