On 2/24/2024 12:58 PM, Chris M. Thomasson wrote:
> On 2/24/2024 4:16 AM, WM wrote:
>> Le 23/02/2024 à 13:23, Richard Damon a écrit :
>>> On 2/23/24 4:04 AM, WM wrote:
[...]
>> All numbers exist as points on the real line.
> [...]
>
> Really? You mind can only think of one axis?
>

I bet 2 ary graph paper blows you mind.....

On 03/22/2024 04:26 PM, Jim Burns wrote:
>
> ℕ⭳ contradicts your darkᵂᴹ and visibleᵂᴹ concept
>
>

Well, there's not an inductive proof
that there are infinitely many integers, ....

base case: not infinite
inductive case: not infinite

There's an inductive proof there's
no largest, ....

base case: not largest
inductive case: not largest

Yet, ....

base case: largest yet
inductive case: largest yet

So, what's "done", "complete", "fixed"?

What is the "definition of done"?

There's usually that "almost all"
arrives at "nowhere not", yet,
there's the Sorites and Heap
about when one's the other.

So, ZF, just "gives" an inductive set.

base case: not an inductive set
inductive case: not an inductive set
fixed case: an inductive set <- it's an axiom

Yet, there's to be arrived at via deduction,
that there's an inductive set.

When WM/MW invokes "infinity"
or ZF invokes "infinity", either way,
either way it's taking something
not given by inductive inference,
except by the exhaustion of induction.

base: not first false, not last true
inductive: not first false, not last true
fixed: not ultimately untrue <- it's a truism

So, running out all the cases,

base: not first false, not last true, not largest, not infinite, largest yet
inductive: same
fixed: true, largest, infinite, largest

takes out "not", "first", "last", and "yet", and "false", ....

<- is it an axiom?

Then, about the "Big-Endian" and "Little-Endian",
either way doesn't result the other,
points do not make lines and lines do not make points,
exhaustion of induction shows,
thus in cases where it _is_ so,
it must be that only exhaustion of induction shows.

That's deduction, "it must be".

So, there there is both a Big End and a Little End,
basically makes for that there's a middle.

The dialectic into the dichotomy the di-lemma,
or di-lem-na, has that geometry and the objects
of mathematics _do_ have completions, the
fixed points in the compact are members of
the space, the least-upper-bound property
does hold, these points make lines and lines
are of points, inference consumes both ends.

So, infinity making all this extra sense,
and to differentiate that, as it were,
from non-sense, it's large a matter of
book-keeping, and conscientious dialectic
throughout, and expansion of comprehension:
then attachment to the objects of geometry,
their form and proportion, and moments and motion.

So, infinite induction, basically goes directly
into line-reals and continuity.

constant monotone strictly increasing
extent density completeness measure

And Zeno's like "well yeah there is both ...".

Truth is regular / geometry is motion.

(Some years ago for like fifty posts,
in replies to Jim, it was "truth is regular"
and "geometry is motion". Here's another.
This is that the regular and rulial is a
thing in both dispersion and density,
and that geometry and motion are global and local.)

Then, when we were talking about Flying
Rainbow Sparkle Ponies, it was an example
of a model of real numbers, i.e. equipped with
properties of real numbers are models of
real numbers, similarly as how the Relephant
is a model of more than one model of real numbers.

It's the Elephant in the middle of the room,
and where the one with the five blind men
is one of the oldest stories about the limits
of inductive inference, in the world.

And Zeno's bridge is like, "doesn't matter
how you cross, it's one bridge at a time."

On 3/22/2024 7:41 PM, Ross Finlayson wrote:
> On 03/22/2024 04:26 PM, Jim Burns wrote:
>>
>> ℕ⭳ contradicts your darkᵂᴹ and visibleᵂᴹ concept
>>
>>
>
> Well, there's not an inductive proof
> that there are infinitely many integers, ....
[...]

Sometimes I like when somebody says I fount the largest integer and
somebody else says that plus one...

Well, there is a book:

https://youtu.be/ja6QSyct6SE

;^)

On 3/22/2024 2:54 PM, WM wrote:
> Le 22/03/2024 à 19:13, Jim Burns a écrit :
>> On 3/22/2024 8:13 AM, WM wrote:
>>> Le 22/03/2024 à 00:04, Jim Burns a écrit :

>>>> You have disabled the not.first.false "telescope"
>>>
>>> I don't believe t all that it is able .
>>
>> We study _claims themselves_ as objects and
>> we have found useful properties of _the claims themselves_
>> which they have, no matter what _the claims are about_
>>
>> Do you not.believe that
>> they have the useful properties we've found?
>>
>> Do you (WM) believe that, somewhere, sometime,
>> there is a finiteⁿᵒᵗᐧᵂᴹ claim.sequence
>> holding a false claim
>> but not.holding a first.false claim?
>
> That is not relevant.

I'll elaborate.

Imagine claims.about.points.in.the.line
being each represented by its own integer ≥ 2

Consider
a finiteⁿᵒᵗᐧᵂᴹ claim.sequence CLA of
claims.about.points.in.the.line.
represented by
a finiteⁿᵒᵗᐧᵂᴹ integer.sequence NUM of
integers ≥ 2

NUM is finiteⁿᵒᵗᐧᵂᴹ, so,
if some integer in NUM is prime,
then some integer in NUM is first.prime.
∃jP⇒∃k₁P

Imagine that NUM is arranged such that
_no integer in NUM is first.prime_
No integer in NUM is prime.
∃jP⇒∃k₁P
¬∃k₁P
⎯⎯⎯⎯⎯⎯
¬∃jP

You see 272566871177 in NUM
You know that 272566871177 is not.prime.

Next, we do the same with
finiteⁿᵒᵗᐧᵂᴹ claim.sequence CLA of
claims.about.points.in.the.line.

There exists a lot of subtle philosophy
about the nature of truth and of reality.
This method tosses all that in the trash bin.
It says truth is simply a property of claims,
like primeness is a property of integers ≥ 2
And that's it.

CLA is finiteⁿᵒᵗᐧᵂᴹ, so,
if some claim in CLA is false,
then some claim in CLA is first.false.
∃ψ⊥⇒∃φ₁⊥

Imagine that CLA is arranged such that
_no claim in CLA is first.false_
No claim in CLA is false.
∃ψ⊥⇒∃φ₁⊥
¬∃φ₁⊥
⎯⎯⎯⎯⎯⎯
¬∃ψ⊥

You see "The points are uncountable" in CLA
You know that "The points are uncountable" is not.false.

>> Do you (WM) believe that, somewhere, sometime,
>> there is a finiteⁿᵒᵗᐧᵂᴹ claim.sequence
>> holding a false claim
>> but not.holding a first.false claim?
>
> That is not relevant.

There is more to arranging CLA such that
no claim in CLA is first.false,
but the reason which we want to do that is
∃ψ⊥⇒∃φ₁⊥ which is
∀φ¬₁⊥⇒∀ψ⊤

> Every finite (potentially infinite)
> claim sequence
> "n is a visible number,
> n+1 is a visible number,
> n+2 is a visible number,
> ..."
> has no first false claim.
> Nevertheless almost all numbers are invisible.

Pleas define "finite" as you use it here.

On 03/23/2024 12:13 PM, Jim Burns wrote:
> On 3/22/2024 2:54 PM, WM wrote:
>> Le 22/03/2024 à 19:13, Jim Burns a écrit :
>>> On 3/22/2024 8:13 AM, WM wrote:
>>>> Le 22/03/2024 à 00:04, Jim Burns a écrit :
>
>>>>> You have disabled the not.first.false "telescope"
>>>>
>>>> I don't believe t all that it is able .
>>>
>>> We study _claims themselves_ as objects and
>>> we have found useful properties of _the claims themselves_
>>> which they have, no matter what _the claims are about_
>>>
>>> Do you not.believe that
>>> they have the useful properties we've found?
>>>
>>> Do you (WM) believe that, somewhere, sometime,
>>> there is a finiteⁿᵒᵗᐧᵂᴹ claim.sequence
>>> holding a false claim
>>> but not.holding a first.false claim?
>>
>> That is not relevant.
>
> I'll elaborate.
>
>
> Imagine claims.about.points.in.the.line
> being each represented by its own integer ≥ 2
>
> Consider
> a finiteⁿᵒᵗᐧᵂᴹ claim.sequence CLA of
> claims.about.points.in.the.line.
> represented by
> a finiteⁿᵒᵗᐧᵂᴹ integer.sequence NUM of
> integers ≥ 2
>
> NUM is finiteⁿᵒᵗᐧᵂᴹ, so,
> if some integer in NUM is prime,
> then some integer in NUM is first.prime.
> ∃jP⇒∃k₁P
>
> Imagine that NUM is arranged such that
> _no integer in NUM is first.prime_
> No integer in NUM is prime.
> ∃jP⇒∃k₁P
> ¬∃k₁P
> ⎯⎯⎯⎯⎯⎯
> ¬∃jP
>
> You see 272566871177 in NUM
> You know that 272566871177 is not.prime.
>
> Next, we do the same with
> finiteⁿᵒᵗᐧᵂᴹ claim.sequence CLA of
> claims.about.points.in.the.line.
>
> There exists a lot of subtle philosophy
> about the nature of truth and of reality.
> This method tosses all that in the trash bin.
> It says truth is simply a property of claims,
> like primeness is a property of integers ≥ 2
> And that's it.
>
> CLA is finiteⁿᵒᵗᐧᵂᴹ, so,
> if some claim in CLA is false,
> then some claim in CLA is first.false.
> ∃ψ⊥⇒∃φ₁⊥
>
> Imagine that CLA is arranged such that
> _no claim in CLA is first.false_
> No claim in CLA is false.
> ∃ψ⊥⇒∃φ₁⊥
> ¬∃φ₁⊥
> ⎯⎯⎯⎯⎯⎯
> ¬∃ψ⊥
>
> You see "The points are uncountable" in CLA
> You know that "The points are uncountable" is not.false.
>
>>> Do you (WM) believe that, somewhere, sometime,
>>> there is a finiteⁿᵒᵗᐧᵂᴹ claim.sequence
>>> holding a false claim
>>> but not.holding a first.false claim?
>>
>> That is not relevant.
>
> There is more to arranging CLA such that
> no claim in CLA is first.false,
> but the reason which we want to do that is
> ∃ψ⊥⇒∃φ₁⊥ which is
> ∀φ¬₁⊥⇒∀ψ⊤
>
>> Every finite (potentially infinite)
>> claim sequence
>> "n is a visible number,
>> n+1 is a visible number,
>> n+2 is a visible number,
>> ..."
>> has no first false claim.
>> Nevertheless almost all numbers are invisible.
>
> Pleas define "finite" as you use it here.
>
>

.. -v
... -v
.... -v
(...) -v
.........

..
^- ..
^- ...
^- (...)
^- ........

With repetition as induction, or vice-versa,
doesn't it seem like it needs both ends to
arrive at either from each other?

On 03/24/2024 11:45 AM, Ross Finlayson wrote:
> On 03/23/2024 12:13 PM, Jim Burns wrote:
>> On 3/22/2024 2:54 PM, WM wrote:
>>> Le 22/03/2024 à 19:13, Jim Burns a écrit :
>>>> On 3/22/2024 8:13 AM, WM wrote:
>>>>> Le 22/03/2024 à 00:04, Jim Burns a écrit :
>>
>>>>>> You have disabled the not.first.false "telescope"
>>>>>
>>>>> I don't believe t all that it is able .
>>>>
>>>> We study _claims themselves_ as objects and
>>>> we have found useful properties of _the claims themselves_
>>>> which they have, no matter what _the claims are about_
>>>>
>>>> Do you not.believe that
>>>> they have the useful properties we've found?
>>>>
>>>> Do you (WM) believe that, somewhere, sometime,
>>>> there is a finiteⁿᵒᵗᐧᵂᴹ claim.sequence
>>>> holding a false claim
>>>> but not.holding a first.false claim?
>>>
>>> That is not relevant.
>>
>> I'll elaborate.
>>
>>
>> Imagine claims.about.points.in.the.line
>> being each represented by its own integer ≥ 2
>>
>> Consider
>> a finiteⁿᵒᵗᐧᵂᴹ claim.sequence CLA of
>> claims.about.points.in.the.line.
>> represented by
>> a finiteⁿᵒᵗᐧᵂᴹ integer.sequence NUM of
>> integers ≥ 2
>>
>> NUM is finiteⁿᵒᵗᐧᵂᴹ, so,
>> if some integer in NUM is prime,
>> then some integer in NUM is first.prime.
>> ∃jP⇒∃k₁P
>>
>> Imagine that NUM is arranged such that
>> _no integer in NUM is first.prime_
>> No integer in NUM is prime.
>> ∃jP⇒∃k₁P
>> ¬∃k₁P
>> ⎯⎯⎯⎯⎯⎯
>> ¬∃jP
>>
>> You see 272566871177 in NUM
>> You know that 272566871177 is not.prime.
>>
>> Next, we do the same with
>> finiteⁿᵒᵗᐧᵂᴹ claim.sequence CLA of
>> claims.about.points.in.the.line.
>>
>> There exists a lot of subtle philosophy
>> about the nature of truth and of reality.
>> This method tosses all that in the trash bin.
>> It says truth is simply a property of claims,
>> like primeness is a property of integers ≥ 2
>> And that's it.
>>
>> CLA is finiteⁿᵒᵗᐧᵂᴹ, so,
>> if some claim in CLA is false,
>> then some claim in CLA is first.false.
>> ∃ψ⊥⇒∃φ₁⊥
>>
>> Imagine that CLA is arranged such that
>> _no claim in CLA is first.false_
>> No claim in CLA is false.
>> ∃ψ⊥⇒∃φ₁⊥
>> ¬∃φ₁⊥
>> ⎯⎯⎯⎯⎯⎯
>> ¬∃ψ⊥
>>
>> You see "The points are uncountable" in CLA
>> You know that "The points are uncountable" is not.false.
>>
>>>> Do you (WM) believe that, somewhere, sometime,
>>>> there is a finiteⁿᵒᵗᐧᵂᴹ claim.sequence
>>>> holding a false claim
>>>> but not.holding a first.false claim?
>>>
>>> That is not relevant.
>>
>> There is more to arranging CLA such that
>> no claim in CLA is first.false,
>> but the reason which we want to do that is
>> ∃ψ⊥⇒∃φ₁⊥ which is
>> ∀φ¬₁⊥⇒∀ψ⊤
>>
>>> Every finite (potentially infinite)
>>> claim sequence
>>> "n is a visible number,
>>> n+1 is a visible number,
>>> n+2 is a visible number,
>>> ..."
>>> has no first false claim.
>>> Nevertheless almost all numbers are invisible.
>>
>> Pleas define "finite" as you use it here.
>>
>>
>
>
>
>
> . -v
> .. -v
> ... -v
> (...) -v
> ........
>
>
> .
> ^- ..
> ^- ...
> ^- (...)
> ^- ........
>
> With repetition as induction, or vice-versa,
> doesn't it seem like it needs both ends to
> arrive at either from each other?
>
>

The Wikipedia "List of philosophical problems"
includes an entry for induction, whether it's circular.

https://en.wikipedia.org/wiki/List_of_philosophical_problems#Problem_of_induction

Then the idea that "well at least it's reflexive,
or reflective, or symmetric, or has a beginning
and an end, might help reduce that it's circular,
or, circulus vitiosus".

https://plato.stanford.edu/entries/definitions/#VicCirPri

Hm. So Russell picks as usual one of the inductive
cases to aggrandize himself and avoid Mirimanoff and
the extra-ordinary, while ignoring that it's just
as circular otherwise the branch he promotes,
instead of resolving it, ignoring it.

It's like, Russell's Vicious Circle principle: it's vicious.

"The paradoxes can also be used to motivate a conclusion
that is the very opposite to Russell’s."

https://plato.stanford.edu/entries/definitions/#CirDef

"The revision rule explains the behavior,
both ordinary and extraordinary, of a circular concept."

https://en.wikipedia.org/wiki/Revision_theory

About the other "problems of philosophy", I mostly
solve them by a sort of re-emergent monism, with
re-equipping the theory with the teleology it must
have to be that "the theory" so theorized is true,
these kinds of things.

It seems though that "revision and relevance logic"
sit together to help dispel "circular and material logic".

On 3/24/2024 10:45 PM, Ross Finlayson wrote:
> On 03/24/2024 11:45 AM, Ross Finlayson wrote:
>> On 03/23/2024 12:13 PM, Jim Burns wrote:

>>> [...]
>>
>> . -v
>> .. -v
>> ... -v
>> (...) -v
>> ........
>>
>>
>> .
>> ^- ..
>> ^- ...
>> ^- (...)
>> ^- ........

I think that what you're saying is:
.. ⮧
... ⮧
.... ⮧
(...) ⮧
.........

..
⮤ ..
⮤ ...
⮤ (...)
⮤ ........

>> With repetition as induction, or vice-versa,
>> doesn't it seem like it needs both ends to
>> arrive at either from each other?
>
> The Wikipedia "List of philosophical problems"
> includes an entry for induction,
> whether it's circular.
>
> https://en.wikipedia.org/wiki/List_of_philosophical_problems#Problem_of_induction
>

Those are different uses of the word "induction".
| | Scottish philosopher David Hume first formulated
| the problem of induction, arguing there is
| no non-circular way to justify inductive reasoning.
| That is, reasoning based on
| inferring conclusions from specific observations.
| This is a problem because everybody uses induction
| on a day to day basis e.g
| The sun rose in the east today therefore
| the sun will rise in the east tomorrow.
[1]

Cisfinite.induction
(AKA induction.but.not.transfinite.induction) is
a theorem of the properties of finiteⁿᵒᵗᐧᵂᴹ sequences.
Cisfinite.induction is not circular.

There is a view expressed that
cisfinite.induction involves a trip to ω in some way.
It doesn't, but the view expressed seems to consider that
a mere quibble, and that view is not deterred.

cisfinite.induction:
( P(0) ∧ ∀β ∈ ⟦0,ω⦆: P(β) ⇒ P(β⁺¹) ) ⟹
( ∀δ ∈ ⟦0,ω⦆: P(δ) )

Notation.
⟦0,ω⦆ is an ordinal.interval.
I want to distinguish ordinal.intervals from
real.number.intervals [0,x), something which would be
ugly and distracting if I limited myself to ASCII.

ω is the first infiniteⁿᵒᵗᐧᵂᴹ ordinal.
For each δ ∈ ⟦0,ω⦆ each non.∅.subset of ⟦0,δ⟧
holds a first and a last,
because finiteⁿᵒᵗᐧᵂᴹ δ

Cisfinite.induction can be re.written to be
a claim about finiteⁿᵒᵗᐧᵂᴹ intervals.
In that form, ω doesn't need to appear at all.

( ∃⟦0,δ⟧ finiteⁿᵒᵗᐧᵂᴹ: P(0) ∧ ¬P(δ) ) ⟹
( ∃⟦β,β⁺¹⟧ ⊆ ⟦0,δ⟧: P(β) ∧ ¬P(β⁺¹) )

Meaning:
If, on the finiteⁿᵒᵗᐧᵂᴹ trip to δ
P(α) changes from ⊤ to ⊥
then there is some β at which
P(α) changes from ⊤ to ⊥ in one step.

That's a theorem.
[2]

The usual formulation of cisfinite.induction
is the contrapositive of that.

If, on the finiteⁿᵒᵗᐧᵂᴹ trip to δ
there is NO β at which
P(α) changes from ⊤ to ⊥ in one step,
then P(α) NOT.changes from ⊤ to ⊥

[1]
| The sun rose in the east today therefore
| the sun will rise in the east tomorrow.

Physical.induction isn't justifiable in
purely logical terms.

But, in about 5,000,000,000 years,
the Sun _won't_ rise in the east.
It will expand and engulf the Earth.
We shouldn't _want_ it to be justifiable in
purely logical terms.
That would give incorrect answers.

I think that physical.induction is justifiable.
I'd better think so, I use it.
I think that the sun will rise tomorrow,
if not in 5,000,000,000 years.

I think that some prior principle is needed.

In astronomy, the principle of mediocrity is used
very effectively. It says, basically, that
we (probably) are not special.

It's the principle of mediocrity which allows us to
turn a mere handful of observations of
an unimaginably huge universe into a confident
prediction of the future of our own sun.

How I imagine the principle of mediocrity working
is as a way to quantify our ignorance, and so,
to be a way to have knowledge of some kind, at which point
we pull ourselves up by our own bootstraps,
and, leaving the details for the engineers to work out,
we create this marvelous picture of our universe.

(I.am.not.an.astronomer.)

[2]
For predicate P(γ) on ⟦0,δ⟧
define super.predicate P⟦0,γ⟧
P⟦0,γ⟧ :⇔ ∀α ∈ ⟦0,γ⟧: P(α)

P⟦0,β⟧ ∧ ¬P⟦0,γ⟧ ⟹ β < γ

Theorem:
( ∃⟦0,δ⟧ finiteⁿᵒᵗᐧᵂᴹ: P(0) ∧ ¬P(δ) ) ⟹
( ∃⟦β,β⁺¹⟧ ⊆ ⟦0,δ⟧: P(β) ∧ ¬P(β⁺¹) )

| Assume P(0) ∧ ¬P(δ)
| such that ⟦0,δ⟧ finiteⁿᵒᵗᐧᵂᴹ
| | P⟦0,0⟧
| {α: P⟦0,α⟧} ⊆ ⟦0,δ⟧ is nonempty.
| β is last in {α: P⟦0,α⟧}
| | ¬P⟦0,δ⟧
| {α: ¬P⟦0,α⟧} ⊆ ⟦0,δ⟧ is nonempty.
| γ is first in {α: ¬P⟦0,α⟧}
| | P⟦0,β⟧ ∧ ¬P⟦0,γ⟧
| β < γ
| No element of ⟦0,δ⟧ is between β and γ
| β⁺¹ = γ
| | P⟦0,β⟧
| ∀α ∈ ⟦0,β⟧: P(α)
| P(β)
| | ¬P⟦0,γ⟧
| ∃α ∈ ⟦0,β⟧∪{γ}: ¬P(α)
| However,
| ¬∃α ∈ ⟦0,β⟧: ¬P(α)
| ¬P(γ)
| | P(β) ∧ ¬P(γ)
| P(β) ∧ ¬P(β⁺¹)

Therefore,
( ∃⟦0,δ⟧ finiteⁿᵒᵗᐧᵂᴹ: P(0) ∧ ¬P(δ) ) ⟹
( ∃⟦β,β⁺¹⟧ ⊆ ⟦0,δ⟧: P(β) ∧ ¬P(β⁺¹) )

....which is equivalent to cisfinite.induction.

On 03/25/2024 11:36 AM, Jim Burns wrote:
> On 3/24/2024 10:45 PM, Ross Finlayson wrote:
>> On 03/24/2024 11:45 AM, Ross Finlayson wrote:
>>> On 03/23/2024 12:13 PM, Jim Burns wrote:
>
>>>> [...]
>>>
>>> . -v
>>> .. -v
>>> ... -v
>>> (...) -v
>>> ........
>>>
>>>
>>> .
>>> ^- ..
>>> ^- ...
>>> ^- (...)
>>> ^- ........
>
> I think that what you're saying is:
> . ⮧
> .. ⮧
> ... ⮧
> (...) ⮧
> ........
>
> .
> ⮤ ..
> ⮤ ...
> ⮤ (...)
> ⮤ ........
>
>>> With repetition as induction, or vice-versa,
>>> doesn't it seem like it needs both ends to
>>> arrive at either from each other?
>>
>> The Wikipedia "List of philosophical problems"
>> includes an entry for induction,
>> whether it's circular.
>>
>> https://en.wikipedia.org/wiki/List_of_philosophical_problems#Problem_of_induction
>>
>>
>
> Those are different uses of the word "induction".
> |
> | Scottish philosopher David Hume first formulated
> | the problem of induction, arguing there is
> | no non-circular way to justify inductive reasoning.
> | That is, reasoning based on
> | inferring conclusions from specific observations.
> | This is a problem because everybody uses induction
> | on a day to day basis e.g
> | The sun rose in the east today therefore
> | the sun will rise in the east tomorrow.
> [1]
>
>
> Cisfinite.induction
> (AKA induction.but.not.transfinite.induction) is
> a theorem of the properties of finiteⁿᵒᵗᐧᵂᴹ sequences.
> Cisfinite.induction is not circular.
>
> There is a view expressed that
> cisfinite.induction involves a trip to ω in some way.
> It doesn't, but the view expressed seems to consider that
> a mere quibble, and that view is not deterred.
>
> cisfinite.induction:
> ( P(0) ∧ ∀β ∈ ⟦0,ω⦆: P(β) ⇒ P(β⁺¹) ) ⟹
> ( ∀δ ∈ ⟦0,ω⦆: P(δ) )
>
> Notation.
> ⟦0,ω⦆ is an ordinal.interval.
> I want to distinguish ordinal.intervals from
> real.number.intervals [0,x), something which would be
> ugly and distracting if I limited myself to ASCII.
>
> ω is the first infiniteⁿᵒᵗᐧᵂᴹ ordinal.
> For each δ ∈ ⟦0,ω⦆ each non.∅.subset of ⟦0,δ⟧
> holds a first and a last,
> because finiteⁿᵒᵗᐧᵂᴹ δ
>
> Cisfinite.induction can be re.written to be
> a claim about finiteⁿᵒᵗᐧᵂᴹ intervals.
> In that form, ω doesn't need to appear at all.
>
> ( ∃⟦0,δ⟧ finiteⁿᵒᵗᐧᵂᴹ: P(0) ∧ ¬P(δ) ) ⟹
> ( ∃⟦β,β⁺¹⟧ ⊆ ⟦0,δ⟧: P(β) ∧ ¬P(β⁺¹) )
>
> Meaning:
> If, on the finiteⁿᵒᵗᐧᵂᴹ trip to δ
> P(α) changes from ⊤ to ⊥
> then there is some β at which
> P(α) changes from ⊤ to ⊥ in one step.
>
> That's a theorem.
> [2]
>
> The usual formulation of cisfinite.induction
> is the contrapositive of that.
>
> If, on the finiteⁿᵒᵗᐧᵂᴹ trip to δ
> there is NO β at which
> P(α) changes from ⊤ to ⊥ in one step,
> then P(α) NOT.changes from ⊤ to ⊥
>
> [1]
> | The sun rose in the east today therefore
> | the sun will rise in the east tomorrow.
>
> Physical.induction isn't justifiable in
> purely logical terms.
>
> But, in about 5,000,000,000 years,
> the Sun _won't_ rise in the east.
> It will expand and engulf the Earth.
> We shouldn't _want_ it to be justifiable in
> purely logical terms.
> That would give incorrect answers.
>
> I think that physical.induction is justifiable.
> I'd better think so, I use it.
> I think that the sun will rise tomorrow,
> if not in 5,000,000,000 years.
>
> I think that some prior principle is needed.
>
> In astronomy, the principle of mediocrity is used
> very effectively. It says, basically, that
> we (probably) are not special.
>
> It's the principle of mediocrity which allows us to
> turn a mere handful of observations of
> an unimaginably huge universe into a confident
> prediction of the future of our own sun.
>
> How I imagine the principle of mediocrity working
> is as a way to quantify our ignorance, and so,
> to be a way to have knowledge of some kind, at which point
> we pull ourselves up by our own bootstraps,
> and, leaving the details for the engineers to work out,
> we create this marvelous picture of our universe.
>
> (I.am.not.an.astronomer.)
>
> [2]
> For predicate P(γ) on ⟦0,δ⟧
> define super.predicate P⟦0,γ⟧
> P⟦0,γ⟧ :⇔ ∀α ∈ ⟦0,γ⟧: P(α)
>
> P⟦0,β⟧ ∧ ¬P⟦0,γ⟧ ⟹ β < γ
>
> Theorem:
> ( ∃⟦0,δ⟧ finiteⁿᵒᵗᐧᵂᴹ: P(0) ∧ ¬P(δ) ) ⟹
> ( ∃⟦β,β⁺¹⟧ ⊆ ⟦0,δ⟧: P(β) ∧ ¬P(β⁺¹) )
>
> | Assume P(0) ∧ ¬P(δ)
> | such that ⟦0,δ⟧ finiteⁿᵒᵗᐧᵂᴹ
> |
> | P⟦0,0⟧
> | {α: P⟦0,α⟧} ⊆ ⟦0,δ⟧ is nonempty.
> | β is last in {α: P⟦0,α⟧}
> |
> | ¬P⟦0,δ⟧
> | {α: ¬P⟦0,α⟧} ⊆ ⟦0,δ⟧ is nonempty.
> | γ is first in {α: ¬P⟦0,α⟧}
> |
> | P⟦0,β⟧ ∧ ¬P⟦0,γ⟧
> | β < γ
> | No element of ⟦0,δ⟧ is between β and γ
> | β⁺¹ = γ
> |
> | P⟦0,β⟧
> | ∀α ∈ ⟦0,β⟧: P(α)
> | P(β)
> |
> | ¬P⟦0,γ⟧
> | ∃α ∈ ⟦0,β⟧∪{γ}: ¬P(α)
> | However,
> | ¬∃α ∈ ⟦0,β⟧: ¬P(α)
> | ¬P(γ)
> |
> | P(β) ∧ ¬P(γ)
> | P(β) ∧ ¬P(β⁺¹)
>
> Therefore,
> ( ∃⟦0,δ⟧ finiteⁿᵒᵗᐧᵂᴹ: P(0) ∧ ¬P(δ) ) ⟹
> ( ∃⟦β,β⁺¹⟧ ⊆ ⟦0,δ⟧: P(β) ∧ ¬P(β⁺¹) )
>
> ...which is equivalent to cisfinite.induction.
>
>

Most of your symbols come through fine, yet here
the -v and ^- "forward and reverse induction"
are just probably in some range not in this machine's
locale's font, or "Supplemental Arrows".

https://en.wikipedia.org/wiki/Arrow_(symbol)#Unicode

I get about half of those, the upper half.

About induction, yeah that kind of is what induction is,
and about what makes for infinite induction which is
the usual consideration here where "induction itself"
as causality of course just always follows.

It does raise the point though, "induction of induction",
from the philosophy and its basic terms, and the reason
and rationality and about how the philosophy of
mathematics and for logicism and positivism the
empirical, yet also for platonism and the metaphysics,
or "stipulated" axiomatics vis-a-vis "true" axioms,
for model theory and proof theory, about whether
the meta-theory is also the theory, it is a thing.

In my podcasts recently it's been a lot about teleology
vis-a-vis ontology as the epistemology, with the usual
sort of idea that a "strong" platonism results a "stronger"
logicism and "stronger" positivism as from some "axiomless
natural deduction: must be true" and "not ultimately untrue",
just noting that for many, many years my stated philosophy
here has been for axiomless natural deduction and a
"strongest mathematical platonism", that the objects
of mathematics exist then that our usual notions in
words have that geometry and words are two different
things, or from two different ends.

Then the idea of the

..
....
......
(...)
........

and whether it's up or down or back and around,
makes for that counting is accumulation and there
results a book-keeping which explains why though
inductive inference carries and borrows freely,
that's only because the numbers are infinite.

It's certainly so for anything that solves related rates
or defines infinite series' infinite limits,
it's a framework and an accounting.

Otherwise an "inductive set" isn't the integers
at all, it's just a notion of free-borrow and free-carry.

Which it provides, ....

On 3/26/2024 9:16 PM, Ross Finlayson wrote:
> On 03/25/2024 11:36 AM, Jim Burns wrote:

>> [...]
>
> About induction,
> yeah that kind of is what induction is,

I use "induction" three ways.
transfinite.induction == "We are ordinals"
cisfinite.induction == "We are finite ordinals"
physical.induction == "We are not special".

I spent most of my time on cisfinite.induction,
demonstrating more strongly than "kind of" that
cisfinite.induction == "We are finite ordinals"

It sounds like we're "kind of" agreeing,
but I can't tell what we're agreeing about.

> and about what makes for infinite induction
> which is the usual consideration here
> where "induction itself" as causality
> of course just always follows.

Induction isn't causality.
Induction(s), transfinite, cisfinite, and physical,
identify potentially.exploitable regularities.

For 4.3 billion years, the sun rose in the east.
Physical.induction says,
tomorrow, the sun will rise in the east.
But it doesn't _cause_ the sun to rise in the east.

What causality is _not_ is correlation.

A well.worn example is ice cream and crime.
Statistics tell us that, more often than not,
crime and ice cream consumption rise together
and fall together.

No one suggests that eating ice cream causes crime.

Instead, it is suggested that
beautiful weather causes people to go outside,
where they are more likely to eat ice cream
and also more likely to get mugged.

What causality _is_ is potentially.effective
points.of.intervention.

Even if we could actually intervene in
the eating of ice cream,
Imagine a media campaign:
"Friends don't let friends eat ice cream."
we would not expect its effect to include
less crime.

Our theory is
eating ice cream is not a potentially.effective
point.of.intervention with respect to crime.

We say, eating ice cream does not cause crime.

On the other hand,
if we could actually intervene in
the weather,
Imagine Wile E. Coyote, Super-Genius,
unpacking an Acme Weather-O-Matic™,
dialing it to "miserable" and switching it on.
we would expect less crime and less ice cream.

Our theory is
weather is a potentially.effective point.of.
..intervention with respect to crime and ice cream.

We say beautiful weather causes more crime and
more ice cream eating.

> It does raise the point though,
> "induction of induction",
....
> about whether
> the meta-theory is also the theory,
> it is a thing.

This is my attempt to answer the question(s):
transfinite.induction == "We are ordinals"
cisfinite.induction == "We are finite ordinals"
physical.induction == "We are not special".

Transfinite and cisfinite are theorems of
"We are ordinals" and "We are finite ordinals".

Physical.induction can be derived from
evidence and Bayes' Theorem and the assumption that
the evidence is a random (not.special) selection
from past and future evidence.collection.

A billion eastern sunrises is a lot of evidence,
and, via Bayes' Theorem, the posterior probability that
tomorrow has an eastern sunrise is close to 1

I'm assuming here that,
prior to observing sunrises,
each hypothesis Hₚ = "P(eastern.sunrise) = p"
has the same probability (Hₚ is not.special).
P(Hₚ) = 1

That's a very crude model,
but it can be elaborated.

Elaborating crude initial models is kind of
how things are done in Physics.land,
home of the adiabatic spherical cows in a vacuum.

On 3/27/2024 1:25 PM, Jim Burns wrote:
> On 3/26/2024 9:16 PM, Ross Finlayson wrote:

>> [...]
>
> Physical.induction can be derived from
> evidence and Bayes' Theorem and the assumption that
> the evidence is a random (not.special) selection
> from past and future evidence.collection.
>
> A billion eastern sunrises is a lot of evidence,
> and, via Bayes' Theorem, the posterior probability that
> tomorrow has an eastern sunrise is close to 1
>
> I'm assuming here that,
> prior to observing sunrises,
> each hypothesis Hₚ = "P(eastern.sunrise) = p"
> has the same probability (Hₚ is not.special).
> P(Hₚ) = 1

Oops.
What I didn't say is that
I'm assuming a continuous distribution for
probability.of.eastern.sunrise p

Prior probability that probability.of.eastern.sunrise ≤ p
∫₀ᵖ P(Hₛ)𝑑s = ∫₀ᵖ 1⋅𝑑s = p

Probability of Eₙ n independent eastern sunrises,
given Hₚ
P(Eₙ|Hₚ) = pⁿ

Probability of Hₚ given Eₙ (Bayes')
P(Hₚ|Eₙ) =
P(Eₙ|Hₚ)⋅P(Hₚ) / P(Eₙ) =
P(Eₙ|Hₚ)⋅P(Hₚ) / ∫₀¹ P(Eₙ|Hₛ)⋅P(Hₛ)⋅𝑑s =
pⁿ⋅1 / ∫₀¹ sⁿ⋅1⋅𝑑s =
(n+1)pⁿ

Probability of E given Eₙ
P(E|Eₙ) =
∫₀¹ P(E∧Hₛ|Eₙ)⋅𝑑s =
∫₀¹ P(E|Hₛ∧Eₙ)⋅P(Hₛ|Eₙ)⋅𝑑s =
∫₀¹ s⋅(n+1)sⁿ⋅𝑑s =
(n+1)/(n+2)

> That's a very crude model,
> but it can be elaborated.
>
> Elaborating crude initial models is kind of
> how things are done in Physics.land,
> home of the adiabatic spherical cows in a vacuum.
>
>

On 03/27/2024 11:23 AM, Jim Burns wrote:
> On 3/27/2024 1:25 PM, Jim Burns wrote:
>> On 3/26/2024 9:16 PM, Ross Finlayson wrote:
>
>>> [...]
>>
>> Physical.induction can be derived from
>> evidence and Bayes' Theorem and the assumption that
>> the evidence is a random (not.special) selection
>> from past and future evidence.collection.
>>
>> A billion eastern sunrises is a lot of evidence,
>> and, via Bayes' Theorem, the posterior probability that
>> tomorrow has an eastern sunrise is close to 1
>>
>> I'm assuming here that,
>> prior to observing sunrises,
>> each hypothesis Hₚ = "P(eastern.sunrise) = p"
>> has the same probability (Hₚ is not.special).
>> P(Hₚ) = 1
>
> Oops.
> What I didn't say is that
> I'm assuming a continuous distribution for
> probability.of.eastern.sunrise p
>
> Prior probability that probability.of.eastern.sunrise ≤ p
> ∫₀ᵖ P(Hₛ)𝑑s = ∫₀ᵖ 1⋅𝑑s = p
>
> Probability of Eₙ n independent eastern sunrises,
> given Hₚ
> P(Eₙ|Hₚ) = pⁿ
>
> Probability of Hₚ given Eₙ (Bayes')
> P(Hₚ|Eₙ) =
> P(Eₙ|Hₚ)⋅P(Hₚ) / P(Eₙ) =
> P(Eₙ|Hₚ)⋅P(Hₚ) / ∫₀¹ P(Eₙ|Hₛ)⋅P(Hₛ)⋅𝑑s =
> pⁿ⋅1 / ∫₀¹ sⁿ⋅1⋅𝑑s =
> (n+1)pⁿ
>
> Probability of E given Eₙ
> P(E|Eₙ) =
> ∫₀¹ P(E∧Hₛ|Eₙ)⋅𝑑s =
> ∫₀¹ P(E|Hₛ∧Eₙ)⋅P(Hₛ|Eₙ)⋅𝑑s =
> ∫₀¹ s⋅(n+1)sⁿ⋅𝑑s =
> (n+1)/(n+2)
>
>> That's a very crude model,
>> but it can be elaborated.
>>
>> Elaborating crude initial models is kind of
>> how things are done in Physics.land,
>> home of the adiabatic spherical cows in a vacuum.
>>
>>
>

Induction the inference is the usual idea of causality,
and what comes around goes around.

The sun rising daily for example, this is a Ptolemaic
concept that's sort of had figured out a Copernican
concept, that due our equilibrium at rest in the
shadow of the gravity well of Earth, the theory goes,
it's causal that the Earth spinning results the image
of the sun rising, also that it rises in the East.

I.e., "physical induction" isn't necessarily confusing
"causation" and "correlation", and just because it's
attached to a physical model according to a scientific
hypothesis doesn't detach it from the logical and
mathematical model that represents the "act" of
induction.

Then, here philosophy's "problem of induction"
is linked with its issues with stipulations at all,
about what are the "true" of the axioms, of course
definitions, vis-a-vis an objective universe of the
mathematical objects, the "model" of the theory
itself, that the theory really does have, according
to model theory, a model of itself, and its own consistency.

(That any axiomatization strong enough to
formalize ordinary arithmetic Goedel shows can't
model its own consistency, here the idea is that
the "objectively existing" or platonistic universe
of mathematical objects is just "true" itself,
while then of course there are many and various
law(s) of large numbers because there are various
and sundry models of infinite quantities, processes,
relations, exhaustions, bounds, magnitudes,
grandnesses, and so on, the potential, practical,
effective, and actual. That is to say, according to
comprehension, there isn't only a "standard" model
of integers, yet fragments and extensions.)

Now, you brought statistics into it, and, we
know that much of the law(s) of probability
are law(s) of large numbers, just so you know
that with regards to Bayes and Jeffries priors
and Bayes and Jeffries and Knight uncertainties,
just so as we've been discussing there are issues
with convergence and divergence and slow convergence
and divergence in infinite series, it is so that statistics
as well, has its own sort issues in large numbers,
and notions of largeness, and what is large.

So, (Sparse) Cantor space, "Square" Cantor space,
and "Signal" Cantor space, help to begin to illustrate
this milieu, where various number-theoretic conjectures
are _independent_ standard ordinary number theory,
that at least having three formalized definitions of
completeness and continuous domains, definitely
supplements a mathematics otherwise rather oddly
hopping on the one sort of leg. Or, given no feet,
a three-legged stool may make a stable seat.

"Borel vis-a-vis Combinatorics", about the complementary
duals and what should be complimentary duels,
each "completing" from "counting" yet diverging
quite altogether, helps show that meeting in the middle,
"the middle of nowhere", really is a relevant concept,
if what you're interested in is mathematics,
and even not knowing it, yet at least knowing it.

Thanks again and as usual your replies are warmly appreciated.

This thread is pretty dull, except we've filled it up a bit.

On 3/27/2024 6:17 PM, Ross Finlayson wrote:
> On 03/27/2024 11:23 AM, Jim Burns wrote:

>> [...]
>
> Induction the inference is the usual idea of causality,
> and what comes around goes around.

I think maybe you (RF) are not alone in thinking so,
but, usual idea or not,
that has its own Latin.named fallacy,
"post hoc ergo propter hoc"

Assume induction is used and
it gives a correct answer.

Even then, what do you have?
One eastern sunrise after another.

It seems to me that causality is more than that,
and, if it isn't thought so, then it should be.

On 03/28/2024 03:43 AM, Jim Burns wrote:
> On 3/27/2024 6:17 PM, Ross Finlayson wrote:
>> On 03/27/2024 11:23 AM, Jim Burns wrote:
>
>>> [...]
>>
>> Induction the inference is the usual idea of causality,
>> and what comes around goes around.
>
> I think maybe you (RF) are not alone in thinking so,
> but, usual idea or not,
> that has its own Latin.named fallacy,
> "post hoc ergo propter hoc"
>
> Assume induction is used and
> it gives a correct answer.
>
> Even then, what do you have?
> One eastern sunrise after another.
>
> It seems to me that causality is more than that,
> and, if it isn't thought so, then it should be.
>
>

McKeon has a great introduction to Aristotle's "Metaphysics".

There really is a strong technical school of philosophy
that Being and Nothing arise as from, "Nothing", Becoming,
Sein and Da Sein, the noumenon at all, the Ding-an-Sich,
Ku and no-mind and these types things, why there really
is a teleology and not just an ontology.

So, Aristotle has Prior Analytics, and, Posterior Analytics.
The usual idea is that one will convince most usual people
of the existence of an argument, while the other one is
there to deconstruct and take that apart.

"It is clear then from what has been said that there is a
substance which is eternal and unmovable and separate
from sensible things."

The sun rises at all....

McKeon, "Beginning with the observation that all men
by nature desire to know, Aristotle inquires into the
nature of the science we seek as "wisdom" by placing
it in the sequence of faculties and purposive operations
of animals - sense, memory, experience, art, and science -
and by examining what the wise man knows. He concludes
that wisdom is concerned with causes as such, and is the
science of first principles and causes, which is architectonic
or most authoritative among the sciences. "

"The examination of earlier conceptions of cause, like the
examinations of theories of the soul as principles of motion
and of life in the opening books of the Physics and the De Anima,
turns on distinctions of matter and form. The earliest philosophers
limited their attention to material causes and explained things
by the matter or the elements of which they are compared. Later
philosophers turned to formal causes or essences, which they
separated from matter and constituted into independent and
self-subsistent Forms. The moving or efficient cause, which intitates
the change, and the final cause or end, in which it eventuates,
which have a prominent place in the inquiries of Aristotle's
physical treatises, wre touched on only sporadically by a
few philoshopers like Empedocles and Anaxagoras. After
criticisms and adaptations or earlier conceptions of cause,
the book ends with a demonstration that there are no more than
the four causes."

Aristotle: "All men by nature desire to know."

On 03/10/2024 10:21 AM, Ross Finlayson wrote:
> On 03/10/2024 01:03 AM, WM wrote:
>> On 09.03.2024 23:52, Tom Bola wrote:
>>>
>>>>>> XXXXXXXXXXXXXXXXXXXXX...
>>>>>>
>>>>>> and
>>>>>> OOOOO ...
>>>>>> OOOOO ...
>>>>>> OOOOO ...
>>>>>> ...
>>>>>>
>>>>>> and we can cover all the Os with an X from the line above it.
>>>>
>>>> Hier bleibt unverstanden, dass die Zeile der Indizes nicht länger als
>>>> jede Zeile der Matrix sein kann.
>>>
>>> Dedekind-Unendlichkeit: Unendliche Mengen haben echte unendliche
>>> Teilmengen.
>>
>> Das hat nichts mit Dedekind zu tun. Jede Zeile der Matrix ist auch ohne
>> Dedekind unendlich. Dedekind behauptet eine Bijektion zwischen Menge und
>> Untermenge. Das ist aber nur für seine Schöpfungen, also potentielle
>> Unendlichkeit richtig.
>>
>> Die aktual unendliche Matrix enthält mehr Elemente als eine ihrer Zeilen
>> oder Spalten. Nach Cantor oder Dedekind müsste aber eine Bijektion
>> zwischen allen Elementen inklusive der ersten Spalte und z.B. der ersten
>> Spalte bestehen. Das ist ausgeschlossen.
>>
>> Gruß, WM
>>
>
> Cantor hat dann noch Unendlichkeit, Umbegrentzheit, das Unzaelhbar,
> das "un-ending, un-bounded, un-countable".
>
> Es kommt wie "Diagonalargument", sprachlicht "Anti-Diagonalargument",
> das Unzaehlbar.
>
> Es is definiert assoziieren Bijektivfunktion.
>
> Meist funktion sind Kartesisch, dieser Funktionen,
> und assoziieren Theorem Cantor-Schroder-Bernstein,
> transitive assoziierent, dieser Domaenen, Funktionsumfang,
> diese besehen sich.
>
> So, ..., nicht alles Funktionen sind Kartesisch. ein
> Stetiggrenze, kann nicht nachbestellt sich machen,
> Element duerfen nicht Ausgetauscht werden.
>
> Grenze: is nicht Ganz, oder es ist? Voltiglich.
>
> So, Stetiggrenze n/d, nicht Karteshisch,
> Zaehlbar, [0,1], Feld Reellezahlen, Unzaehlbar,
> Signale-Reellezahlen, Stetiggrenze Stetigganze,
> nicht Karteshisch, wieder Zaehlbar.
>
>
> Klar, hein?
>
>

Drei Ecken....