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Re: Ordinals

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  Re: Ordinals "markus...@gmail.com" <markusklyver@gmail.com> - 2024-02-20 11:35 -0800
    Re: Ordinals Mild Shock <janburse@fastmail.fm> - 2024-02-20 20:54 +0100
      Re: Ordinals "mitchr...@gmail.com" <mitchrae3323@gmail.com> - 2024-02-20 11:59 -0800
    Re: Ordinals Ross Finlayson <ross.a.finlayson@gmail.com> - 2024-02-20 12:15 -0800
      Re: Ordinals Ross Finlayson <ross.a.finlayson@gmail.com> - 2024-02-20 12:27 -0800
      Re: Ordinals Jim Burns <james.g.burns@att.net> - 2024-02-20 17:02 -0500
        Re: Ordinals Ross Finlayson <ross.a.finlayson@gmail.com> - 2024-02-20 19:36 -0800
          Re: Ordinals Mild Shock <bursejan@gmail.com> - 2024-02-21 05:47 -0800
            Re: Ordinals Mild Shock <bursejan@gmail.com> - 2024-02-21 05:50 -0800
              Re: Ordinals Mild Shock <bursejan@gmail.com> - 2024-02-21 06:16 -0800
          Re: Ordinals Ross Finlayson <ross.a.finlayson@gmail.com> - 2024-02-21 10:24 -0800
            Re: Ordinals Mild Shock <janburse@fastmail.fm> - 2024-02-22 09:03 +0100
              Re: Ordinals Mild Shock <janburse@fastmail.fm> - 2024-02-22 09:11 +0100
              Re: Ordinals Ross Finlayson <ross.a.finlayson@gmail.com> - 2024-02-22 10:16 -0800
                Re: Ordinals Mild Shock <janburse@fastmail.fm> - 2024-02-22 19:20 +0100
                  Re: Ordinals Mild Shock <janburse@fastmail.fm> - 2024-02-22 19:22 +0100
                    Re: Ordinals Mild Shock <janburse@fastmail.fm> - 2024-02-22 19:40 +0100
                      Re: Ordinals Mild Shock <janburse@fastmail.fm> - 2024-02-22 20:00 +0100
                    Re: Ordinals Ross Finlayson <ross.a.finlayson@gmail.com> - 2024-02-22 10:55 -0800
                      Re: Ordinals Mild Shock <janburse@fastmail.fm> - 2024-02-22 20:08 +0100
                        Re: Ordinals Mild Shock <janburse@fastmail.fm> - 2024-02-22 20:13 +0100
                  Re: Ordinals Ross Finlayson <ross.a.finlayson@gmail.com> - 2024-02-22 11:01 -0800
                    Re: Ordinals Mild Shock <janburse@fastmail.fm> - 2024-02-22 20:11 +0100
                      Re: Ordinals Mild Shock <janburse@fastmail.fm> - 2024-02-22 20:13 +0100
              Re: Ordinals Jim Burns <james.g.burns@att.net> - 2024-02-22 14:34 -0500
                Re: Ordinals Ross Finlayson <ross.a.finlayson@gmail.com> - 2024-02-22 11:39 -0800
                Re: Ordinals Mild Shock <janburse@fastmail.fm> - 2024-02-23 00:41 +0100
        Re: Ordinals WM <wolfgang.mueckenheim@tha.de> - 2024-02-21 08:33 +0000
          Re: Ordinals Jim Burns <james.g.burns@att.net> - 2024-02-21 12:59 -0500
            Re: Ordinals WM <wolfgang.mueckenheim@tha.de> - 2024-02-22 13:00 +0000
              Re: Ordinals Jim Burns <james.g.burns@att.net> - 2024-02-22 11:13 -0500
                Re: Ordinals WM <wolfgang.mueckenheim@tha.de> - 2024-02-23 08:47 +0000
                  Semanticists candy (Was: Ordinals) Mild Shock <janburse@fastmail.fm> - 2024-02-26 12:58 +0100
                  Re: Ordinals Jim Burns <james.g.burns@att.net> - 2024-02-27 14:25 -0500
                    Re: Ordinals Ross Finlayson <ross.a.finlayson@gmail.com> - 2024-02-27 11:59 -0800
                    Re: Ordinals WM <wolfgang.mueckenheim@tha.de> - 2024-02-27 20:05 +0000
                      Re: Ordinals Jim Burns <james.g.burns@att.net> - 2024-02-27 17:24 -0500
                        Re: Ordinals WM <wolfgang.mueckenheim@tha.de> - 2024-02-28 09:48 +0000
                          Re: Ordinals Jim Burns <james.g.burns@att.net> - 2024-02-28 06:52 -0500
                            Re: Ordinals WM <wolfgang.mueckenheim@tha.de> - 2024-02-28 17:24 +0000
                              Re: Ordinals Richard Damon <richard@damon-family.org> - 2024-02-28 17:07 -0500
                                Re: Ordinals WM <wolfgang.mueckenheim@tha.de> - 2024-02-29 08:23 +0000
                                  Re: Ordinals Richard Damon <richard@damon-family.org> - 2024-02-29 07:35 -0500
                                    Re: Ordinals WM <wolfgang.mueckenheim@tha.de> - 2024-02-29 19:25 +0000
                                      Re: Ordinals "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2024-02-29 13:48 -0800
                                      Re: Ordinals Richard Damon <richard@damon-family.org> - 2024-02-29 22:12 -0500
                                        Re: Ordinals Ross Finlayson <ross.a.finlayson@gmail.com> - 2024-02-29 19:36 -0800
                                          Re: Ordinals Mild Shock <janburse@fastmail.fm> - 2024-03-01 20:38 +0100
                                            Re: Ordinals Mild Shock <janburse@fastmail.fm> - 2024-03-01 20:52 +0100
                                              Re: Ordinals Mild Shock <janburse@fastmail.fm> - 2024-03-01 21:08 +0100
                                                Re: Ordinals Jim Burns <james.g.burns@att.net> - 2024-03-01 15:56 -0500
                                                  Re: Ordinals Mild Shock <janburse@fastmail.fm> - 2024-03-01 22:53 +0100
                                                    Re: Ordinals Mild Shock <janburse@fastmail.fm> - 2024-03-01 23:11 +0100
                                                      Re: Ordinals Mild Shock <janburse@fastmail.fm> - 2024-03-01 23:33 +0100
                                                      Re: Ordinals Jim Burns <james.g.burns@att.net> - 2024-03-01 19:41 -0500
                                                        Re: Ordinals Ross Finlayson <ross.a.finlayson@gmail.com> - 2024-03-01 20:15 -0800
                                                          Re: Ordinals Ross Finlayson <ross.a.finlayson@gmail.com> - 2024-03-02 10:49 -0800
                                                        Re: Ordinals Mild Shock <janburse@fastmail.fm> - 2024-03-02 20:40 +0100
                                            Re: Ordinals Ross Finlayson <ross.a.finlayson@gmail.com> - 2024-03-01 14:45 -0800
                                        Re: Ordinals WM <wolfgang.mueckenheim@tha.de> - 2024-03-01 08:47 +0000
                                          Re: Ordinals Richard Damon <richard@damon-family.org> - 2024-03-01 09:44 -0500
                                            Re: Ordinals WM <wolfgang.mueckenheim@tha.de> - 2024-03-01 18:29 +0000
                                              Re: Ordinals Richard Damon <richard@damon-family.org> - 2024-03-01 14:00 -0500
                                                Re: Ordinals WM <wolfgang.mueckenheim@tha.de> - 2024-03-02 12:36 +0000
                                                  Re: Ordinals Richard Damon <richard@damon-family.org> - 2024-03-02 09:22 -0500
                                                    Re: Ordinals WM <wolfgang.mueckenheim@tha.de> - 2024-03-02 15:25 +0000
    Re: Ordinals "mitchr...@gmail.com" <mitchrae3323@gmail.com> - 2024-02-20 17:21 -0800
      RSemanticists candy (Re: Ordinals) [Addendum] Mild Shock <janburse@fastmail.fm> - 2024-02-26 13:01 +0100

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#625725

On 2/28/24 12:24 PM, WM wrote:
> Le 28/02/2024 à 12:52, Jim Burns a écrit :
>> On 2/28/2024 4:48 AM, WM wrote:
>>> Le 27/02/2024 à 23:24, Jim Burns a écrit :
>>>> On 2/27/2024 3:05 PM, WM wrote:
>>>>> Le 27/02/2024 à 20:25, Jim Burns a écrit :
>>>>>> On 2/23/2024 3:47 AM, WM wrote:
>>
>>>>>> We know that Peano induction not.ceases in
>>>>>> the Peano (final) ordinals.
>>>>>
>>>>> We know that every visible step is reversible.
>>>>
>>>> A one.ended ascent is reversible,
>>>> but is not a descent.
>>>
>>> Every reversion of an ascent is a descent.
>>
>> A one.ended ascent starts and not.stops.
> 
> As long as it runs through visible numbers it is finite and reversible.
> 
> Regards, WM
> 
> 

In other words, your definition of "Visible Numbers" are finite sub-sets 
of the actual set of values.

And this is because your logic can only handle finite sets.

Your "Dark" numbers are just the numbers that you can not handle with 
your restricted finite limited logic.

There is actually no problem with those numbers, except you incorrectly 
limited logic can't deal with them.

They are a product of YOUR limitations, not of the inability of the 
proper logic to deal with them.

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#625733

Le 28/02/2024 à 23:07, Richard Damon a écrit :
> On 2/28/24 12:24 PM, WM wrote:
>> Le 28/02/2024 à 12:52, Jim Burns a écrit :
>>> On 2/28/2024 4:48 AM, WM wrote:
>>>> Le 27/02/2024 à 23:24, Jim Burns a écrit :
>>>>> On 2/27/2024 3:05 PM, WM wrote:
>>>>>> Le 27/02/2024 à 20:25, Jim Burns a écrit :
>>>>>>> On 2/23/2024 3:47 AM, WM wrote:
>>>
>>>>>>> We know that Peano induction not.ceases in
>>>>>>> the Peano (final) ordinals.
>>>>>>
>>>>>> We know that every visible step is reversible.
>>>>>
>>>>> A one.ended ascent is reversible,
>>>>> but is not a descent.
>>>>
>>>> Every reversion of an ascent is a descent.
>>>
>>> A one.ended ascent starts and not.stops.
>> 
>> As long as it runs through visible numbers it is finite and reversible.
>> 
> 
> In other words, your definition of "Visible Numbers" are finite sub-sets 
> of the actual set of values.

Visible narural numbers are FISONs {1, 2, 3, ..., n}.
> 
> And this is because your logic can only handle finite sets.

This because there is no infinite natural number. 
> 
> Your "Dark" numbers are

the only possibility to have completed infinity. Note: "Going on and on" 
is not completed infinity but potential infinity.

> There is actually no problem with those numbers,

What numbers? Infinite natural numbers?

Regards, WM

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#625735

On 2/29/24 3:23 AM, WM wrote:
> Le 28/02/2024 à 23:07, Richard Damon a écrit :
>> On 2/28/24 12:24 PM, WM wrote:
>>> Le 28/02/2024 à 12:52, Jim Burns a écrit :
>>>> On 2/28/2024 4:48 AM, WM wrote:
>>>>> Le 27/02/2024 à 23:24, Jim Burns a écrit :
>>>>>> On 2/27/2024 3:05 PM, WM wrote:
>>>>>>> Le 27/02/2024 à 20:25, Jim Burns a écrit :
>>>>>>>> On 2/23/2024 3:47 AM, WM wrote:
>>>>
>>>>>>>> We know that Peano induction not.ceases in
>>>>>>>> the Peano (final) ordinals.
>>>>>>>
>>>>>>> We know that every visible step is reversible.
>>>>>>
>>>>>> A one.ended ascent is reversible,
>>>>>> but is not a descent.
>>>>>
>>>>> Every reversion of an ascent is a descent.
>>>>
>>>> A one.ended ascent starts and not.stops.
>>>
>>> As long as it runs through visible numbers it is finite and reversible.
>>>
>>
>> In other words, your definition of "Visible Numbers" are finite 
>> sub-sets of the actual set of values.
> 
> Visible narural numbers are FISONs {1, 2, 3, ..., n}.

And so SUBSETS of the Natural Number

>>
>> And this is because your logic can only handle finite sets.
> 
> This because there is no infinite natural number.

No, but there are an infinte number of them.

You logic can't handle sets of infinite/unbounded size.

>>
>> Your "Dark" numbers are
> 
> the only possibility to have completed infinity. Note: "Going on and on" 
> is not completed infinity but potential infinity.

Only because your logic can't handle it.

> 
>> There is actually no problem with those numbers,
> 
> What numbers? Infinite natural numbers?

The unbounded set of Natural Numbers that go on and on and on.

> 
> Regards, WM


You are just proving your logic is broken and your brain is two sizes 
too small for what you are trying to do.

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#625741

Le 29/02/2024 à 13:35, Richard Damon a écrit :
> On 2/29/24 3:23 AM, WM wrote:

>> 
>> Visible narural numbers are FISONs {1, 2, 3, ..., n}.
> 
> And so SUBSETS of the Natural Number

Of course. What else? Every natural number belongs to the set ℕ.

>>> There is actually no problem with those numbers,
>> 
>> What numbers? Infinite natural numbers?
> 
> The unbounded set of Natural Numbers that go on and on and on.

Visible numbers do never complete infinity.

Regards, WM

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#625751

On 2/29/2024 11:25 AM, WM wrote:
> Le 29/02/2024 à 13:35, Richard Damon a écrit :
>> On 2/29/24 3:23 AM, WM wrote:
> 
>>>
>>> Visible narural numbers are FISONs {1, 2, 3, ..., n}.
>>
>> And so SUBSETS of the Natural Number
> 
> Of course. What else? Every natural number belongs to the set ℕ.
> 
>>>> There is actually no problem with those numbers,
>>>
>>> What numbers? Infinite natural numbers?
>>
>> The unbounded set of Natural Numbers that go on and on and on.
> 
> Visible numbers do never complete infinity.

Huh? Visible? Oh god... You are a 100% hyper hardcore ultra finite odd 
ball... Infinity makes your brain bleed?

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#625753

On 2/29/24 2:25 PM, WM wrote:
> Le 29/02/2024 à 13:35, Richard Damon a écrit :
>> On 2/29/24 3:23 AM, WM wrote:
> 
>>>
>>> Visible narural numbers are FISONs {1, 2, 3, ..., n}.
>>
>> And so SUBSETS of the Natural Number
> 
> Of course. What else? Every natural number belongs to the set ℕ.

And evvery natural number is finite and thus namable and thus visible.

> 
>>>> There is actually no problem with those numbers,
>>>
>>> What numbers? Infinite natural numbers?
>>
>> The unbounded set of Natural Numbers that go on and on and on.
> 
> Visible numbers do never complete infinity.

Right, but no individual Natural Number does either, so they all can be 
visible.

It is the SET that "completes infinity", and the set isn't any of the 
individual number.

> 
> Regards, WM
> 
> 

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#625755

On 02/29/2024 07:12 PM, Richard Damon wrote:
> On 2/29/24 2:25 PM, WM wrote:
>> Le 29/02/2024 à 13:35, Richard Damon a écrit :
>>> On 2/29/24 3:23 AM, WM wrote:
>>
>>>>
>>>> Visible narural numbers are FISONs {1, 2, 3, ..., n}.
>>>
>>> And so SUBSETS of the Natural Number
>>
>> Of course. What else? Every natural number belongs to the set ℕ.
>
> And evvery natural number is finite and thus namable and thus visible.
>
>>
>>>>> There is actually no problem with those numbers,
>>>>
>>>> What numbers? Infinite natural numbers?
>>>
>>> The unbounded set of Natural Numbers that go on and on and on.
>>
>> Visible numbers do never complete infinity.
>
> Right, but no individual Natural Number does either, so they all can be
> visible.
>
> It is the SET that "completes infinity", and the set isn't any of the
> individual number.
>
>>
>> Regards, WM
>>
>>
>

Why do you keep repeating this if everybody here knows that,
and it's always the case the it just tit-for-tat = for-i-a.
For-git-a-bout-tat, instead of get-tit-for-tat-bout.

A usual idea about the inductive set is that it includes
itself, and yes I know that Russell says it doesn't after
he proves that it does.

The continuum limit of f(n) = n/d in the continuum limit
as d goes to infinity, has that ran(f) has extent, density,
completeness, measure, implies it's a continuous domain,
implies f isn't a Cartesian function, sort of results a
special result between discrete and continuous,
kind of like sliced bread, ....

It's a most usual notion of ordering theory, which is
the theory where ordinals are primary elements instead
of sets being primary elements, to support the time-ordering-property
for the line integral and path integral and the [0,1] of
category theory, and so on, Jordan content, "iota-values".

That is to say, besides you know a usual Riemann, Lebesgue,
Stieltjes, there's Jordan, in measure theory, it's these, ....

Ordinals are primary elements in ordering theory,
sets then implemented in those instead of those in sets,
put them together is sort of how it is, "ubiquitous ordinals",
order type being powerset being successor, this kind of thing.

I imagine and hope you knew that ordering theory has that
ordinals are primary objects in the theory and it's different
than set theory where sets are primary objects in the theory,
and that they're usually enough used to model each other
because they're both so ubiquitous.

Then with regards ".999, read it again", it's like,
you're welcome to read it either way, it being both,
as long as you keep it sorted and straight both ways.

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#625779

Somehow I am quite new to the ordinal analysis
that Alan Turing started. Nice find Gödel introduces
a nice notion in his incompletness paper,

namely he says a function is primitive recursive,
to degree N, if the primitive recursion schema is
applied N times. So I guess this definition of

factorial would then have degree 3, since 'R' is used 3 times:

/* The R combinator */
natrec(_, 0, X, X) :- !.    % attention, not steadfast
natrec(F, N, X, Z) :- M is N-1, natrec(F, M, X, Y), call(F, Y, Z).

plus(X, Y, Z) :- natrec(succ, X, Y, Z).

mult(X, Y, Z) :- natrec(plus(X), Y, 0, Z).

step((X,Y),(Z,T)) :- succ(X,Z), mult(Z,Y,T).

factorial(X,Y) :- natrec(step,X,(0,1),(_,Y)).

Works fine, although eats quite some computing resources,
i.e. 9_303_219 inferences, since it must form
3268800 successors:

?- factorial(10,X).
X = 3628800.

?- time(factorial(10,X)).
% 9,303,219 inferences, 0.578 CPU in 0.617 seconds
(94% CPU, 16092054 Lips)
X = 3628800.

Ross Finlayson schrieb:
> ... fourier my ass, what has it to do with ordinals ...
> 

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#625781

Know that one is the secret and source of all the cardinals.
-- Abraham ibn Ezra (1153)

But have mercy to me. So far I thought in ordinal
anlysis of programs, we would simply take the
tree of execution, and this is somehow an ordinal

for a terminating program? But whats the tree
repectively ordinal for for example 3! = 6 ?
Are all finite ordinals the same or not?

For example the ordinals 0, 1, 2, 3, with von
Neumann succeessor are:

0 = {}
1 = {{}}
2 = {{},{{}}}
3 = {{},{{}},{{{},{{}}}}}.

Or as trees, * = empty set, o = non-empty set:

0 =            *

1 =            o
                |
                *

2 =            o
               / \
              *   o
                  |
                  *


3 =            o
              / |  \
            *   o    o
                |   / \
                *  *   o
                       |
                       *


But what about this tree, it has also no infinite decend,
but what property is missing to make it an ordinal?

? =             o
               / | \
              *  o  o
                 |  |
                 *  o
                    |
                    *

Or as a set:

? = {{},{{}},{{{}}}}.

Why is it not an ordinal?

P.S.: I tried to find an answer here, but I guess
I am too lazy to read it. Its starts with the funny
quote and has funny pictures in it:

Trees, ordinals and termination
N Dershowitz · 1993
https://link.springer.com/content/pdf/10.1007/3-540-56610-4_68.pdf

Mild Shock schrieb:
> Somehow I am quite new to the ordinal analysis
> that Alan Turing started. Nice find Gödel introduces
> a nice notion in his incompletness paper,
> 
> namely he says a function is primitive recursive,
> to degree N, if the primitive recursion schema is
> applied N times. So I guess this definition of
> 
> factorial would then have degree 3, since 'R' is used 3 times:
> 
> /* The R combinator */
> natrec(_, 0, X, X) :- !.    % attention, not steadfast
> natrec(F, N, X, Z) :- M is N-1, natrec(F, M, X, Y), call(F, Y, Z).
> 
> plus(X, Y, Z) :- natrec(succ, X, Y, Z).
> 
> mult(X, Y, Z) :- natrec(plus(X), Y, 0, Z).
> 
> step((X,Y),(Z,T)) :- succ(X,Z), mult(Z,Y,T).
> 
> factorial(X,Y) :- natrec(step,X,(0,1),(_,Y)).
> 
> Works fine, although eats quite some computing resources,
> i.e. 9_303_219 inferences, since it must form
> 3268800 successors:
> 
> ?- factorial(10,X).
> X = 3628800.
> 
> ?- time(factorial(10,X)).
> % 9,303,219 inferences, 0.578 CPU in 0.617 seconds
> (94% CPU, 16092054 Lips)
> X = 3628800.
> 
> Ross Finlayson schrieb:
>> ... fourier my ass, what has it to do with ordinals ...
>>
> 

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#625782

I think the key are these terminological definitions:

"In order theory, a partial order is called well-founded if the 
corresponding strict order is a well-founded relation. If the order is a 
total order then it is called a well-order."
https://en.wikipedia.org/wiki/Well-founded_relation

So my tree "?" in question might have no infinite descend,
but it might not belong to the same total order, as the
other sets. But how exclude "?" ? The criteria of

transitive set is not violated:
trans(A) :<=> ∀ x , y : x ∈ A ∧ y ∈ x ⇒ y ∈ A
https://de.wikipedia.org/wiki/Transitive_Menge

One can easily verify that the above is satisfied by
the set "?". So what is violated? Well this
here is violate, namely each elememt should be

transitive as well, and so on:

"hereditarily transitive sets"
h-trans(A) :<=> trans(A) & ∀x(x ∈ A => h-trans(A))

Because the third branch is not transitive:

 > ? =             o
 >                / | \
 >               *  o  o
 >                  |  |
 >                  *  o
 >                     |
 >                     *
 >
 > Or as a set:
 >
 > ? = {{},{{}},{{{}}}}.

So I remember Jim Burns when he posited a more
general approach, he said transfinite induction
must be satisfied.

Otherwise we can take this Quine atom x = {x},
https://math.stackexchange.com/a/2874533

And by a suitable interpretation of the circular
h-trans definition, a definition that is not well-defined
since it has no unique interpretation,

we might judge this Quine atom an ordinal.

LoL

Mild Shock schrieb:
> 
> Know that one is the secret and source of all the cardinals.
> -- Abraham ibn Ezra (1153)
> 
> But have mercy to me. So far I thought in ordinal
> anlysis of programs, we would simply take the
> tree of execution, and this is somehow an ordinal
> 
> for a terminating program? But whats the tree
> repectively ordinal for for example 3! = 6 ?
> Are all finite ordinals the same or not?
> 
> For example the ordinals 0, 1, 2, 3, with von
> Neumann succeessor are:
> 
> 0 = {}
> 1 = {{}}
> 2 = {{},{{}}}
> 3 = {{},{{}},{{{},{{}}}}}.
> 
> Or as trees, * = empty set, o = non-empty set:
> 
> 0 =            *
> 
> 1 =            o
>                 |
>                 *
> 
> 2 =            o
>                / \
>               *   o
>                   |
>                   *
> 
> 
> 3 =            o
>               / |  \
>             *   o    o
>                 |   / \
>                 *  *   o
>                        |
>                        *
> 
> 
> But what about this tree, it has also no infinite decend,
> but what property is missing to make it an ordinal?
> 
> ? =             o
>                / | \
>               *  o  o
>                  |  |
>                  *  o
>                     |
>                     *
> 
> Or as a set:
> 
> ? = {{},{{}},{{{}}}}.
> 
> Why is it not an ordinal?
> 
> P.S.: I tried to find an answer here, but I guess
> I am too lazy to read it. Its starts with the funny
> quote and has funny pictures in it:
> 
> Trees, ordinals and termination
> N Dershowitz · 1993
> https://link.springer.com/content/pdf/10.1007/3-540-56610-4_68.pdf
> 
> Mild Shock schrieb:
>> Somehow I am quite new to the ordinal analysis
>> that Alan Turing started. Nice find Gödel introduces
>> a nice notion in his incompletness paper,
>>
>> namely he says a function is primitive recursive,
>> to degree N, if the primitive recursion schema is
>> applied N times. So I guess this definition of
>>
>> factorial would then have degree 3, since 'R' is used 3 times:
>>
>> /* The R combinator */
>> natrec(_, 0, X, X) :- !.    % attention, not steadfast
>> natrec(F, N, X, Z) :- M is N-1, natrec(F, M, X, Y), call(F, Y, Z).
>>
>> plus(X, Y, Z) :- natrec(succ, X, Y, Z).
>>
>> mult(X, Y, Z) :- natrec(plus(X), Y, 0, Z).
>>
>> step((X,Y),(Z,T)) :- succ(X,Z), mult(Z,Y,T).
>>
>> factorial(X,Y) :- natrec(step,X,(0,1),(_,Y)).
>>
>> Works fine, although eats quite some computing resources,
>> i.e. 9_303_219 inferences, since it must form
>> 3268800 successors:
>>
>> ?- factorial(10,X).
>> X = 3628800.
>>
>> ?- time(factorial(10,X)).
>> % 9,303,219 inferences, 0.578 CPU in 0.617 seconds
>> (94% CPU, 16092054 Lips)
>> X = 3628800.
>>
>> Ross Finlayson schrieb:
>>> ... fourier my ass, what has it to do with ordinals ...
>>>
>>
> 

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#625786

On 3/1/2024 3:08 PM, Mild Shock wrote:

> So I remember Jim Burns when he posited a more
> general approach, he said transfinite induction
> must be satisfied.

In a phrase I'm a little proud of, I said
transfinite.induction is well.order in drag.
One is a simple re-write of the other.

> Otherwise we can take this Quine atom x = {x},
> https://math.stackexchange.com/a/2874533
>
> And by a suitable interpretation of the circular
> h-trans definition, a definition that is not well-defined
> since it has no unique interpretation,
>
> we might judge this Quine atom an ordinal.
>
> LoL

Ah.
But an ordinal is
a _regular_ transitive set of transitive sets.
So, not x = {x}

A regular non.empty set A holds
a disjoint element B.
∃B ∈ A: A∩B = ∅

But, if A is transitive.transitive,
each element is a subset, and
∀B ∈ A: A∩B = B
Transitive.transitive A can only be regular
if one of its elements is 0

∅ ∈ A  ∧  ∅ ∈ A′ ties all the ordinals together.

It's a beautiful thing.

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#625788

Lets work without regularity axiom, and
examine this naive attempt, hereditary =
my ancestors satisfied it as well:

"hereditarily transitive sets"
h-trans(A) :<=> trans(A) & ∀x(x ∈ A => h-trans(A))

Otherwise when regularity is present,
this excludes Quine atom q = {q}. When regularty is
not present, we can prove:

~h-trans(q)


Jim Burns schrieb:
> On 3/1/2024 3:08 PM, Mild Shock wrote:
> 
>> So I remember Jim Burns when he posited a more
>> general approach, he said transfinite induction
>> must be satisfied.
> 
> In a phrase I'm a little proud of, I said
> transfinite.induction is well.order in drag.
> One is a simple re-write of the other.
> 
>> Otherwise we can take this Quine atom x = {x},
>> https://math.stackexchange.com/a/2874533
>>
>> And by a suitable interpretation of the circular
>> h-trans definition, a definition that is not well-defined
>> since it has no unique interpretation,
>>
>> we might judge this Quine atom an ordinal.
>>
>> LoL
> 
> Ah.
> But an ordinal is
> a _regular_ transitive set of transitive sets.
> So, not x = {x}
> 
> A regular non.empty set A holds
> a disjoint element B.
> ∃B ∈ A: A∩B = ∅
> 
> But, if A is transitive.transitive,
> each element is a subset, and
> ∀B ∈ A: A∩B = B
> Transitive.transitive A can only be regular
> if one of its elements is 0
> 
> ∅ ∈ A  ∧  ∅ ∈ A′ ties all the ordinals together.
> 
> It's a beautiful thing.
> 
> 

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#625789

Corr.: we cannot prove:

~h-trans(q)

See also the remark here by Andrés E. Caicedo:

Note that in the absence of foundation (= regularity),
this is a bit peculiar. For instance, if x={x}, then x
is hereditarily finite, although it does not belong to Vω.)
https://math.stackexchange.com/a/2874533

About digging into "transfinite.induction is well.order in drag"
by Jim Burns. You probably mean transfinite.induction
follows from well.order. What about the other direction?

Now my question, is assume we have no foundation,
but epsilon induction, what will happen. epsilon
induction is usually not an axiom. But what

if we stipulate it as an axiom?

Considered as an axiomatic principle, it is
called the axiom schema of set induction.
∀ x . ( ( ∀ ( y ∈ x ) . ψ ( y ) ) → ψ ( x ) ) → ∀ z . ψ ( z )
https://en.wikipedia.org/wiki/Epsilon-induction

Mild Shock schrieb:
> Lets work without regularity axiom, and
> examine this naive attempt, hereditary =
> my ancestors satisfied it as well:
> 
> "hereditarily transitive sets"
> h-trans(A) :<=> trans(A) & ∀x(x ∈ A => h-trans(A))
> 
> Otherwise when regularity is present,
> this excludes Quine atom q = {q}. When regularty is
> not present, we can prove:
> 
> ~h-trans(q)
> 
> 
> Jim Burns schrieb:
>> On 3/1/2024 3:08 PM, Mild Shock wrote:
>>
>>> So I remember Jim Burns when he posited a more
>>> general approach, he said transfinite induction
>>> must be satisfied.
>>
>> In a phrase I'm a little proud of, I said
>> transfinite.induction is well.order in drag.
>> One is a simple re-write of the other.
>>
>>> Otherwise we can take this Quine atom x = {x},
>>> https://math.stackexchange.com/a/2874533
>>>
>>> And by a suitable interpretation of the circular
>>> h-trans definition, a definition that is not well-defined
>>> since it has no unique interpretation,
>>>
>>> we might judge this Quine atom an ordinal.
>>>
>>> LoL
>>
>> Ah.
>> But an ordinal is
>> a _regular_ transitive set of transitive sets.
>> So, not x = {x}
>>
>> A regular non.empty set A holds
>> a disjoint element B.
>> ∃B ∈ A: A∩B = ∅
>>
>> But, if A is transitive.transitive,
>> each element is a subset, and
>> ∀B ∈ A: A∩B = B
>> Transitive.transitive A can only be regular
>> if one of its elements is 0
>>
>> ∅ ∈ A  ∧  ∅ ∈ A′ ties all the ordinals together.
>>
>> It's a beautiful thing.
>>
>>
> 

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#625790

My numb nut rewriting faculty, after spying
wikipeda, takes the contrapositive (i.e replace
A->B by ~B -> ~A) of the set induction axiom:

¬ ∀ z ψ ( z )  →  ¬ ∀ x ( ( ∀ y ( y ∈ x  → ψ ( y ) ) → ψ ( x ) )

Now use for ψ ( z ) the formula ¬ z ∈ u, and one gets:

∃ z z ∈ u   →  ∃ x ¬ ( ∀ y ( y ∈ x  → ¬ y ∈ u ) → ¬ x ∈ u )

Again some contrapositive:

∃ z z ∈ u   →  ∃ x ¬ ( x ∈ u  →  ¬ ∀ y ( y ∈ x  → ¬ y ∈ u ))

And hence:

∃ z z ∈ u   →  ∃ x ( x ∈ u  ∧  ∀ y ( y ∈ x  → ¬ y ∈ u ))

Some last quantifier switch, and we got the regularity axiom:

∃ z z ∈ u   →  ∃ x ( x ∈ u  ∧  ∃ y ( y ∈ x  ∧ y ∈ u ))

Usually written as:

u ≠ ∅  →  ∃ x (x ∈ u  ∧  x ∩ u ≠ ∅)

Mild Shock schrieb:
> Corr.: we cannot prove:
> 
> ~h-trans(q)
> 
> See also the remark here by Andrés E. Caicedo:
> 
> Note that in the absence of foundation (= regularity),
> this is a bit peculiar. For instance, if x={x}, then x
> is hereditarily finite, although it does not belong to Vω.)
> https://math.stackexchange.com/a/2874533
> 
> About digging into "transfinite.induction is well.order in drag"
> by Jim Burns. You probably mean transfinite.induction
> follows from well.order. What about the other direction?
> 
> Now my question, is assume we have no foundation,
> but epsilon induction, what will happen. epsilon
> induction is usually not an axiom. But what
> 
> if we stipulate it as an axiom?
> 
> Considered as an axiomatic principle, it is
> called the axiom schema of set induction.
> ∀ x . ( ( ∀ ( y ∈ x ) . ψ ( y ) ) → ψ ( x ) ) → ∀ z . ψ ( z )
> https://en.wikipedia.org/wiki/Epsilon-induction
> 
> Mild Shock schrieb:
>> Lets work without regularity axiom, and
>> examine this naive attempt, hereditary =
>> my ancestors satisfied it as well:
>>
>> "hereditarily transitive sets"
>> h-trans(A) :<=> trans(A) & ∀x(x ∈ A => h-trans(A))
>>
>> Otherwise when regularity is present,
>> this excludes Quine atom q = {q}. When regularty is
>> not present, we can prove:
>>
>> ~h-trans(q)
>>
>>
>> Jim Burns schrieb:
>>> On 3/1/2024 3:08 PM, Mild Shock wrote:
>>>
>>>> So I remember Jim Burns when he posited a more
>>>> general approach, he said transfinite induction
>>>> must be satisfied.
>>>
>>> In a phrase I'm a little proud of, I said
>>> transfinite.induction is well.order in drag.
>>> One is a simple re-write of the other.
>>>
>>>> Otherwise we can take this Quine atom x = {x},
>>>> https://math.stackexchange.com/a/2874533
>>>>
>>>> And by a suitable interpretation of the circular
>>>> h-trans definition, a definition that is not well-defined
>>>> since it has no unique interpretation,
>>>>
>>>> we might judge this Quine atom an ordinal.
>>>>
>>>> LoL
>>>
>>> Ah.
>>> But an ordinal is
>>> a _regular_ transitive set of transitive sets.
>>> So, not x = {x}
>>>
>>> A regular non.empty set A holds
>>> a disjoint element B.
>>> ∃B ∈ A: A∩B = ∅
>>>
>>> But, if A is transitive.transitive,
>>> each element is a subset, and
>>> ∀B ∈ A: A∩B = B
>>> Transitive.transitive A can only be regular
>>> if one of its elements is 0
>>>
>>> ∅ ∈ A  ∧  ∅ ∈ A′ ties all the ordinals together.
>>>
>>> It's a beautiful thing.
>>>
>>>
>>
> 

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#625797

On 3/1/2024 5:11 PM, Mild Shock wrote:

> About digging into
> "transfinite.induction is well.order in drag"
> by Jim Burns. You probably mean
> transfinite.induction follows from well.order.
> What about the other direction?

Both directions.

well.order

∃ᵒʳᵈγ:p(γ)  ⟹  ∃#1ᵒʳᵈβ:p(β)

∃ᵒʳᵈγ:p(γ)  ⟹  ∃ᵒʳᵈβ:(p(β) ∧ ¬∃ᵒʳᵈα<β:p(α))

¬∃ᵒʳᵈβ:(p(β) ∧ ¬∃ᵒʳᵈα<β:p(α))  ⟹  ¬∃ᵒʳᵈγ:p(γ)

∀ᵒʳᵈβ:(¬p(β) ∨ ¬∀ᵒʳᵈα<β:¬p(α))  ⟹  ∀ᵒʳᵈγ:¬p(γ)

∀ᵒʳᵈβ:(​̅p(β) ∨ ¬∀ᵒʳᵈα<β:​̅p(α))  ⟹  ∀ᵒʳᵈγ:​̅p(γ)

∀ᵒʳᵈβ:(∀ᵒʳᵈα<β:​̅p(α) ⇒ ​̅p(β))  ⟹  ∀ᵒʳᵈγ:​̅p(γ)

∀ᵒʳᵈβ:(​̅p[0,β) ⇒ ​̅p(β))  ⟹  ∀ᵒʳᵈγ:​̅p(γ)

∀ᵒʳᵈβ:(​̅p[0,β) ⇒ ​̅p[0,β)∧​̅p(β))  ⟹  ∀ᵒʳᵈγ:​̅p(γ)

∀ᵒʳᵈβ:(​̅p[0,β) ⇒ ​̅p[0,β⁺¹))  ⟹  ∀ᵒʳᵈγ:​̅p(γ)

transfinite.induction

> Now my question, is assume we have no foundation,
> but epsilon induction, what will happen. epsilon
> induction is usually not an axiom. But what
> if we stipulate it as an axiom?
>
> Considered as an axiomatic principle, it is
> called the axiom schema of set induction.
> ∀x.((∀(y∈x).ψ(y))→ψ(x))→∀z.ψ(z)
> https://en.wikipedia.org/wiki/Epsilon-induction

That wiki.page assumes regular sets.
I'm not sure it does so explicitly,
but it defines ordinals as
transitive sets of transitive sets
which, without regularity, include x = {x}

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#625798

On 03/01/2024 04:41 PM, Jim Burns wrote:
> On 3/1/2024 5:11 PM, Mild Shock wrote:
>
>> About digging into
>> "transfinite.induction is well.order in drag"
>> by Jim Burns. You probably mean
>> transfinite.induction follows from well.order.
>> What about the other direction?
>
> Both directions.
>
> well.order
>
> ∃ᵒʳᵈγ:p(γ)  ⟹  ∃#1ᵒʳᵈβ:p(β)
>
> ∃ᵒʳᵈγ:p(γ)  ⟹  ∃ᵒʳᵈβ:(p(β) ∧ ¬∃ᵒʳᵈα<β:p(α))
>
> ¬∃ᵒʳᵈβ:(p(β) ∧ ¬∃ᵒʳᵈα<β:p(α))  ⟹  ¬∃ᵒʳᵈγ:p(γ)
>
> ∀ᵒʳᵈβ:(¬p(β) ∨ ¬∀ᵒʳᵈα<β:¬p(α))  ⟹  ∀ᵒʳᵈγ:¬p(γ)
>
> ∀ᵒʳᵈβ:(​̅p(β) ∨ ¬∀ᵒʳᵈα<β:​̅p(α))  ⟹  ∀ᵒʳᵈγ:​̅p(γ)
>
> ∀ᵒʳᵈβ:(∀ᵒʳᵈα<β:​̅p(α) ⇒ ​̅p(β))  ⟹  ∀ᵒʳᵈγ:​̅p(γ)
>
> ∀ᵒʳᵈβ:(​̅p[0,β) ⇒ ​̅p(β))  ⟹  ∀ᵒʳᵈγ:​̅p(γ)
>
> ∀ᵒʳᵈβ:(​̅p[0,β) ⇒ ​̅p[0,β)∧​̅p(β))  ⟹  ∀ᵒʳᵈγ:​̅p(γ)
>
> ∀ᵒʳᵈβ:(​̅p[0,β) ⇒ ​̅p[0,β⁺¹))  ⟹  ∀ᵒʳᵈγ:​̅p(γ)
>
> transfinite.induction
>
>> Now my question, is assume we have no foundation,
>> but epsilon induction, what will happen. epsilon
>> induction is usually not an axiom. But what
>> if we stipulate it as an axiom?
>>
>> Considered as an axiomatic principle, it is
>> called the axiom schema of set induction.
>> ∀x.((∀(y∈x).ψ(y))→ψ(x))→∀z.ψ(z)
>> https://en.wikipedia.org/wiki/Epsilon-induction
>
> That wiki.page assumes regular sets.
> I'm not sure it does so explicitly,
> but it defines ordinals as
> transitive sets of transitive sets
> which, without regularity, include x = {x}
>
>
>




Like a process in time with no beginning?

The other day James Webb Space Telescope,
roundly paintcanned inflationary cosmology,
yet, even before that, the sky survey was measuring
the age of the universe every few years, and every
few years, it got hundreds of millions of years older.

Then the JWST sort of arrived at "you know it
really looks like we might have to start counting
over".

"Epoch".

A complementary notion to "Big Bang cosmology",
is, "Steady State cosmology", or, sitting next to
"Big Bang cosmology", "cyclic cosmology".

Two principles of theories of physics include
the dichotomy of unitarity and complementarity,
which is funny because one is without dichotomy
and the other is with.

2020/1/1 - 18262 = 1970/1/1

Time, then, reflects upon these foundational theories,
and anti-foundational theories, in simile, to Big Bang
theory, and Steady State theory.

Now, since scientism and logical positivism, Compte
and Boole and the Vienna circle, and Zermelo Fraenkel
and Le Maitre, with ZF set theory and Big Bang theory,
it was exactly about a century ago.  1920: a century
of hindsight, retrospect, from 2020.

So, the idea for delta-epsilonics toward zero, but
not crossing it, and least upper bound, then relying
on symmetry, sort of has the symmetry about the
origin exists for the symmetry about the origin to
exist.

Before DesCartes, one might aver the Euclid's theory,
geometry, was a bit free-er, geometry:  do it anywhere
you want.  Then, the notion of equipping the Euclidean
space, with a Cartesian space, makes exactly for the
notion of the ordinate itself, the ordinates and the
abscissae, which run or drop from the origin its axes,
to the curve its intercept, in these discussions of ordinals,
see arrive the notion of the ordinate, and the co-ordinates,
and that in all our real analysis, it's always based on the
co-ordinates.


So, how can there be negative numbers when first
the numbers must go all the way to infinity, if they
never reach it?  The usual idea is that the comprehension
just goes ... around.

Then for DesCartes that introduces his notions of the
vortex everywhere, as anticipating particle/wave duality
with an implicit atomism, and for example:  the spiral,
space-filling curve, an aspect of a:  continuum.

One might imagine a faithful as possible simulacrum
of Einstein, a Zweistein:  "I'm not making you say
the universe is infinite, but Space-Time is in a
continuous manifold".

So, unitarity and complementarity, the principles of
physics, sort of have the same notions in mathematics.





Ordinals:  meet ordinates.

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#625811

On 03/01/2024 08:15 PM, Ross Finlayson wrote:
> On 03/01/2024 04:41 PM, Jim Burns wrote:
>> On 3/1/2024 5:11 PM, Mild Shock wrote:
>>
>>> About digging into
>>> "transfinite.induction is well.order in drag"
>>> by Jim Burns. You probably mean
>>> transfinite.induction follows from well.order.
>>> What about the other direction?
>>
>> Both directions.
>>
>> well.order
>>
>> ∃ᵒʳᵈγ:p(γ)  ⟹  ∃#1ᵒʳᵈβ:p(β)
>>
>> ∃ᵒʳᵈγ:p(γ)  ⟹  ∃ᵒʳᵈβ:(p(β) ∧ ¬∃ᵒʳᵈα<β:p(α))
>>
>> ¬∃ᵒʳᵈβ:(p(β) ∧ ¬∃ᵒʳᵈα<β:p(α))  ⟹  ¬∃ᵒʳᵈγ:p(γ)
>>
>> ∀ᵒʳᵈβ:(¬p(β) ∨ ¬∀ᵒʳᵈα<β:¬p(α))  ⟹  ∀ᵒʳᵈγ:¬p(γ)
>>
>> ∀ᵒʳᵈβ:(​̅p(β) ∨ ¬∀ᵒʳᵈα<β:​̅p(α))  ⟹  ∀ᵒʳᵈγ:​̅p(γ)
>>
>> ∀ᵒʳᵈβ:(∀ᵒʳᵈα<β:​̅p(α) ⇒ ​̅p(β))  ⟹  ∀ᵒʳᵈγ:​̅p(γ)
>>
>> ∀ᵒʳᵈβ:(​̅p[0,β) ⇒ ​̅p(β))  ⟹  ∀ᵒʳᵈγ:​̅p(γ)
>>
>> ∀ᵒʳᵈβ:(​̅p[0,β) ⇒ ​̅p[0,β)∧​̅p(β))  ⟹  ∀ᵒʳᵈγ:​̅p(γ)
>>
>> ∀ᵒʳᵈβ:(​̅p[0,β) ⇒ ​̅p[0,β⁺¹))  ⟹  ∀ᵒʳᵈγ:​̅p(γ)
>>
>> transfinite.induction
>>
>>> Now my question, is assume we have no foundation,
>>> but epsilon induction, what will happen. epsilon
>>> induction is usually not an axiom. But what
>>> if we stipulate it as an axiom?
>>>
>>> Considered as an axiomatic principle, it is
>>> called the axiom schema of set induction.
>>> ∀x.((∀(y∈x).ψ(y))→ψ(x))→∀z.ψ(z)
>>> https://en.wikipedia.org/wiki/Epsilon-induction
>>
>> That wiki.page assumes regular sets.
>> I'm not sure it does so explicitly,
>> but it defines ordinals as
>> transitive sets of transitive sets
>> which, without regularity, include x = {x}
>>
>>
>>
>
>
>
>
> Like a process in time with no beginning?
>
> The other day James Webb Space Telescope,
> roundly paintcanned inflationary cosmology,
> yet, even before that, the sky survey was measuring
> the age of the universe every few years, and every
> few years, it got hundreds of millions of years older.
>
> Then the JWST sort of arrived at "you know it
> really looks like we might have to start counting
> over".
>
> "Epoch".
>
> A complementary notion to "Big Bang cosmology",
> is, "Steady State cosmology", or, sitting next to
> "Big Bang cosmology", "cyclic cosmology".
>
> Two principles of theories of physics include
> the dichotomy of unitarity and complementarity,
> which is funny because one is without dichotomy
> and the other is with.
>
> 2020/1/1 - 18262 = 1970/1/1
>
> Time, then, reflects upon these foundational theories,
> and anti-foundational theories, in simile, to Big Bang
> theory, and Steady State theory.
>
> Now, since scientism and logical positivism, Compte
> and Boole and the Vienna circle, and Zermelo Fraenkel
> and Le Maitre, with ZF set theory and Big Bang theory,
> it was exactly about a century ago.  1920: a century
> of hindsight, retrospect, from 2020.
>
> So, the idea for delta-epsilonics toward zero, but
> not crossing it, and least upper bound, then relying
> on symmetry, sort of has the symmetry about the
> origin exists for the symmetry about the origin to
> exist.
>
> Before DesCartes, one might aver the Euclid's theory,
> geometry, was a bit free-er, geometry:  do it anywhere
> you want.  Then, the notion of equipping the Euclidean
> space, with a Cartesian space, makes exactly for the
> notion of the ordinate itself, the ordinates and the
> abscissae, which run or drop from the origin its axes,
> to the curve its intercept, in these discussions of ordinals,
> see arrive the notion of the ordinate, and the co-ordinates,
> and that in all our real analysis, it's always based on the
> co-ordinates.
>
>
> So, how can there be negative numbers when first
> the numbers must go all the way to infinity, if they
> never reach it?  The usual idea is that the comprehension
> just goes ... around.
>
> Then for DesCartes that introduces his notions of the
> vortex everywhere, as anticipating particle/wave duality
> with an implicit atomism, and for example:  the spiral,
> space-filling curve, an aspect of a:  continuum.
>
> One might imagine a faithful as possible simulacrum
> of Einstein, a Zweistein:  "I'm not making you say
> the universe is infinite, but Space-Time is in a
> continuous manifold".
>
> So, unitarity and complementarity, the principles of
> physics, sort of have the same notions in mathematics.
>
>
>
>
>
> Ordinals:  meet ordinates.
>
>

Of course ordinals are about the slenderest,
leanest, most minimal sort of sets that establish
a course-of-passage, as what it's usually called
passing through ordinals, of infinite induction,
according to their structure, their form, their
content, their model.

For example they have almost none of the
modular character of the integers, though
each is different and they're ordered, it
involves counting back-and-forth and
up-and-down and building a mememory,
a structure the content the relation embodied,
form their scaffold a model, to relate a
model of _some_ ordinals, to a model
of _some_ integers.

So, in this way building in a usual way
models of abstract algebra's finite fields,
then on up, then on out, and for each of
those all theirs, sort of results that
_eventually_ then there's a huge structure
of all their intra-relation, "integers".

You might wonder that cardinals, in
set theory, maybe they're a little fuller?
A cardinal is an equivalence class of
_all the sets in the set-theoretic universe_
that in _all the models of functions among
those in the set-theoretic universe_ that
_all those functions indicating 1-1 and onto_
that those existing from a proto-typical
set with an element after zero's:  equals
cardinal "1".

So, ordinals, basically got nothing, in
structure, except next, while cardinals,
are all the structure there is that in any
way relates anything at all a set, to a
set of prototypical elements, after
cardinal "0", as it were, cardinal "1".


Then, _ordinates_, are, way, way after that,
or for analytical geometers in their theory
kind of before, that way, way after there's
Euclidean geometry, then a Cartesian basis,
an origin and axes, with a metric on those
so it's a space and norm among those so
it's an orthonormal basis, the ordinates
are arrayed on out and down all those,
also implementing ordinals.

So, ordinals, meet ordinates, ordinates, ordinals.

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#625812

Jim Burns schrieb:
>> Considered as an axiomatic principle, it is
>> called the axiom schema of set induction.
>> ∀x.((∀(y∈x).ψ(y))→ψ(x))→∀z.ψ(z)
>> https://en.wikipedia.org/wiki/Epsilon-induction
> 
> That wiki.page assumes regular sets.

Well I showed classicaly that the set induction
axiom schema implies the regularity axiom:

Mild Shock schrieb:
 > My numb nut rewriting faculty, after spying
 >
 > /* set induction */
 > ¬ ∀ z ψ ( z )  →  ¬ ∀ x ( ( ∀ y ( y ∈ x  → ψ ( y ) ) → ψ ( x ) )
 >
 > |
 > |
 > v
 >
 > /* set induction */
 > u ≠ ∅  →  ∃ x (x ∈ u  ∧  x ∩ u ≠ ∅)

The only problem with this derivation, it
might not be intuitionistically valid. I might
have used a propositional or quantifier rules,

which are not accepted intuitionistically.

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#625794

On 03/01/2024 11:38 AM, Mild Shock wrote:
> Somehow I am quite new to the ordinal analysis
> that Alan Turing started. Nice find Gödel introduces
> a nice notion in his incompletness paper,
>
> namely he says a function is primitive recursive,
> to degree N, if the primitive recursion schema is
> applied N times. So I guess this definition of
>
> factorial would then have degree 3, since 'R' is used 3 times:
>
> /* The R combinator */
> natrec(_, 0, X, X) :- !.    % attention, not steadfast
> natrec(F, N, X, Z) :- M is N-1, natrec(F, M, X, Y), call(F, Y, Z).
>
> plus(X, Y, Z) :- natrec(succ, X, Y, Z).
>
> mult(X, Y, Z) :- natrec(plus(X), Y, 0, Z).
>
> step((X,Y),(Z,T)) :- succ(X,Z), mult(Z,Y,T).
>
> factorial(X,Y) :- natrec(step,X,(0,1),(_,Y)).
>
> Works fine, although eats quite some computing resources,
> i.e. 9_303_219 inferences, since it must form
> 3268800 successors:
>
> ?- factorial(10,X).
> X = 3628800.
>
> ?- time(factorial(10,X)).
> % 9,303,219 inferences, 0.578 CPU in 0.617 seconds
> (94% CPU, 16092054 Lips)
> X = 3628800.
>
> Kimono Rictus erupted while counterfeiting:
>> ... fourier my ass, what has it to do with ordinals ...
>>
>


Joseph Fourier?  Fourier is famed for the Fourier-style
analysis, particularly the particular Fourier analysis.
Dirichlet and Fejer employ trigonometry.

One of it's most well-known counterparts is the
Taylor-style analysis, up after l'Hospitale's rule,
Rolle's theorem, up into Rodriguez formula.

The idea of putting them together about a
sort of Clairaut-MacLaurin for Fourier-Taylor,
about the zeros, is for that it's quite modern,
these sorts approaches.


When one mentions ordinals, as we've been
discussing ordering theory and the order-type
of ordinals and ORD the order-type of ordinals,
Cesare Burali-Forti's as it were, that the order,
the rulial, the regular, also lends to association,
the ordinance, or local laws, and the ordnance,
or, local laws.


So, when you say ORD, is it, loaded?  The term?

Here it's got all the ordinals in it and contains itself.



If you didn't already know the theory of ordinals,
it's its own sort of primary element in the universe
of mathematical objects as for a theory by there being
a structure, a model theory, a model of same - ordinals,
and I suppose I've used the phrase "ubiquitous ordinals"
at least for twenty years.

It's a theory with numbers in it?

The ordering theory the axiomatic sub-field, has picked
up a lot of ground over the past decades, I don't much
know its particulars, except as with regards to that
the clock-arithmetic it makes for the modular, just like
the iota-values going zero to one, is about exactly how
it's done, here with regards usually enough to category
theory [0,1].

Some people use it instead of set theory,
it suffices for their needs.


Hey Burse, don't be counterfeiting.  It's not just that
I don't like it, but governments and syndicates don't like it.
The law don't like it.

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#625766

Le 01/03/2024 à 04:12, Richard Damon a écrit :

> And evvery natural number is finite and thus namable and thus visible.

That concerns potential infinity only.

> It is the SET that "completes infinity", and the set isn't any of the 
> individual number.

The set is nothig but the collection of its elements. The complete set 
requires that no element is missing. That proves, via ∀n ∈ ℕ: 1/n - 
1/(n+1) = d_n > 0, the existence of a smallest unit fraction.

Regards, WM
 

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