On 10/7/2022 9:59 AM, Timothy Golden wrote:
> On Thursday, October 6, 2022
> at 2:48:55 PM UTC-4, Jim Burns wrote:
>> On 10/6/2022 9:56 AM, Timothy Golden wrote:
>>> On Wednesday, October 5, 2022
>>> at 4:09:01 PM UTC-4, Jim Burns wrote:

>>>> It's a non-circular description of
>>>> _being finite_
>>>
>>> I don't know that you can find any instance
>>> that breaks your system of splits though.
>>
>> You're asking what _doesn't_ have
>> last-before and first-after for each split,
>> do I have that right?

>> Also also,
>> the finite and transfinite ordinals
>> 0,1,2,3,...;ω,ω+1,...
>>
>> ω is first-after all the finite ordinals,
>> but, for finite k, k+1 is finite too,
>> so there is no last-before ω
>
> I am amazed that you accept this omega
> interpretation at all.

I speculate that what you (TG) call the halting
problem of '...' is
the problem of accepting that there is no
last natural number, or last Fibonaccit number,
or ..., or ..., or ...

> I suppose where it falls apart is on the
> assumption that there is a first infinite value.

What reason do we have to think that
there is a first infinite value?

Let 𝐹ₙ be a finite sequence of naturals,
skipping none, from 0 to n.

𝐹ₙ is ordered by '<'
𝐹ₙ begins at 0 and ends at n.
For each split 𝐵,𝐴 of 𝐹ₙ
some j ends 𝐵
some k begins 𝐴
and k = j+1

When I say that 𝐹 is
a finite sequence of naturals,
skipping none, from 0 to somewhere,
that is what I mean.

Let 𝓕 be the set of all 𝐹
the set of all finite sequences 𝐹 of naturals,
skipping none, from 0 to somewhere.

⋃𝓕 is the union of all finite sequences.
If n ∈ ⋃𝓕 then n ends 𝐹ₙ ∈ 𝓕
If n ends 𝐹ₙ ∈ 𝓕 then n ∈ ⋃𝓕

> I suppose where it falls apart is on the
> assumption that there is a first infinite value.

⋃𝓕 is the first infinite set after (⊂)
each 𝐹 in 𝓕, and each 𝐹 is finite.

By "first after", I mean
each 𝐹 is a subset of ⋃𝓕
and
no 'earlier' subset 𝐶 of ⋃𝓕 exists
for which each 𝐹 is a subset of 𝐶

> The sort of breakage that allows for
> many infinities denies a first infinity.
> As if ω-1 does not exist.

ω-1 does not exist.

> I do think it is possible to implement
> a largest natural value that then forms a 'school'.

A largest natural does not exist.

> For instance those concerned with large numbers
> should not define large numbers with large numbers.

We can define the set ℕ of natural numbers this way:

∀k : k ≠ 0
∀k : (∀j ≠ k : k+1 ≠ j+1)

Inductive(𝐶) ⟺
0 ∈ 𝐶 ∧ ∀k ∈ 𝐶 : k+1 ∈ 𝐶

Inductive(ℕ) ∧
~∃𝐶 ⫋ ℕ : Inductive(𝐶)

I don't know why that would be
"defining with large numbers" but
I don't know what that would mean, so ?

>> The idea of internal notation sounds nice.
>> I don't think I can implement it on USEnet,
>> but maybe on a website of some kind.
>
> Certainly, and the disclosure of the algorithm
> really should come clean informationally,
> which I guess means notationally.

| "In this paper we have proposed a mechanism for
| allowing ellipsis in automatic proofs."

This suggests to me that the authors are using
a _declarative_ programming language, and not
an _imperative_ programming language.

Your use of the word "algorithm" and other
comments you've made suggest to me that you are
stuck in imperative-programming-language mode.

https://en.wikipedia.org/wiki/Declarative_programming
| | Many languages that apply this style attempt to
| minimize or eliminate side effects by describing
| what the program must accomplish in terms of the
| problem domain, rather than describe how to accomplish
| it as a sequence of the programming language
| primitives (the how being left up to the language's
| implementation). This is in contrast with imperative
| programming, which implements algorithms in explicit
| steps.

> The natural numbers have a halting problem.

The natural numbers do not halt.
Since this is not a problem,
they do not have a halting problem.

> It is funny really, that to defeat the Peano axioms
> we need only present a large value and
> ask whether it is natural.

That's not how anything works.

If i, j and k refer to natural numbers,
then the Peano axioms are true of them.
because
that (or something leading to that)
is _what we mean_ by "natural number".

On Friday, October 7, 2022 at 9:59:19 AM UTC-4, Timothy Golden wrote:
> On Thursday, October 6, 2022 at 2:48:55 PM UTC-4, Jim Burns wrote:
> > On 10/6/2022 9:56 AM, Timothy Golden wrote:
> > > On Wednesday, October 5, 2022
> > > at 4:09:01 PM UTC-4, Jim Burns wrote:
> > >> On 10/5/2022 9:23 AM, Timothy Golden wrote:
> >
> > >>> It's crystal clear that 333...34 is not
> > >>> a finite value.
> > >>> I think that you think that you've short-cut
> > >>> that process with this 'split' claim,
> > >>> which I still don't really understand.
> > >>
> > >> It's a non-circular description of
> > >> _being finite_
> > >
> > > I don't know that you can find any instance
> > > that breaks your system of splits though.
> > You're asking what _doesn't_ have
> > last-before and first-after for each split,
> > do I have that right?
> >
> > Infinite things. 333...34, for example.
> >
> > 333...34 might be a less-than-satisfying example
> > because I only know it because you told me.
> >
> > But how I read your "infinite" is that,
> > for some split of 0,1,...,333...34,
> > there ISN'T a last-before and a first-after.
> > You might like to know that.
> >
> > Also,
> > the rationals from 0 to 1, in their standard
> > order, for another example.
> >
> > Consider any split 𝐵,𝐴 of ℚ[0,1]
> > There isn't a last-before p/q of 𝐵
> > and a first-after r/s of 𝐴
> > If there were, then their midpoint (p/q+r/s)/2
> > would not be in either 𝐵 or 𝐴
> > and 𝐵,𝐴 would not be a split.
> >
> > I specified _standard order_ because
> > there is a non-standard order of ℚ⁺
> > (due to Georg Cantor) in which splits _do_
> > have last-before and first-after.
> > 1/1 1/2 2/1 1/3 2/2 3/1 1/4 ...
> >
> > However, in Cantor's order, there is no second
> > end. Finite requires steps across splits
> > and two ends. ℚ⁺ is not finite in that order,
> > either, but for a different reason.
> >
> > Also also,
> > the finite and transfinite ordinals
> > 0,1,2,3,...;ω,ω+1,...
> >
> > ω is first-after all the finite ordinals,
> > but, for finite k, k+1 is finite too,
> > so there is no last-before ω
> I am amazed that you accept this omega interpretation at all.
> I suppose where it falls apart is on the assumption that there is a first infinite value.
> The sort of breakage that allows for many infinities denies a first infinity.
> As if ω-1 does not exist.
> I do think it is possible to implement a largest natural value that then forms a 'school'.
> For instance those concerned with large numbers should not define large numbers with large numbers.
> In this regard constituents of large numbers that exceed 9, which is the last singular glyph, should not be allowed in.
> for instance the polynomial/number relationship should not go past 9x10^9.. This then yields a largest value of 9999999999.
> This then establishes an ω=10000000000. The logic of disambiguation still shows the usage of '10' within the polynomial interpretation, which is arguably a large number, but that the string of glyphs contains an unusable termination character 1234567890A, and this 'A' is for 'aleph'; this would form a consistent language of number. The invisible a and the omega all in use here. The polynomial can now read:
> Sum( d[n] A^n ) where n ranges from 0 to A. This first school only needs to graduate upon the need of a larger digit.
I wonder whether the usage of the exponential operator should be suspended since we're trying for simplicity. I don't think we can do without the product though unless we literally engage interconnected wheels of digits, which is very Peano. Maybe through harmonics some sort of primitive wave version of number is available here. The more digits the more harmonics. Then at least we can get the Piano axioms going. That will be confusing. Mathematicians will love it.

>
> This does interact with infinite interpretations. The desire to philosophically define the natural value as unbounded, as for instance some greek who determines that materials must be made of atoms but that the atoms must be infinite in quantity in order to explain the world. Meanwhile the modern interpretation puts the number of atoms in the uiniverse at 10^82. Here is a value that would require the next school of interpretation, which will use the largest value of the first school as its constructive limit; an aleph two; maybe a B using the notation that I've provided:
> 1,2,3, ..., 9, A. school one.
> 1,2,3,...,10, 11, 12, ... 21, 22, 23, ..., 97, 98, 99, ... , 9999999997,9999999998, 9999999999=10^A-1, B. school two.
> 1,2,3..., 9999999999, 10000000000, 10000000001, ... 10^B - 1, C. school three.
> As we breach the first school and realize that the ambiguity of large numbers defined with large numbers does not actually break anything we see that the modern system can hold up but for the ambiguity of defining large numbers with large numbers. Still, as mathematicians attempting to approach perfection this schooling is necessary for the numbers, and in practice working directly with Peano will never get out of the first school

Errr. Second school, I mean, by the notation above.

> anyways. Well, with gigahertz processors the machines can, but the human cannot. This of course is due to the successor problem, which is the slow way to infinity. I doubt if our modern processors will make it out of the third school any time soon. As we slice things up into discrete chunks it will always make sense to use the least amount that gets reasonable accuracy. It would be foolhardy to demand a fourth school system for instance that nobody will use. It may exist, but that does not make it usable.
>
I wonder if it could be admitted that all works which lack instantiation could be carried out in the first school. Literally on a playing field of nine glyphs. Since the variables go uninstantiated this may be possible. Those who concern themselves with large numbers will be very impressive to say the least.

> Now the large number ambiguity has been brought back to a practical analysis. Maybe my alephs are your omegas; fine. Those who dodge the fundamental ambiguities of number as if they have risen beyond school two are fibbing. Especially those holding so tightly to Peano's method.
> So this forms a new joke in mathematics: What is broken that does not need fixing? Numbers. The interpretation goes back to physical correspondence: you can present me with a count of some sheep, but that is completely different than presenting me the sheep. That this is the ultimate lie of the number is actionable. It is almost a call to mathematicians to drop you paper and pencil and get out into the real world a bit. Work with some material in your hands and witness the puzzles of existence at a more direct level.
> >
> > > https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.108.3102&rep=rep1&type=pdf
> > > "The key idea is to use an internal notation
> > > in which the ambiguity inherent in elliptic
> > > notation is resolved."
> > Nice. I wish them the best with their proposal.
> >
> > I tried to do something similar here, though in a
> > fairly kludged fashion.
> >
> > For a while, until I decided it wasn't helping,
> > I wrote the finite natural sequence ⟨0⋯i,i+1⋯k⟩
> > The idea is that we need to specify the start and
> > stop of the sequence, _and how to cross the splits_
> >
> > In some _other_ sequence,
> > we would specify a different way to cross
> > the splits.
> > Someone familiar with the Fibonacci sequence
> > ⟨ 0 1 1 2 3 5 8 13 ⋯ ⟩
> > would not need to be told what '⋯' means here,
> > but we could also state it explicitly
> > ⟨ 0 1 ⋯ fᵢ₋₁ fᵢ fᵢ₋₁+fᵢ ⋯ ⟩
> >
> > The idea of internal notation sounds nice.
> > I don't think I can implement it on USEnet,
> > but maybe on a website of some kind.
> Certainly, and the disclosure of the algorithm really should come clean informationally, which I guess means notationally.
> some want to dodge notation and convention as if they are not a part of the actual workings of the system.
> The adaptive human is capable of adopting deeply nested exceptions. It is perhaps in the DNA of the social animal to do this.
> It is highly unmathematical, however. To claim to be a mathematician is to write yourself a ticket away from it, but this claim is a lie.
> Spock, who we used to worship as kids, for instance. I'd rather deal with Fibonacci than a fibbing Nazi, but it seems we are going to have to deal with the latter soon. They already are adapting us into it.
> >
> > > [...], whereas our numbers as sequences of digits
> > > start with their largest first.
> > Note that Arabic and Hebrew read right-to-left,
> > so it would be the other way around for readers
> > of those languages.
> > > suppose the digits of pi were satisfied by
> > > this format. Then we'd be chasing digits from
> > > both directions from some ultimate digit
> > > computational system. We know the head would
> > > read 31415927... d7 d6 d5 d4 d3 d2 d1, and
> > > these last digits are still unknown today,
> > It is known today that there are no last digits
> > of pi.
> >
> >
> > I speculate that what you (TG) call the halting
> > problem of '...' is
> > the problem of accepting that there is no
> > last natural number, or last Fibonaccit number,
> > or ..., or ..., or ...
> >
> > Consider another Halting Problem, the question of
> > whether a given Turing machine with a given input
> > halts or does not halt.
> >
> > For some Turning machines, for some inputs,
> > the machine does not halt.
> > NOT that _we don't know_ its final state.
> > _There is no final state of the machine_
> It is obvious that every sequence that ends in an ellipsis dopes not halt..
> The natural numbers have a halting problem.
> My number schools do address this.
> Making school B is modern, but making school C is not.
> >
> > We know that _there is no final natural_
> > from the description of a natural as
> > something we can count to.
> > But, if we can count to k,
> > then we can count to k+1.
> > k can't be the final natural,
> > and that is true no matter which natural
> > k refers to.
> > No natural is the final natural.
> > >>> If the digits are in the known library of digits
> > >>> then all is well.
> > >>
> > >> It's clear by now that you think that all is well.
> > >> I'm explaining why that's not correct.
> > >
> > > And I have explained why it is correct.
> > >
> > > No matter how many splits you put in a natural
> > > number the only thing that is going to fail it
> > > as a number, or numbers, I suppose after your split,
> > > is a non-digit in the sequence.
> > No. Only for _finite_ digit sequences.
> > > You are willing to
> > > allow infinity in the definition of natural numbers
> > > via the ellipsis in {1,2,3,...} where there is
> > > no k at the end. Claims are that the natural
> > > numbers are infinite.
> > For each natural number k,
> > there are finitely-many natural numbers before k
> > This is a matter of what it is to be a
> > natural number, to be something we can count to.
> >
> > For each natural number k,
> > there is another natural number k+1 after it.
> > Once more, because k is something we can count to:
> > If we can count to k, we can also count to k+1
> >
> > There is no count _with two ends_ which
> > encompasses all which we can count to:
> > _All_ is not finite.
> > > I have provided one of these in the value
> > > 333...34.
> > You have told me 333...34 is infinite.
> > The most reasonable reading of what you say
> > is that 333...34 is not one of the things
> > we can count to.
> > > The fact that we can even square the value
> > > is quite a big deal, yeah?
> > Does defining a square operation make it
> > one of the things you wish to discuss?
> >
> > That's fine. The clearer you are about what
> > you wish to discuss, the better.
> >
> > There are very different things which also
> > have square operations. Matrices, for example.
> >
> >
> > What is a natural number?
> >
> > My answer is: you can count to it.
> >
> > What is your answer?
> > Ordered with only 0123456789 ?
> > That would include things you can't count to.
> > Why would they be natural numbers?
> The schooling architecture does hit this list and is consistent with your own requirement.
> It is also true that our modern computers work consistently (somewhat) with such sizing requirements.
> Products are problematic in modern mathematics, and to some extent modern computing languages.
> Using digital analysis we see that the product of two large numbers is a large-large number.
> This offends closure some of the time, especially within a schooled architecture.
> Division as an inverse operator has its own problems that cause a refutation of the rational value.
>
> It is funny really, that to defeat the Peano axioms we need only present a large value and ask whether it is natural.
> 3333334
> will hold off the discussion with a Peano advocate for some time. As to whether this makes way for digital analysis: I would think so.
> The rapidity with which the digits work was implemented for some good reason wasn't it? How fascinating that modulo mechanics are at the bottom of it, and interesting that those modulo mechanics have not yet been fully leveraged by status quo mathematics.
> That they've all eaten the rational value as well is an ambiguity worth noting.
> Again, that the discussion does in fact go over to human psychology is instructive.
> We have been taught many things under threat of failure.
> That they are all correct is an intangible problem; just like the number.
>
> I confirm that mistakes have been made and point them out as I see them.
> No doubt this places me in a marginalized community.
> I have been censored off of every well known platform that does that sort of thing.
> That again is the failure mode. As to what is failing us now: it is the status quo.
> The inability of those drenched in its assumptions to confront their own adopted ambiguities is by definition the status quo.
> The state of things as they stand. That curricular systems are under attack now by a new truth seeking class: this will always be true.
> It is as well true that Nazis fled to Argentina and remain seated in such places; that the U.S. essentially bought up Nazi rocket scientists and imported them directly into its high tech system: true. These details make dark numbers seem innocent in hindsight.
> The whole thing takes on a sick level of cacophony when you try to consider whether they were socialists too?
> Whether the U.S. military is a socialist organization, for instance? One that feels entitled to knock off and down other socialist organizations.
> On the grounds that they are socialist. No wonder the suicide rate is so high.
> Truth seekers will at least attempt to address the concerns within a discussion that another repeatedly puts before them.
> With your 'split' approach I am a bit guilty of not applying myself.
> I'd like to falsify it, but I sense that the lack of productivity that it engages in will not allow it.
> Dedekind cuts are no doubt nearby. There the digital approach makes itself felt as digit chasing.
> The endless digits of the irrational values are about as close as we come to the form 333...34
> in a more traditional form, and of course the 1/3=0.333... as well.
> Without the rational falsification the path will go cold, and that is just status quo ignorance.
> That the decimal point is a structural augmentation with a unital interpretation that then extends back onto the rational value is the disambiguated path. Again, Jim, I am grateful of your participation and where you halt is pretty well where the system halts I'd say; as it stands today.

On 10/8/2022 8:26 AM, Timothy Golden wrote:

> I wonder whether the usage of the exponential
> operator should be suspended since we're
> trying for simplicity.

The exponential operator is simple.

Define the successor '⁺⁺'
∀k : k⁺⁺ ≠ 0
∀k : (∀j ≠ k : k⁺⁺ ≠ j⁺⁺)

Define addition '+'
∀k : k+0 = k
∀k : (∀j : k+j⁺⁺ = (k+j)⁺⁺

Define multiplication '*'
∀k : k*0 = 0
∀k : (∀j : k*j⁺⁺ = (k*j)+k

We've built the machinery for '⁺⁺' '+' '*'
At this point, defining exponentiation '^'
is turning an already-built crank once more.
∀k : k^0 = 1 = 0⁺⁺
∀k : (∀j : k^j⁺⁺ = (k^j)*k

| A physicist, a mathematician and an engineer
| stay in a hotel.
| | The engineer is awakened by a smell and gets up to
| check it. He finds a fire in the hallway, sees
| a nearby fire extinguisher and after extinguishing
| it, goes back to bed.
| | Later that night, the physicist gets up, again
| because of the smell of fire. He quickly gets up
| and sees the fire in the hallway. After calculating
| air pressure, flame temperature and humidity as well
| as distance to the fire and projected trajectory, he
| extinguishes the fire with the least amount of fluid.
| | Then the mathematician awakens, and finds that the
| embers of the fire are still burning. After giving
| much thought to the problem, he gets up and lights it
| up to an actual fire. Then he goes back to sleep,
| satisfied that the problem has been reduced to a
| previously solved one.
| https://proofwiki.org/wiki/ProofWiki:Jokes/Physicist_Mathematician_and_Engineer_Jokes/Burning_Hotel

On Saturday, October 8, 2022 at 9:27:44 AM UTC-7, Jim Burns wrote:
> On 10/8/2022 8:26 AM, Timothy Golden wrote:
>
> > I wonder whether the usage of the exponential
> > operator should be suspended since we're
> > trying for simplicity.
> The exponential operator is simple.
>
> Define the successor '⁺⁺'
> ∀k : k⁺⁺ ≠ 0
> ∀k : (∀j ≠ k : k⁺⁺ ≠ j⁺⁺)
>
> Define addition '+'
> ∀k : k+0 = k
> ∀k : (∀j : k+j⁺⁺ = (k+j)⁺⁺
>
> Define multiplication '*'
> ∀k : k*0 = 0
> ∀k : (∀j : k*j⁺⁺ = (k*j)+k
>
> We've built the machinery for '⁺⁺' '+' '*'
> At this point, defining exponentiation '^'
> is turning an already-built crank once more.
> ∀k : k^0 = 1 = 0⁺⁺
> ∀k : (∀j : k^j⁺⁺ = (k^j)*k
>
>
> | A physicist, a mathematician and an engineer
> | stay in a hotel.
> |
> | The engineer is awakened by a smell and gets up to
> | check it. He finds a fire in the hallway, sees
> | a nearby fire extinguisher and after extinguishing
> | it, goes back to bed.
> |
> | Later that night, the physicist gets up, again
> | because of the smell of fire. He quickly gets up
> | and sees the fire in the hallway. After calculating
> | air pressure, flame temperature and humidity as well
> | as distance to the fire and projected trajectory, he
> | extinguishes the fire with the least amount of fluid.
> |
> | Then the mathematician awakens, and finds that the
> | embers of the fire are still burning. After giving
> | much thought to the problem, he gets up and lights it
> | up to an actual fire. Then he goes back to sleep,
> | satisfied that the problem has been reduced to a
> | previously solved one.
> |
> https://proofwiki.org/wiki/ProofWiki:Jokes/Physicist_Mathematician_and_Engineer_Jokes/Burning_Hotel

So, you build multiplication from addition about the same way?

The repeated addition is multiplication and repeated multiplication is
exponentiation, is also a similar repetition, of ... addition, that
builds outward, and makes moduli, then, there's also that
roots and factors and otherwise the results of inverse, division, roots,
exponential roots and power roots, has only exact roots exist,
then for example implementing partition and division primarily,
what define roots.

Generally, ....

The "arithmetizing the multiplication in addition", and "arithmetizing
the exponentiation in multiplication", is made much easier from that
it's again some counting for some numbering.

This is where Presbuger arithmetic, it's complete, while, Peano, ...,
is an "incomplete", an arithmetization, that it defines _two_ operations.
This is for Goedel "the order of the model of the arithmetization must at
least be having closed all its operators in one operator".

Otherwise per Goedel it's directly not so, in terms of Goedel's theories
on the completeness and incompleteness of theories, in their consistency.

Which is for constructivists an opportunity to build one, .... Then,
that if it's not "can't be built won't complete", that's unbounded,
and variables are independent or dependent, in their space of values
which shares one arithmetized bound.

Then even larger theories like "ZF set theory" much follow in terms
whatever's defined in sets and what structure would entail, that the
existence of the sets, at least has a numbers.

On 10/8/2022 3:45 PM, Ross A. Finlayson wrote:

> This is where Presbuger arithmetic, it's complete,
> while, Peano, ..., is an "incomplete",
> an arithmetization, that it defines _two_ operations.

Yes,
Presburger arithmetic (induction, addition)
is complete,
while
Peano arithmetic (induction, addition, multiplication)
and
Robinson arithmetic (addition, multiplication)
are incomplete.

I know that that has given me much to think about,
at times.

Perhaps it is useful to think of Peano and Robinson as
fishing in a much larger pond than Presburger,
large enough that it is not possible to account
for all of its dark crannies.

So, the difference is not that Presburger has
better nets. That would seem odd, since Peano
has everything Presburger has, and more.

An important line seems to be crossed
when a pond is large enough to contain
self-portraits of nets
(and of fishermen? I'm wobbly on my metaphor.)

For that, it is _sufficient_ to be able to
express recursive definitions without recursion.
I'm not sure what would be _necessary_

On Saturday, October 8, 2022 at 3:45:13 PM UTC-4, Ross A. Finlayson wrote:
> On Saturday, October 8, 2022 at 9:27:44 AM UTC-7, Jim Burns wrote:
> > On 10/8/2022 8:26 AM, Timothy Golden wrote:
> >
> > > I wonder whether the usage of the exponential
> > > operator should be suspended since we're
> > > trying for simplicity.
> > The exponential operator is simple.
> >
> > Define the successor '⁺⁺'
> > ∀k : k⁺⁺ ≠ 0
> > ∀k : (∀j ≠ k : k⁺⁺ ≠ j⁺⁺)
> >
> > Define addition '+'
> > ∀k : k+0 = k
> > ∀k : (∀j : k+j⁺⁺ = (k+j)⁺⁺
> >
> > Define multiplication '*'
> > ∀k : k*0 = 0
> > ∀k : (∀j : k*j⁺⁺ = (k*j)+k
> >
> > We've built the machinery for '⁺⁺' '+' '*'
> > At this point, defining exponentiation '^'
> > is turning an already-built crank once more.
> > ∀k : k^0 = 1 = 0⁺⁺
> > ∀k : (∀j : k^j⁺⁺ = (k^j)*k
> >
> >
> > | A physicist, a mathematician and an engineer
> > | stay in a hotel.
> > |
> > | The engineer is awakened by a smell and gets up to
> > | check it. He finds a fire in the hallway, sees
> > | a nearby fire extinguisher and after extinguishing
> > | it, goes back to bed.
> > |
> > | Later that night, the physicist gets up, again
> > | because of the smell of fire. He quickly gets up
> > | and sees the fire in the hallway. After calculating
> > | air pressure, flame temperature and humidity as well
> > | as distance to the fire and projected trajectory, he
> > | extinguishes the fire with the least amount of fluid.
> > |
> > | Then the mathematician awakens, and finds that the
> > | embers of the fire are still burning. After giving
> > | much thought to the problem, he gets up and lights it
> > | up to an actual fire. Then he goes back to sleep,
> > | satisfied that the problem has been reduced to a
> > | previously solved one.
> > |
> > https://proofwiki.org/wiki/ProofWiki:Jokes/Physicist_Mathematician_and_Engineer_Jokes/Burning_Hotel
> So, you build multiplication from addition about the same way?
>
> The repeated addition is multiplication and repeated multiplication is
> exponentiation, is also a similar repetition, of ... addition, that
> builds outward, and makes moduli, then, there's also that
> roots and factors and otherwise the results of inverse, division, roots,
> exponential roots and power roots, has only exact roots exist,
> then for example implementing partition and division primarily,
> what define roots.
>
> Generally, ....
>
> The "arithmetizing the multiplication in addition", and "arithmetizing
> the exponentiation in multiplication", is made much easier from that
> it's again some counting for some numbering.

Well it does wrap around to the OP usage of pow().
In some regards Chris is vindicated.

>
> This is where Presbuger arithmetic, it's complete, while, Peano, ...,
> is an "incomplete", an arithmetization, that it defines _two_ operations.
> This is for Goedel "the order of the model of the arithmetization must at
> least be having closed all its operators in one operator".
>
> Otherwise per Goedel it's directly not so, in terms of Goedel's theories
> on the completeness and incompleteness of theories, in their consistency.
>
> Which is for constructivists an opportunity to build one, .... Then,
> that if it's not "can't be built won't complete", that's unbounded,
> and variables are independent or dependent, in their space of values
> which shares one arithmetized bound.

Isn't this related to the usage of the ellipsis? Upon engaging even just the sum of
{ 1, 2, 3, ... } + { 1, 2, 3, ... }
engages two ellipses. I understand my notation may not be accepted, but the idea is pretty clearly the same job as the recursive definitions.
Defining an adder as a counter is not actually a very efficient strategy, so it leaves me wondering if there is a superior definition. That notation and superposition can be melded: keeping with modulo-one characteristics the usual product notation would make better summation notation:
Let a = 111 (three in modulo-ten)
Let b = 1111 (four in modulo-ten)
Define a + b = a b = 111 1111 = 1111111 (seven in modulo-ten)
and so the obnoxious recursive form can go away. That summation does not call for counting, though it can be interpreted that way, is desirable in that from a physical perspective all objects in reality do lay in superposition with one another. We do not count up their bits in order to account for this. It simply pre-exists. In this regard the count might be regarded as a decomposition of the sum as a more fundamental aspect of 'things'. Our ability to count specifically what those things are, such as sheep in a field that is well walled with stones. They exist there in superposition, and our ability to come up with a transcription or representation is the abstraction: counting via whatever mechanism is the advanced technology rather than the sum being a more advanced form. In effect this is addressing the meaning of number differently than the successor method. That modulo-two and modulo-ten forms are as well consistent does not hold up to this superposition notation as superposition of their digits and so the sum as an operator is necessary. The notion of those numbers as representatives has more mechanistic qualities. Yes, counting will support them, but since hardware addition is possible it exposes what we mean by operation: computation is possible. In the modulo-one form it can be said that computation is not even necessary. That the operator has to do with the numerical form rather than with the physical form is accurate isn't it? The product suffers even more greatly in that you can take five oranges and multiply them by three oranges, but you will still have just eight oranges. These issues of physical correspondence as wiped off of the map of mathematics does seem to take this awareness very well. Where mathematics holds its physical correspondence best is with the sum; the integral; superposition. Necessitating counting is philosophically interesting, but it is not necessarily accurate to reality. Likewise then, verifying a large value as natural; such as 33333345; will not then necessitate counting up to this value via the successors of zero. Yes, the possibility is philosophically available, especially for a finite constant, but the idea that this is the only way to arrive here is despicable.

>
> Then even larger theories like "ZF set theory" much follow in terms
> whatever's defined in sets and what structure would entail, that the
> existence of the sets, at least has a numbers.

On 10/10/2022 10:12 AM, Timothy Golden wrote:

> In this regard the count might be regarded as
> a decomposition of the sum as a more fundamental
> aspect of 'things'. Our ability to count specifically
> what those things are, such as sheep in a field
> that is well walled with stones.

The _usefulness_ of counting how many sheep
are in a field depends upon a property which
sheep in a field have, pebbles in a bag have,
stars in the sky have, but which not all other
sets have.

Some sets cannot have their elements matched
to a proper subset or to a proper superset.

Some sets, if one matches all the sheep headed
out in the morning, and it matches all the
sheep coming back in the evening, then no sheep
are unaccounted for.

A set like a pastureful of sheep can be
ordered so that there is a first sheep,
there is a last sheep, and, for each split,
there is a last-before sheep and a first-after
sheep.

Not all sets are like a pastureful of sheep.

From 0 to any natural number k, there is a first
number 0, there is a last number k, and,
for each split, there is a last-before i,
a first-after j, and j = i⁺⁺

To any natural number, the set of those numbers
is like a pastureful of sheep. Those numbers in
the morning will match those numbers in the
evening.

However, the set of _all_ the natural numbers
is not like that. It can match proper subsets
and proper supersets of itself.

It would be understandable to treat "finite"
sets like pasturefuls of sheep like the rule,
and treat sets _unlike_ pasturefuls of sheep
as defective in some way, as a problem
needing a solution. Our lived experience
is apparently constructed more often of sheep
than of numbers.

My suggestion is that it's actually the other
way around. Not that there is anything _wrong_
with finite sets. But they are the exception,
not the rule, sets with the useful property of
not matching proper subsets and proper supersets.

https://en.wikipedia.org/wiki/Unum_(number_format)
https://en.wikipedia.org/wiki/Elias_gamma_coding
https://archive.org/details/tim_golden_and_tersymmetrical_suppression_conspiracy

On Saturday, October 15, 2022 at 4:06:53 PM UTC-4, Tomatzio Von Relish wrote:
> https://en.wikipedia.org/wiki/Unum_(number_format)
> https://en.wikipedia.org/wiki/Elias_gamma_coding
> https://archive.org/details/tim_golden_and_tersymmetrical_suppression_conspiracy

So we see now who Von Picklish is!
Looks like you had quite a bit of fun while writing this.
You've totally obfuscated polysign numbers!
And I'm getting a good chuckle out of it.

Now, really, in case you've drawn in some new reader here, please consider the familiar behavior of the real numbers:
- x + x = 0 .
There is a balance in these signs. This is the two-signed system P2. To engage P3 we simply write:
- x + x * x = 0 ,
and you are off into polysign numbers now. The meanings of the signs in P3 are new. Yet the algebra will work out the same, with some care for notation. Geometry comes along for the ride as well and is born of this simple balance.

On Sunday, October 16, 2022 at 9:53:10 AM UTC-4, Timothy Golden wrote:
> On Saturday, October 15, 2022 at 4:06:53 PM UTC-4, Tomatzio Von Relish wrote:
> > https://en.wikipedia.org/wiki/Unum_(number_format)
> > https://en.wikipedia.org/wiki/Elias_gamma_coding
> > https://archive.org/details/tim_golden_and_tersymmetrical_suppression_conspiracy
> So we see now who Von Picklish is!
> Looks like you had quite a bit of fun while writing this.
> You've totally obfuscated polysign numbers!
> And I'm getting a good chuckle out of it.
>
> Now, really, in case you've drawn in some new reader here, please consider the familiar behavior of the real numbers:
> - x + x = 0 .
> There is a balance in these signs. This is the two-signed system P2. To engage P3 we simply write:
> - x + x * x = 0 ,
> and you are off into polysign numbers now. The meanings of the signs in P3 are new. Yet the algebra will work out the same, with some care for notation. Geometry comes along for the ride as well and is born of this simple balance.
I'll have to voice my criticisms more carefully. You state:
"One can observe that for the two-signed case, his structure is different compared to the real numbers."
This is not really coming clean though. The law of sign balance states:
- x + x = 0
and this is totally consistent with the traditional real value. Any value z in P2 is unmistakably the same as the real value x in R.
While it is true that structurally, and in particular in actual C++ code there is magnitudinal storage for each sign, thus requiring two magnitudes to store a P2 value, this is because we are about to generalize sign! Because of the balance the reduction that most think in is still available, and so this claim of a discrepancy is hollow. Here I am filling in what you have left out.

Next paragraph: "Golden describes the signon[10], the building block of polysigned numbers, a sort of miniature of signed successors."
As flattered as I am to see my name in your print, I have to express that the signon needn't be treated as a building block. This certainly is not my own frame of thought. I find the signon only after substantial work in generalizing sign. It is upon studying lattice concepts and spatial packing that the signon comes to light. Plenty of math is available without the signon, which is so named after the polygon, but is sign-based and is simply stepping in every sign direction once and with every permutation. In P3 this establishes a hexagon and in P4 a rhombic dodecahedron, though because the steps are directed and unidirectional in nature, and as paths emanate from the center of these objects, there is more structure to the signon than these simple shapes that are named. The signon is general dimensional, and that in P2 we are looking at double segments is of interest as well. This is the one stage where what is generally unidirectional takes a bidirectional geometry simply due to the exact opposition of the two signed system -1+1=0.

What a strange interpretation you've evolved here. Somehow you feel you can overlook the balance of the signs which is the basis of the development of polysign numbers. Yet I see you have a section 'P3 equivalence' which sort of covers it.
So it is a matter of presenting a power notation alongside the sum notation?
It would be really great to see you publish the division algorithm. That to me is a significant gain that you've managed and I can barely understand.
Anyway, I mean no insult though I am taken aback at this presentation. Hopefully this winter I'll get back into the code some and manage to expose subP4 and subP5 and so forth. And of course the little scoundrel P1 gets so little traction, yet it is at the bottom of it all.