Philosophical Speculations

SanteriSatama wrote: ↑Thu Mar 25, 2021 3:38 am
https://math.stackexchange.com/question ... =1&lq=1%29

Tweak: redefine "pairwise disjoint" as applying only to non-empty families of sets.

ScottRoberts wrote: ↑Thu Mar 25, 2021 11:24 pm

SanteriSatama wrote: ↑Thu Mar 25, 2021 3:38 am
https://math.stackexchange.com/question ... =1&lq=1%29

Tweak: redefine "pairwise disjoint" as applying only to non-empty families of sets.

Would that suffice, or as is the usual way of fixes, create a dozen more problems? I honestly don't know. Any case, trying to patch a leaking raft of wire and bubblegum is not my idea of mathematical beauty. We can do better.

SanteriSatama wrote: ↑Thu Mar 25, 2021 11:33 pm

ScottRoberts wrote: ↑Thu Mar 25, 2021 11:24 pm

SanteriSatama wrote: ↑Thu Mar 25, 2021 3:38 am
https://math.stackexchange.com/question ... =1&lq=1%29

Tweak: redefine "pairwise disjoint" as applying only to non-empty families of sets.

Would that suffice, or as is the usual way of fixes, create a dozen more problems? I honestly don't know. Any case, trying to patch a leaking raft of wire and bubblegum is not my idea of mathematical beauty. We can do better.

How could it raise problems? Why would anyone be concerned with pairwise disjointness except in non-empty sets of sets?

ScottRoberts wrote: ↑Fri Mar 26, 2021 12:20 am How could it raise problems? Why would anyone be concerned with pairwise disjointness except in non-empty sets of sets?

I don't know. ZFC has not originated from the coherence theory of truth, internally to pure math. Historically it's been motivated from the coherence theory of justification, to give applied mathematics of calculus a justification in pure mathematics. A concern could be, would the original aim and motivation be still sufficiently served?

The way how the numerical theory of real numbers is derived from ZFC seems very complex and goes over my head. I stumbled on the pairwise disjoint issue when trying to make (in vain) some sense of that bridge of construction.

ScottRoberts wrote: ↑Tue Mar 23, 2021 2:15 am Go for it. When you can produce the equivalent of the Euler equation, get back to me.

OK, here comes:

<< >>

Perhaps not the full story yet (ie. a valid proof), but enough to start to demonstrate the relation in this new language. In the numerical approach, this is a problematic implication:

e^2πi = 1 = e^0.

On the degree of abstract, Relop palindromes can be interpreted to be analogues on the same level as adeles, ideles etc., which are very difficult to think and discuss in the numerical languages and their ways of thinking. AFAIK nobody has ever called cohomology easy, I understand nearly nothing of its language.

Let's first look at the structure of most common identity elements, where Relop and numerical start to meet: < >, <0>, <1>. Opening a space between an open interval creates space to write also numeral values for identity element. SQR -1, antinumber of identity element of exponentiation and its subarithmetic of multiplication, is an overly complex way of writing an empty space, the i of the i-dea of i-dentity element. But as we now see, "i" gives too much form to the formless. Imaginary number i-magines an i in an empty space.

Adding an empty space to a string of characters, ie. a word, does not change the word, but it can divide a word into two words. Likewise, removing an empty space (in the role of a closed interval) from between words combines them into a new word. Under the condition of palindromic writing, i-dentity of an empty space both halves and doubles a word. In Relop, the process of halving and doubling can be expressed thusly:

<><<<> : <>>><>
delete >< (neither more nor less aka "=") :
<<<> : <>>>

Numerically, counting the relation of "heads" and "beaks (<>< head with a tail, <<> head with a beak):

From
2:2 : 2:2
to fractions
2:1 : 1:2

Similarly,
<<><>< ><><>>
delete >< :
<< >>
Open interval with empty identity element in the middle < > corresponds with linear growth/slide, and << >> with idea of logarithmic growth, the interdependent relations of ln, e and Euler-Mascheroni constant γ.

The relations <><<<> : <>>><> and <<><>< ><><>> can be derived as AB and BA (A-bove and B-elow) combinations from the edges
The edges play their role in the first Stern-Brocot type of derivation of (meta)rational numbers, which I call Pillar.


Generating SB structure from the middle with < >, and counting the relations of heads and beaks/tails in each word, the numerical writing of rationals remains palindromic, e.g.:

1/1, 2/3, 1/2, 1/3 < > 3/1, 2/1, 3/2, 1/1

Worth noting, the metarationals of Relop have more internal structure than the numerical rationals, suggesting some deep connection with the concept of Polynumber in Wildberger's construction, where it is analogous to polynomials.

As said, still WIP, but hope this little demonstration gives some pleasure.

PS: Note that a Turing machine does not presuppose any logical system as such, only a tape of symbols, reading, and rules for manipulating the symbols on a tape. LNC enters the picture as the presupposition of the global proof that the Halting problem is undecidable.

In this core stage development of Relop language, the first syntactic rule of generating is to write only palindromic strings with < >, which then can be given also various numerical interpretations, as the Reading reads/writes from Center, Left and Right, and further along also from Above and Below. Instead of a single reading head, in Relop computation of palindromic strings is fundamentally distributed.

I lost you here:

<< >>

With regard to Relop, I am like a 4-year old who needs to learn what "2+2=4" means. What does "<< >>" mean?

ScottRoberts wrote: ↑Tue Mar 30, 2021 10:55 pm I lost you here:

<< >>

With regard to Relop, I am like a 4-year old who needs to learn what "2+2=4" means. What does "<< >>" mean?

As said, general idea of logarithmicity. Relop way of writing e and i, as they relate to eiγ / γie. Euler's number "<<", imaginary number written as empty space " ", and Euler-Mascheroni constant ">>". Or vice versa, as Relop is being written as palindromes. Mathologer's presentation gives nice visuals of the gamma constant.

I keep on struggling with the numerical language, so I have not yet found an exact translation, but ln(n+1)+γ seems very meaningful, when trying to figure out a better way to speak "e and -e", which was my first interpretation of << >>. That was not intended as undefined "primitive notion", to stop reading, but as a form which the following demonstration and discussion sought give some general meaning. Translating from Relop to standard numerical language, Relop expressions are open to multiple interpretations, which, if we are lucky, may be given more exact definitions. The general way to read is from whole(s) to parts, starting from the idea of an open interval <> as a transfinite whole. So, I'm not presenting a paper of theorems, lecturing ex cathedra, but new language open to various interpretation and definitions by any reader, and hopefully meaningful discussion between those.

PS: The Pillar middle continuum that generates an order of rational numbers (and more) from <> to <<>> when read from A to B already expresses the general notion and relation of Cauchy sequences and exponentiation. Therefore it seemed redundant to interpret << >> as ln i exp (or vice versa).

SanteriSatama wrote: ↑Wed Mar 31, 2021 12:00 am

ScottRoberts wrote: ↑Tue Mar 30, 2021 10:55 pm I lost you here:

<< >>

With regard to Relop, I am like a 4-year old who needs to learn what "2+2=4" means. What does "<< >>" mean?

As said, general idea of logarithmicity. Relop way of writing e and i, as they relate to eiγ / γie. Euler's number "<<", imaginary number written as empty space " ",

Sorry, but the idea that you can get to the conventional meaning of e from "more-more" is completely confounding to me. Do you really expect to be able to say that the derivative of e^x is e^x in this way? Or that the space character can come to mean a rotation of 90 degrees of the complex plane?

I don't see any point in pursuing this. I'll wait until there is a textbook available.