_Mathematics with the G structure_

[I.V. Serov]

Vaughan Pratt states in:

https://cs.nyu.edu/pipermail/fom/2023-January/023735.html
https://cs.nyu.edu/pipermail/fom/2023-January/023723.html

(1) "So far it seems to me that every objection to my proposal (repeated 
below) concerning gapless geodesics depends on the 
assumption/axiom/whatever that a nonempty linear order with no greatest 
(or least) element must be an actual infinity."

(2) "What is bothering me is the idea that because the sequence is 
infinite, it is therefore not something we can experience."

Nevertheless, indeed:

(3) "there is nothing inconsistent about dropping the axiom that a 
nonempty linear order with no
least element is impossible to experience in practice."

(4) "One might complain that writing a real in binary is an actual 
infinity.
To get around this, allow L and U to each be arbitrary finite sequences 
of
bits, with U differing from L by adding 1 to the last bit of each 
element
of U and doing the usual carry propagation in the finitely many bits of
that element."

No complains out here!

The "arbitrary finite sequences of bits" concatenated one after another 
form an (actually) infinite string
- a model of a state of the universe G.
To be more precise: each string-state is infinite at an open side;
and it is finite at the other - at the closed side; and that means, that
each string-state of the universe G is well-founded with the least 
(first) element at the closed side;
each string-state is an infinite sequence itself.

The difference between "actuality" and "potentiality" is a matter of 
interpretation in a certain sense:
Is there actually no last bit at the open side in each state-string of 
the universe G?
Or, is there only potentially no bit with no bit behind it at the open 
side?

It also means, that each string-state is an instantiation of an infinite 
sequence of natural numbers.
Do all these sequences form a set, whose elements, all-together, define 
the Natural Numbers?

Is there actually no first state in the universe G, so that no state is 
a non-successor state?
Or, is there only potentially no state with no state preceding it?

Whatever one experiences as finite is finite, because one experiences a 
finite change from one state to another:
- like adding, removing, flipping or carry-propagating a bit of the 
string.
That does not mean at all, that states of G themselves are finite or 
only potentially infinite.

A move from one string to another - from one state of the universe to 
the other - requires only a finite number of bit operations and that is 
good indeed.

Compare to: *An infinite electron jumps from one infinite orbit to 
another one and experiences a finite change so that the universe is now 
in another string-state.*

It is imperative to confirm that no state-string in G can consist of 
0-bits only, or 1-bits only;
no string-state is unary; all string-states are binary; this is an 
actual universal dichotomy.

The structure G can be translated into a structure that is described by 
all infinite continued fractions, such as [a0;a1,a2, ...],
where ai are natural numbers: 
https://en.wikipedia.org/wiki/Continued_fraction.
This brings us right to the Euclidean algorithm, which is at the core of 
all mathematics.

In this regard it is worth noting, that "no pattern has ever been found 
in this representation" of number pi.
Compare to: 
https://en.wikipedia.org/wiki/Generalized_continued_fraction#π.
Does this mean, that one needs more than two letters of the foundational 
alphabet?

The proposal of G is an excellent one; there are no objections to it out 
there,
except that it is necessary to recover enumeration as well, and that is:
- the structure is not only to be a dense order;
- it is also to be a discrete order; so that,
- there is a cross of the two orders - the cross order of the dense and 
of the discrete.

And this means the following:
- there is no first/least string-state in the universe of string-states:
-- neither with respect to the enumerative discrete order of the 
*universal time*,
-- nor with respect to the dense *matter* order;
- the structure is - actually (or at least potentially) - 
non-well-founded with respect to both orders and this is a fundamental 
requirement indeed.

Does one need to consider the structure G as an actual and/or as a 
potential infinity?
Again, in a certain sense it is a matter of interpretation:
- actual infinity appears to be not all too different from the potential 
one;
- and finiteness appears to be a special (periodic) case of infinity.

What is by far more important is that the universe of string-states is 
to be countable in some generalized sense
and this requires crossing of the dense and the discrete orders.
Without enumeration the mathematical G is to place dice, would need to 
make arbitrary choices and would be unable to justify own deeds. Is it 
the genuine mathematical G then?

Here a model:

Actually, or,
potentially,

what
- there was;
- there is and
- there will be,

is the deterministic universal rule (nous, or Greek νοῦς), which makes

- a being in the prior string-state of the universe
to become
- the being of the posterior string-state of the universe;

by the model force of:
- the carry;
- the heredity and
- the flip,

and so, that offspring states are situated between the parent states
by the model force of
- love.

Not to forget to recurse!

Then and only then, the structure G is to be nothing else (up to an 
isomorphism) than the structure X, the universal chain hereditary form;
and the theories of G and X are to survive together under the promising 
name of "the current history of the future".

I.V. Serov