On the set of all unique genuine numbers

[I.V. Serov]

Dear Collegae,

THANK YOU! YOUR ANSWERS ARE VERY HELPFUL AND INSPIRING.

**********

The idea of this note is to construct a basis structure of the set X of 
genuine numbers.

(A x) is a shortcut for (A x in X), where A - all - denotes the 
universal quantifier;
(B x) is a shortcut for (B x in X), where B - be - denotes the 
existential quantifier.

(X Q) denotes a unary relation Q over the set X;
(x Q) denotes the logical value of the characteristic unary predicate Q 
with x in X as the argument;
An expression (A (X Q)) reads as follow: (for) any unary predicate Q 
over the set X.

(X W X) denotes a binary relation W over the set X;
(x W y) denotes the logical value of the characteristic binary predicate 
W with x and y in X as arguments.

(Q X) denotes a unary operation Q over the set X;
(Q x) denotes the result value in X of the unary operation Q with x in X 
as argument.

N+ denotes the set of the positive natural numbers;
Z denotes the set of the integers.

The logical connectives are denoted as:

"+" - exclusive disjunction;
"|" - non-exclusive disjunction;
"^" - conjunction;
"<->" - biconditional;
"~" - negation;
"->" - material implication.

[A.j] is a shortcut for Axiom.j;
[D.j] is a shortcut for Definition.j;
[R.j] is a shortcut for Reference.j.

Some axioms below can be proved from other axioms below, so we currently 
make no distinction between axioms and theorems and regard them all as 
axioms.

For this post, the explicit references are:

Reference.1. Sat Jan 7 05:28:00 EST 2023 by Thomas Klimpel; 
https://cs.nyu.edu/pipermail/fom/2023-January/023687.html
Reference.2. Sat Jan 7 08:34:23 EST 2023 by Monroe Eskew; 
https://cs.nyu.edu/pipermail/fom/2023-January/023688.html
Reference.3. https://www.chf.nu/fom_x_chf_cross.png
Reference.4. Sat Jan 28 03:13:40 EST 2023 by Vaughan Pratt; 
https://cs.nyu.edu/pipermail/fom/2023-January/023720.html
Reference.5. https://oeis.org/A105025
Reference.6. Mon Nov 7 20:55:44 EST 2022 by Vaughan Pratt 
https://cs.nyu.edu/pipermail/fom/2022-November/023616.html

**********

Yes! The axioms given in [R.1] by Thomas Klimpel indeed aim at defining 
the ground structure as requested in the metaphysical example by means 
of repeated successions:

Axiom.1. (A x) (A y): (x = y) + (x = (P ... (P y))) + (x = (S ... (S 
y))); [see [R.1] and below].

The exclusive disjunction is imperative to prohibit cyclicality.

In order to be a united whole, the ground structure of X is Cartesian, 
it is explicitly defined for (A x) (A y).

And indeed, one of the crossing orders underlying the structure of the 
set X is the "discrete linear order without endpoints" as proposed in 
[R.2] by Monroe Askew.

The approaches in [R.1] and [R.2] provide essential basic features of 
the structure of the set X:

- non-well-foundedness;
- generalized enumerability;
- all-to-all connectivity/unity;
- cyclicity-free infiniteness.

This is not all there is to say about the set X, as the signature of the 
set X is to be omnipotent.
Let us start constructing a set X that "has no beginning" with ... this 
beginning.

**********

Axiom.2. (B (X U X)): ((A x) (A y) (x U y)); [co-existential membership; 
this is the universal relation U over the set X];
Axiom.3. (A (X W X)): ((B x) (B y) (x W y)); [non-existence of an empty 
relation over the set X; the empty set is not a subset of X].

The universal relation of co-existential membership is the only one 
trivial relation over the set X.


Axiom.4. (A (X Q)) (A x) (A y): <-> (x = y) <-> ((x Q) <-> (y Q)); 
[=-sameness; every element of X is related by this relation to itself 
only];
Axiom.5. (A (X Qo)) (A (X Qi)): ((Qo x) = (Qi y)) <-> ((Qo x) = (Qi y)); 
[identity uniqueness].

The symbol "=" above and elsewhere denotes the binary relation (X = X) 
of total =-identity over the set X.

For, the set X satisfies Leibniz's ontological "identity of 
indiscernibles," which states, when applied to the set X, that there are 
no elements x in X that have all their properties in common.
Sameness of x is its' difference from anything else in X in at least one 
property.

That is, for two genuine numbers x and y to be identical, the same or 
equal (in all senses), it is necessary and sufficient for any unary 
predicate Q on elements of X to have the same logical value on them.

The structure of X must guarantee that no two elements of the set X are 
identical in this strongest sense.
It is for this reason that uniqueness is emphasized in the title of this 
post.

The identity relation (X = X) is an equivalence relation, so that:

Axiom.6. (A x): (x = x); [=-reflexivity];
Axiom.7. (A x) (A y): (x = y) <-> (y = x); [=-symmetry];
Axiom.8. (A x) (A y) (A z): ((x = y) ^ (y = z)) -> (x = z); 
[=-transitivity];
Axiom.9. (A x) (B y): ~(x = y); [=-difference];
Axiom.10. (A x) (A y) (B (X Q)): ~ ((x Q) = (y Q)); [=-uniqueness; 
uniqueness guarantees non-automorphism].

**********

There exists, over the set X, a total binary relation (X # X) of strict 
#-chainability, which is an asymmetric, transitive and strongly 
trichotomous:

Axiom.11. (A x) (A y): (x # y) -> ~ (y # x); [#-asymmetry];
Axiom.12. (A x) (A y) (A z): ((x # y) ^ (y # z)) -> (x # z); 
[#-transitivity];
Axiom.13. (A x) (A y): (x # y) + (y # x) + (x == y); [#-strong 
trichotomy].

When (x # y), x is called an #-ancestor of y, while y is called a 
#-descendant of x.

Here, the binary relation (X == X) of #-equivalence (reflexive, 
symmetric and transitive) over the set X is determined by

Definition.1. (A x) (A y): (x == y) <-> (~ (x # y) ^ ~ (y # x)); 
[==-derivability of #-equivalence], so that

Axiom.14. (A x): (x == x); [==-reflexivity];
Axiom.15. (A x) (A y): (x == y) <-> (y == x); [==-symmetry];
Axiom.16. (A x) (A y) (A z): ((x == y) ^ (y == z)) -> (x == z); 
[==-transitivity].

At this point we unambiguously require, that #-equivalence (X == X) is 
nothing else than the identity relation (X = X).

Axiom.17. (A x) (A y): (x = y) <-> (x == y); [logical equivalence of 
#-equivalence and =-sameness].

 From now on, we do not make difference between the symbols "=" and "==".
For two elements to be #-equivalent is exactly the same as being 
identical.

**********

Let us further require:

Axiom.18. (A x) (B y): (y # x); #-left-endlessness;
Axiom.19. (A x) (B y): (y # x); #-right-endlessness.

Next, there exists a unary operation (S x) of S-succession over the set 
X, such that:

Axiom.20. (A x) (B y): (y = (S x)) ^ (x # (S x)); [existence of 
S-succession];
Axiom.21. (A x) (A y): ((S x) = (S y)) <-> (x = y); [bijectivity of 
succession];
Axiom.22. (A x) (A y): (x # y) <-> ((S x) = y) + ((S x) # y); [immediacy 
of succession; (S x) is called the successor of x];
Axiom.23. (A x) (B y): (x = (S y)) ^ (y # x); [existence of precedence; 
such y is called the predecessor of x and is denoted as (P x)].

The unary operation (P x) of P-precedence is the inverse operation of 
the S-succession, so that

Axiom.24. (A x): (P (S x)) = (S (P x)); [see [R.1] and [A.23]].

Definition.2. Any two genuine numbers x and y, such that y = (S x) are 
said to be in love.

**********

We will follow the line of Euclid.

An even genuine number x in X is that which is *divisible into two equal 
genuine parts*,
or, equivalently, which differs by a natural (or integer) unit from an 
odd genuine number.

An odd genuine number x in X is that which is not *divisible into two 
equal genuine parts*,
or, equivalently, which differs by a natural (or integer) unit from an 
even number.

In other words, we *go binary*, for the concept of existence cannot be 
imagined without thinking about non-existence.
Think of yin-yang. "To be or not to be" - is there a question?

Let us enrich our metaphysical example with the following dialectical 
extension.
Let every x in X to be a *thesis*.
Then somewhere in X there is to be an *antithesis* of this *thesis*.
Recursion then?
Well, yes, before recursion: somewhere out there there is to be their 
*anti-synthesis*.

There exists a unary operation (C x) of current-even C-succession over 
the set X, such that:

Axiom.25. (A x) (B y): (y = (C x)) ^ (x # (C x)); [existence of 
current-even C-succession; such y is called the current-even C-successor 
of x and x is called the prior of y];
Axiom.26. (A x) (A y): ((C x) = (C y)) -> (x = y); [injectivity of 
current-even C-succession];
Axiom.27. (A x) (A y): (x # y) <-> (((C x) = y) + ((C x) # y)); 
[immediacy of current-even C-succession].

There exists a unary operation (H x) of history-odd H-succession over 
the set X, such that:

Axiom.28. (A x) (B y): (y = (H x)) ^ (x # (H x)); [existence of 
history-odd H-succession; such y is called the history-odd H-successor 
of x and x is called the future parent of y];
Axiom.29. (A x) (A y): ((H x) = (H y)) -> (x = y); [injectivity of 
history-odd H-succession];
Axiom.30. (A x) (A y): (x # y) <-> (((H x) = y) + ((H x) # y)); 
[immediacy of history-odd H-succession].

There exists a unary operation (F x) of future-odd F-succession over the 
set X, such that:

Axiom.31. (A x) (B y): (y = (F x)) ^ (x # (F x)); [existence of 
future-odd F-succession; such y is called the future-odd F-successor of 
x and x is called the history parent of y];
Axiom.32. (A x) (A y): ((F x) = (F y)) -> (x = y); [injectivity of 
future-odd F-succession];
Axiom.33. (A x) (A y): (x # y) <-> (((F x) = y) + ((F x) # y)); 
[immediacy of future-odd F-succession].

Axiom.34. (A x) (S (H x)) = (C x);
Axiom.35. (A x) (S (C x)) = (F x);
Axiom.36. (A x) (H (S x)) = (F x);
Axiom.37. (A x) (F (P x)) = (H x);
Axiom.38. (A x) (S (S (H x))) = (F x).

Axiom.39. (A x) (S x) # (H x);
Axiom.40. (A x) (H x) # (F x);
Axiom.41. (A x) (S x) # (F x).

Let us denote an even genuine number x in X as (x E);
Let us denote an odd genuine number x in X as (x O).

There by definition:

Definition.3. (A x): (x E) <->
(
( ~(B z): (x = (H z))) ^
( (B y): (x = (C y))) ^
( ~(B z): (x = (F z)))
); [An even genuine number has no parents and is an current-even 
C-successor of some genuine number].

Definition.4. (A x): (x O) <->
(
( (B z): (x = (H z))) ^
(~ (B w): (x = (C w))) ^
( (B y): (x = (F y)))
); [An odd genuine number has no prior and has a history parent and a 
future parent].

Odd and even genuine numbers *alternate* with respect to "S":

Axiom.42. (A x): (x E) <-> ((S x) O);
Axiom.43. (A x): ~ ((x E) = (x E));
Axiom.44. (A x): (x O) <-> ((S x) E).

By [A.36] we can agree to write (O x) instead of (F x) or (H (S x)) and 
call the genuine number (O x) the offspring of x and its successor (S 
x).

Definition.5. (A x): ((O x) = (F x)) ^ ((O x) = (H (S x))); [pedigree].

Every offspring (O x) is odd and has two parents x and (S x) in love.
One of the parents is odd, the other is even.

Definition.6. When x in X is odd, we say it is an idea.

Definition.7. When x in X is even, we say it is an eidos.

One of the parents is an idea, the other is an eidos.

Axiom.45. (A x): (x # (S x)) ^ ((S x) # (O x)); [[A.18] and [A.39]].

**********

There exists, over the set X, a total binary relation (X @< X) of 
#-non-strict total denseness, which is reflexive, antisymmetric, 
transitive and strongly connected:

Axiom.46. (A x): (x @< x); [@<-reflexivity];
Axiom.47. (A x) (A y): ((x @< y) ^ (y @< x)) <-> (x @ y); 
[@<-antisymmetry];
Axiom.48. (A x) (A y) (A z): ((x @< y) ^ (y @< z)) -> (x @< z); 
[@<-transitivity];
Axiom.49. (A x) (A y): ((x @< y) | (y @< x)); [@<-strong connectivity];
Axiom.50. (A x) (B y): (y @< x); [@<-left-endlessness];
Axiom.51. (A x) (B y): (y @< x); [@<-right-endlessness],

where the binary @-equivalence relation (x @ y) is defined over the set 
X by:

Axiom.52. (A x): (x @ x); [@-reflexivity];
Axiom.53. (A x) (A y): (x @ y) <-> (y @ x) [@-symmetry];
Axiom.54. (A x) (A y) (A z): ((x @ y) ^ (y @ z)) -> (x @ z); 
[@-transitivity];
Axiom.55. (A x): (x @ (C x)); [essence of "@"-equivalence].

For the non-strict order "@<" there is an associated relation "<" called 
strict denseness over the set X, that is defined by the reflexive 
reduction:

Definition.8. (A x) (A y): (x < y) <-> ((x @< y) ^ ~ (x @ y)).

Axiom.56. (A x) (A y): (x @ y) -> (x < (O x));
Axiom.57. (A x): (x < (O x)).

Axiom.58. (A x): ((x) < (S x));
Axiom.59. (A x): ((P x) < (O x));
Axiom.60. (A x): ((O x) < (S x)).

Axiom.61. (A x) (A y): ((x < y) -> (B z): (x < z) ^ (z < y) ^ (z = (O (P 
y)))); [O-P-denseness];
Axiom.62. (A x) (A y): ((x < y) -> (B z): (x @ z) ^ (z < y) ^ (z = (C 
x))); [C-denseness].

The order (X @< X) is a dense genuine generalization of the natural 
lexicographical order.

***********

To which extent is the structure of an element x in X is different from 
the structure of the set X itself?
We call X a set and this post is called "on the set of all unique 
genuine numbers".
What is an element x of X? An element x of X is not an urelement.
An element x of X is a genuine number which possesses an internal 
hereditary structure and which is different from the structure of the 
set X.
The element x in X may require set-theoretic axiomatics to describe it, 
which differs from the axiomatics required to describe the set X.
So let us come up with a *get* name (from genuine *gathering*) for the 
set-theoretic of the element x of X.
We will distinguish between (theories of) sets and gets, unless and 
until they prove to be equivalent in some general sense.

Axiom.63. (A x): get x is non-empty and there is a choice function 
chf(x) defined on X, such that chf(x) is some element of get x; 
[x-choiceness].

***********

In [R.1] and [R.2] it was asked whether the *distance* between two 
genuine numbers is finite.
Various distances between genuine numbers can be defined.

For example:
The distance between x and (S x) with respect to (X # X) is a natural 
unit and is *finite*;
The distance between x and (S x) with respect to (X @< X) is *infinite*;
The distance between x and (C x) is *finite* with respect to (X @< X);
The distance between x and (C x) is *infinite* with respect to (X # X).

The three worlds - the natural, the integer and the genuine - are 
connected with each other by the following:

Axiom.64.
(((A x) in X) ((A y) in X)):
((B n) in N+) ((B m) in N+) ((B i) in Z) ((B xn) in X) ((B ym) in X):
(xn = (x * n)) ^ (ym = (y * m)) ^ (xn # ym) ^ (xn = ym & i) and such n, 
m, i, xn and ym are unique;
[strong archimedianity].

Here (x * n) denotes *scalar product* of genuine x and positive integer 
n;
and (x & i) denotes *additive shift* of genuine x by an integer i.

***********

Axiom.65.
While the set X with the signature {U,=,#,S,H,C,F,<,@<,@} is not 
well-ordered,
for any element x of X there is an x-well-order of X, such that x is the 
least element in this x-well-order.
Any element x is *equally equivalent* to any other element of X in this 
respect and can be chosen freely as the least for this x-well-order.
Respectively any x-well-order is *equally equivalent* to any other 
x-well-order of X in this respect.
[arbitrariness of x-well-ordering].

***********

The set X is to be a metamathematical universe in the strongest 
metalogical, metaphysical and generally philosophical sense.

Axiom.66.
For any ontological, epistemological, gnoseological or any other 
discourse
or an idea or an eidos of a discourse,
if anything, something or nothing,
or an idea or an eidos,
or an idea of an idea,
or an eidos of an idea,
or an idea of an eidos,
or an eidos of an eidos
exists or does not exist, then
it is an element of the set X;
[allness; as in [R.6]: if it is not yet there, it is already there].

The abstract structure with the signature {U,=,#,S,H,C,F,<,@<,@} as 
defined by [A.1 ... A.66] is called the universal chain hereditary form.

***********

In our metaphysical example, the structured set X with the signature 
{U,=,#,S,H,C,F,<,@<,@}
of all unique *days* ordered by (X # X) and discretely enumerated by (S 
X) is also the set of all unique *things* which are *densely inserted* 
between each other by (X @< X).

This can be compared to a construction of a well-founded structure 
similar to the *crossing* of analogs of (X # X) and (X @< X) orders with 
a non-successor element.
In the illustration [R.3], the binary codes of the first 2^7-1 natural 
numbers in (X # X)-like well-founded order are also ordered 
lexicographically by (X @< X)-like *inserts* and value the codes as 
unreduced Stern-Brocot fractions.
The latter structure, when extended to infinity, either potentially or 
actually, can be compared to structure G in [R.4] by Vaughan Pratt.
Minkowski's question-mark function can also be mentioned in this 
connection.

What is the prime difference between the structures of G and of X?
Recall, that X is not only a dense (X >@ X) order; it is also a 
non-well-founded discrete order by (X # X) and (S X), so it allows 
generalized or genuine counting, the genuine enumeration.

With regard to the questions asked in [R.4], take a look at [R.5].
When the structure in [R.5] is attempted to be taken to infinity 
(potentially or actually), both in terms of numbers of rows and in terms 
of the length of each row, then there would be no finite rows and 
therefore there would be no first row, i.e., there would be no 
non-successor element in the structure.
It would become a non-well-founded binary "Hilbert's hotel" and would 
contain (potentially or actually) all-infinite binary rows.

There are two definitions of uncountability, which are usually accepted 
and/or proved as being equivalent in the well-founded world.
Let us turn to the set X.
Is there a bijective function from X to the set of natural numbers? Yes, 
it can be defined due to [A.65].
Then the set X is said to be genuinely enumerable, or genuinely 
countable.
At the same time, the *cardinality* of X is about 2^|x|, where |x| is 
the *cardinality* of a typical element x in X.
Is the set X uncountable then? As it is genuinely enumerable, it is 
perhaps more accurate to call it genuinely powerful and genuinely 
countable instead of calling it uncountable.

Is there any interference of the induction axiom and the power axiom in 
set theories of non-well-founded sets?

Can Cantor's *squarish* (|x| times |x|) diagonal argument play its 
*uncountable* role for this *rectangular* (|x| times 2^|x|) structure?
Would the *sloping diagonals* of [R.5] in the non-well-founded binary 
"Hilbert's hotel" be of relevance to settle the (un)countability issues?

Is it possible to reunite the structures G and X? Consider two options:

X goes to G:
-X accepts that potentially it has a non-successor element, while 
actually it has a predecessor for each x;
-X actually does not accept that it has a non-successor element, it 
accepts this only potentially;
-X actually does not have a non-successor element, it has a 
non-successor element only potentially.
The theory of X remains intact in a potentially well-founded and 
actually in the non-well-founded from.

G goes to X:
- G accepts actual or potential existence of a predecessor for each 
element of G;
- G extends its discrete well-founded countability and rational-like 
denseness to non-well-founded genuine-like powerfulness and generalized 
genuine-like countability.
The G structure becomes genuine!

The offspring-truth is in between these two options, see [D.5].

In the non-well-founded world potential infinities *converge* to actual 
infinities, and both approaches friendly coexist with each other.

***********

The life of the universal chain hereditary form is *co-being* of all its 
genuine numbers ordered by {U,=,#,S,H,C,F,<,@<,@}.

One can define *sequences* of genuine numbers, each being an 
instantiation of a genuine-based natural-like number sequence.
The sequences are filtered out of the universal chain hereditary form.
The four operations from {S,H,C,F} and their inverses form the *basic 
alphabet* to construct *directive words* for sequences.
Sequences represent *lives* of individual genuine numbers.

For example:
For each idea x, there exists an infinite sequence {(x O) C} = [x, (C 
x), ..., (C ... (C x)), ...] with a directive word [CCC ...] for this 
idea x.
The sequence {(x O) C} is called the C-life of the idea x.
Each element of this sequence except the idea x itself is an eidos; it 
is an embodiment of the idea x during its *eternal* C-life.

***********

What about allness as in [A.66], is the universal chain hereditary form 
genuinely universal?
What about a genuine induction principle?
What genuine algebraic structures are there to be defined yet on the set 
X?
What is the set-theoretical structure of the set X?
What is the get-theoretical structure of the get x?
Which axioms of the set theory are necessary and sufficient to develop 
the theory of the set X?
Which axioms of the get theory are necessary and sufficient to develop 
the theory of a get x?
What is the logically causal, hereditary structure of the set X and of 
the gets x in X?
Can genuine numbers be seen as propositional tautologies, so that each 
one can be derived from any other by a rule of inference?
Is the theory of the structured set X categorical?
There are 66 axioms formulated above; some of them can be derived from 
others; what is the minimalistic set-up?
Finally, once again referring to [A.66]: does the set X exist or not?

**********

LET US CALL THIS THEORY 'THE CURRENT HISTORY OF THE FUTURE'.
THANK YOU ONCE MORE!

Sincerely yours,
I.V. Serov