1 Foundation
Let the concept of time and space be mutually equivalent structures, which we
call domains. A domain is an n-dimensional coordinate system whose elements
are causal sets. Causal sets are characterized by their locally invariant coor-
dinates, meaning their positions do not change along individual axes, so the
non-existence of some already existing axes can occur without altering the co-
ordinates of the given causal set. In fact, it is possible to establish the origin of
a new axis, which assigns a specific constant scalar to the causal set and is not
considered a change in the position of the relevant causal set.
Temporal causal sets (TCS) are elements of time, and spatial causal sets (SCS)
are elements of space. SCS has the property of evolving in time; that is, if
SCS is characterized by TCS, then, according to this principle, it spontaneously
shifts to another TCS, which we will call a neighboring TCS. An object is a
body characterized by its position, which is a set of SCS with a non-empty
intersection with this object; we call these SCS its proper SCS. If we assume
that an object is a closed set.
Each object is characterized by a unique sequence of TCS, which are its subsets.
If we say that object A traveled through time, it means that after the end of
the time travel process, the proper SCS of object A are characterized by non-
adjacent TCS.
Time travel will be referred to as a jump. When object A jumps to non-adjacent
TCS, a new ”time path” is created, in which time decreases in parallel with the
original time path in which object A did not jump.
You define the notation ”time path”:
∀k, τ
k
, k ∈ Z
≥0
(1)
”then it is important to establish that there are no time paths that have no
non-empty intersection with the original time path, or we will refer to this
statement as letter A.” Then let for all n, the relationship A
n
= A, hold where
A
n+1
= A+A
n
, then lim
n→k
A
n
= B, where B is the final statement of this axiom
description, for k for which this statement is true, and at the same time, there
is no number h smaller than k for which this statement is also true.
1. A1: ∀k, τ
k
, ∃n, τ
n
, τ
n
∩ τ
k
≠ ∅, ∄l, τ
l
, ∃k, τ
k
, k > l; ∃k
c
, τ
k
c
, ∃!, kτ
k
, k
c
= k ∧
∄k, τ
k
, k > k
c
,
∀n,∀k
τ
k
∩ τ
n
= M, ∣V
M
∣ = l Ô⇒ l = k
c
Then it is necessary to define the hypothetical distance of the time path from
the original one:
1. A2: ∃τ
k
2
, k
2
< k
1
, τ
k
2
∩ τ
k
1
≠ ∅ ∧ ... ∧ ∃τ
k
l
, k
l
< k
l−1
, τ
k
l
∩ τ
k
l−1
≠ ∅ Ô⇒
(τ
k
1
(t))
∗
= l, Ψ
l
∶= {k
1
, ..., k
l
} ⇐⇒ ∄τ
k
l+1
, k
k+1
< k
l
, τ
k
l+1
∩ τ
k
l
≠ ∅
2