The Temporal Continuum
Mohammad Ardeshir 111 Sharif University of Technology, 11365-9415, Tehran,
Iran<EMAIL_ADDRESS>Rasoul Ramezanian 222University of Lausanne,
Internef, 1015 Lausanne, Switzerland<EMAIL_ADDRESS>
###### Abstract

The continuum has been one of the most controversial topics in mathematics
since the time of the Greeks. Some mathematicians, such as Euclid and Cantor,
held the position that a line is composed of points, while others, like
Aristotle, Weyl and Brouwer, argued that a line is not composed of points but
rather a matrix of a continued insertion of points. In spite of this
disagreement on the structure of the continuum, they did distinguish the
temporal line from the spatial line. In this paper, we argue that there is
indeed a difference between the intuition of the spatial continuum and the
intuition of the temporal continuum. The main primary aspect of the temporal
continuum, in contrast with the spatial continuum, is the notion of
orientation.

The continuum has usually been mathematically modeled by _Cauchy_ sequences
and the _Dedekind_ cuts. While in the first model, each point can be
approximated by rational numbers, in the second one, that is not possible
constructively. We argue that points on the temporal continuum cannot be
approximated by rationals as a temporal point is a _flow_ that sinks to the
past. In our model, the continuum is a collection of constructive _Dedekind_
cuts, and we define two topologies for temporal continuum: 1. _oriented_
topology and 2. the _ordinary_ topology. We prove that every total function
from the _oriented_ topological space to the _ordinary_ one is continuous.

##  1 Introduction

Mathematicians have long been divided into two philosophical camps regarding
the structure of the continuum. Some, like Euclid [12] and Cantor [7], view
the continuum as a composition of points. On the other hand, others, including
Aristotle [2], Weyl [4, 19], and Brouwer [5], consider it as a _whole_ , not
composed of points.

In this paper, we aim to highlight another distinction, namely, between what
we refer to as _the spatial continuum_ and _the temporal continuum_. We will
model the temporal continuum using a specific type of Dedekind cuts that
illustrate this difference.

The main distinction between the spatial continuum and the temporal continuum
lies in the notion of _orientation_ : the temporal continuum is oriented,
moving exclusively from the past to the future, and it is impossible to move
in the _opposite_ direction333In this paper, we _assume_ that ‘time is
directed’ without delving into detailed arguments. Numerous works have
explored the direction of time from both physical and philosophical
perspectives; for further reading, refer to [15, 16].. Our subjective
experience reinforces this clear differentiation between the future and the
past. We can vividly remember the past, whereas the future remains uncertain.
Our ignorance of future events, including our own choices and actions,
contributes to the concept of _free will_. Conversely, we lack the ability to
alter the past due to the absence of free will in that direction.

> We can affect the future by our actions: so why can we not by our actions
> affect the past? The answer that springs to mind is this: you cannot change
> the past; if a thing has happened, it has happened, and you cannot make it
> not to have happened [10].

The primary objective of this paper is to propose a model for the continuum
that incorporates the concept of orientation. To achieve this, we will
introduce a formalization of orientation through a specific type of Dedekind
cuts, which we will refer to as oriented Dedekind cuts.

Brouwer and Weyl emphasized two essential aspects of the intuitive continuum,
which are _inexhaustibility_ and _non-discreteness_ (also referred to as _non-
atomicity_) ([3], page 86). The concept of choice sequences is motivated by
these characteristics. Non-lawlike sequences signify the continuum’s
inexhaustibility, and identifying points with unfinished sequences of nested
intervals reflects its non-discreteness ([3], page 87). Defining points as
choice sequences of nested intervals captures the notion that a point on the
continuum is not a dimension-less atom but rather a _halo_.

As choice sequences are non-predetermined and unfinished entities, the value
of every total function from choice sequences to natural numbers relies on an
initial segment of sequences. Brouwer leverages this property to demonstrate
that every total function over the continuum is continuous.

Both the spatial continuum and the temporal continuum share the aforementioned
aspects. However, a defining characteristic of the temporal continuum is the
concept of _orientation_. ”Duration” is a continuous _becoming_ , always
moving towards the future. In the spatial continuum, movement is possible in
both directions. To introduce witnesses for points on the spatial line, one
may proceed by introducing witnesses for all points in the segment $[-1,0]$,
then for points in the segment $[1,2]$, and subsequently for points in the
segment $[0,1]$, and so on. There is no requirement to adhere to any specific
direction. However, for the temporal line, this approach is not feasible. Let
the locations of a moving ball in space serve as witnesses for moments. If we
are at the moment $t_{0}$, and $t_{1}$ is a future moment, we cannot determine
the location of the ball at time $t_{1}$ at the present moment. We must wait
until we reach that moment, and only then will the locations of the ball in
all moments before $t_{1}$ be determined. _We are constantly moving towards
the future_. The following _assertion_ may provide further clarity regarding
our understanding of the notion of orientation. We leverage notion of
orientation to demonstrate that every total function over the temporal
continuum is continuous.

In considering how we experience a point on the temporal continuum, we can
conceptualize a point $t$ as the moment of occurrence of an event $E$:

t:=”The moment of the occurrence of the event E”

For instance, assume it is currently 8:00 am, and I am aware that the event
$E$ will occur sometime after 8:00 am but before 12:00 pm. As time passes, I
experience the point $t$ in the following manner: I examine the occurrence of
$E$ at several consecutive times. To do this, I plan a strategy and choose
specific moments to observe the event. The choice of these moments is up to
me.

For example, I may decide to examine the occurrence of $E$ at times like 10:15
am, 10:20 am, 10:40 am, 11:30 am, and so on. Suppose it is now 11:30 am, and
the last time I observed the occurrence of $E$ was at 10:40 am, where I found
no evidence of it happening yet. Now, suppose it is 11:30 am again, and I
observe the event and find evidence of its occurrence. However, at this point,
I _cannot move to the past_ to examine its occurrence at any previous moments.

All the information I have about $t$ is that it lies _sometime_ within the
interval $(10:40,11:30]$. I cannot distinguish the moment $t$ from other
moments within this interval. In other words, _the point $t$ has ”sunk back”
to the past, and I cannot experience it any further_. It is _absolutely
undecidable_ whether $t$ is before 11:00 am or after it. The moment $t$ cannot
be estimated more accurately, and consequently, I cannot approximate $t$ using
rational moments.

Furthermore, if I have a constructive method to introduce witnesses for all
moments on the temporal line, then the witness for the moment $t$ is also the
witness for all moments within the interval $(10:40,11:30]$. Thus, a point on
the temporal continuum is like a _flow_ that sinks back to the past
eventually.

We consider the above _explanation_ as our understanding of the notion of
_orientation_ and attempt to formalize it in this paper. In our perspective,
we regard $t$ as a _flow_ from the past, meaning it encompasses all moments
that have occurred before $t$. If $t$ is a past moment, we cannot obtain any
additional information about which moments belong to this collection and which
ones do not. If $t$ is a future moment, we still have time to gather data
about the elements of the collection before $t$ eventually sinks back to the
past. During this time, we may examine the membership status of certain
moments. However, as soon as $t$ sinks back to the past, we can no longer
acquire any new data.

This passage discusses the experience of a point on the temporal continuum,
emphasizing the concept of _orientation_ and how it is related to the passage
of time and our ability to observe and distinguish moments. It also introduces
the idea of a point being a _flow_ from the past, capturing the continuous and
irreversible nature of time.

Brouwer’s perspective on the continuum is that it is intuitively given as a
flowing medium of cohesion between two events, not comprised of points
(events) itself, but rather an inexhaustible matrix allowing for a continued
insertion of points. Originally, there are no points on the continuum, but we
can construct points on it or indicate a position within it. Brouwer
emphasized that the intuition of the continuum is the intuition of the medium
of cohesion between two events. He distinguishes between two things: the
medium of cohesion (first thing) and the continuum (second thing), or _primum_
and _secundum_ as he puts it ([14], page 70). Brouwer utilized the category of
choice sequences and the continuity principle to provide a mathematical
analysis of continua. Using choice sequences to define points on the
continuum, a point is modeled as a _halo_. Consequently, to construct a
witness for a point on the continuum, the witness is constructed for a halo
around it. Brouwer proposed his famous continuity principle for choice
sequences and used it to prove that every total function over the real line is
continuous ([17], page 305).

Motivated by the characteristic aspect of the temporal continuum, i.e., the
_orientation_ , as explained above, we introduce oriented cuts to model points
on the temporal continuum as _flows_. The traditional modeling of the
continuum using the _Cauchy_ fundamental sequences of rationals [17] allows
every real number to be approximated by rational numbers. However, since
moments on the temporal continuum sink back to the past, they _cannot_ be
approximated by rationals. Consequently, the _Cauchy_ sequences appear
unsuitable for modeling the temporal continuum. Instead, Dedekind cuts prove
to be more appropriate for this purpose. Among the constructive Dedekind lines
introduced in [17], only $\mathbb{R}^{d}$ can be positively approximated by
rationals due to its _locatedness_ property ([17], page 270), while
$\mathbb{R}^{e}$ and $\mathbb{R}^{be}$ can be approximated by rationals only
up to the double negation via the _strong monotonicity_ property. We
demonstrate that the collection of oriented Dedekind cuts cannot be
approximated by rationals, as desired in this case. As discussed earlier,
points on the temporal continuum are like flows that sink back to the past.
Once they have sunk back, we cannot acquire new data about them, and thus,
they cannot be approximated by rationals.

In addition to our main objective of introducing a mathematical model for the
temporal continuum, we also aim to demonstrate that every total function over
the temporal line is continuous, similar to the Brouwerian real line. For this
purpose, we require a _continuity principle_ for the Dedekind cuts. Hence, we
propose a principle called the _oriented continuity principle_ (OCP ) and
define two topologies for the temporal continuum: 1. the _oriented topology_
and 2. the _ordinary topology_. Utilizing OCP , we establish that every total
function from the oriented topological space to the ordinary one is
continuous.

The sequel of the paper is organized as follows. In section 2, we define the
notion of the oriented cuts, and justify a continuity principle for them. The
oriented continuity principle, OCP, expresses formally the feature we have in
mind about a continuity principle. We show that the oriented reals cannot be
approximated by rationals. in section 3, the _oriented topology_ and the
_ordinary topology_ are defined, and then some consequences of OCP for the
temporal continuum are demonstrated.

We argue in the context of constructive logic. As far as possible, the
standard notations in [17] are used in this paper.

##  2 Oriented Cuts

In this section, we introduce a type of left cuts of $\mathbb{Q}$, named
_oriented cuts_ , and state the oriented continuity principle.

###### Definition 2.1

We let $\mathbb{R}^{o}$ be the set of all strictly increasing bounded
sequences of rational numbers, i.e. $\alpha\in\mathbb{R}^{o}$ if and only if
$\exists M\forall n(\alpha(n)<\alpha(n+1)<M)$. For all
$\alpha,\beta\in\mathbb{R}^{o}$ we define:

  * •

$\alpha\leq\beta$ if and only if $\forall m\exists n(\alpha(m)<\beta(n))$,

  * •

$\alpha<\beta$ if and only if $\exists n\forall m(\alpha(m)<\beta(n))$, and

  * •

$\alpha=^{o}\beta$ if and only if $\alpha\leq\beta\wedge\beta\leq\alpha$.

We call the set $\mathbb{R}^{o}$, regarding equality $=^{o}$, the set of all
oriented reals _(_ cuts _)_.

Oriented reals, bounded strictly increasing sequences of rationals, are
supposed to model the passing of time. The increase of sequences reflects the
passing of time and the strictness of increase ensures that time cannot rest.

###### Definition 2.2

For a rational number $q\in\hbox{\bf$\mathbb{Q}$}{}$, let $\hat{q}$ be the
oriented real defined by $\hat{q}(n)=q-\frac{1}{n+1}$, for all $n$. Let
$q\rightarrow\hat{q}$ be the mapping which assigns the oriented real $\hat{q}$
to $q$. For each $q\in\mathbb{Q}$ and $\alpha\in\mathbb{R}^{o}$, we say

  * a.

$q<\alpha$ if and only if $\hat{q}<\alpha$,

  * b.

$\alpha\leq q$ if and only if $\alpha\leq\hat{q}$,

  * c.

$\alpha<q$ if and only if $\alpha<\hat{q}$, and

  * d.

$q\leq\alpha$ if and only if $\hat{q}\leq\alpha$.

###### Proposition 2.3

For $q\in\hbox{\bf$\mathbb{Q}$}{}$ and $\alpha\in\mathbb{R}^{o}$,

  * $(1)$

$q<\alpha$ if and only if $\exists n(q<\alpha(n))$,

  * $(2)$

$\alpha\leq q$ if and only if $\forall n(\alpha(n)<q)$,

  * $(3)$

$\alpha<q$ if and only if $\exists p\in\mathbb{Q}(p<q\wedge\alpha\leq p)$

  * $(4)$

$q\leq\alpha$ if and only if $\forall p\in\mathbb{Q}(p<q\rightarrow
p<\alpha)$.

Proof. For each $q\in\mathbb{Q}$ consider its map $\hat{q}$ in oriented reals.
Items (1) and (2) are straight forward.

(3). If $\alpha<\hat{q}$ then by definition 2.1, there exists $m$ such that
for all $n$, $\alpha(n)<q-\frac{1}{m+1}$. Let $p=q-\frac{1}{m+1}$. By (2), we
have $\alpha\leq p$. The converse is straight forward.

(4). If $\hat{q}\leq\alpha$ then for all $m$ there exists $n_{m}$ such that
$q-\frac{1}{m+1}<\alpha(n_{m})$. For $p<q$, there exists $m_{0}$ such that
$p<q-\frac{1}{m_{0}+1}<\alpha(n_{m_{0}})$. By (1), $p<\alpha$. For the
converse assume $\forall p\in\mathbb{Q}(p<q\rightarrow p<\alpha)$. Then for
all $m$, $q-\frac{1}{m+1}<\alpha$. By (1), there exists $n_{m}$ such that
$q-\frac{1}{m+1}<\alpha(n_{m})$. $\dashv$

For each $\alpha\in\mathbb{R}^{o}$, let
$A_{\alpha}=\\{q\in\mathbb{Q}\mid\exists n(q<\alpha(n))\\}$ be the _cut_
specified by $\alpha$. Note that $\alpha=^{o}\beta$ if and only if
$A_{\alpha}=A_{\beta}$ as sets.

###### Proposition 2.4

For $\alpha\in\mathbb{R}^{o}$,

  1. 1.

$\forall q\in\mathbb{Q}~{}(q\in A_{\alpha}\rightarrow\exists
p\in\mathbb{Q}~{}(p>q\wedge p\in A_{\alpha}))$ $(\emph{openness})$,

  2. 2.

$\forall p,q\in\mathbb{Q}~{}(p<q\wedge q\in A_{\alpha}\rightarrow p\in
A_{\alpha})$ $(\emph{monotonicity})$,

  3. 3.

$\exists p,q\in\mathbb{Q}~{}(p\in A_{\alpha}\wedge q\not\in A_{\alpha})$
$(\emph{boundedness}).$

Proof. It is straightforward. $\dashv$

###### Lemma 2.5

For all $\alpha,\beta\in\mathbb{R}^{o}$, there exists
$\gamma\in\mathbb{R}^{o}$, which specifies the cut $A_{\alpha}\cap A_{\beta}$.

Proof. The proof is easy. $\dashv$

###### Lemma 2.6

Let $\gamma:\mathbb{N}\rightarrow\mathbb{Q}$ be an upper bounded sequence,
i.e., $\exists M\in\hbox{\bf$\mathbb{Q}$}{}\forall n(\gamma(n)<M)$, then there
exists $\alpha\in\mathbb{R}^{o}$ which specifies
$A=\\{q\in\hbox{\bf$\mathbb{Q}$}{}\mid\exists n\in\mathbb{N}(q<\gamma(n))\\}$.

Proof. Let
$\alpha(n)=max\\{\gamma(0),\gamma(1),...,\gamma(n)\\}-\frac{1}{n+1}$. The
sequence $\alpha$ is strictly increasing and the set $A$ would be equal to
$\\{q\in\hbox{\bf$\mathbb{Q}$}{}\mid\exists n\in\mathbb{N}(q<\alpha(n))\\}$.
$\dashv$

The main difference between the collection of oriented reals $\mathbb{R}^{o}$
and other collections of Dedekind reals such as $\mathbb{R}^{d}$, extended
reals $\mathbb{R}^{be}$ and classical reals $\mathbb{R}^{e}$ is that the
former satisfies the _monotonicity_ property
($\forall\alpha\in\mathbb{R}^{o}\forall
p,q\in\hbox{\bf$\mathbb{Q}$}{}(p<q\wedge q\in A_{\alpha}\rightarrow p\in
A_{\alpha})$), whereas the others satisfy _strong monotonicity_ (for each
Dedekind cut $A$, $\forall p,q\in\hbox{\bf$\mathbb{Q}$}{}(p<q\wedge\neg\neg
q\in A\rightarrow p\in A)$). It is known that
$\mathbb{R}^{d}\subset\mathbb{R}^{be}\subset\mathbb{R}^{e}$ ([17], page 270).
In the following proposition, by using the Markov Principle ([17], page 204):

$\hbox{\bf MP}{}\ \ \ \ \
\forall\beta\in\mathbb{N}^{\mathbb{N}}(\neg\neg\exists
k\beta(k)=0\rightarrow\exists k\beta(k)=0)$,

we show that $\mathbb{R}^{o}\subseteq\mathbb{R}^{be}$.

###### Proposition 2.7

$(\hbox{\bf MP}{})$ For $\alpha\in\mathbb{R}^{o}$, the cut $A_{\alpha}$
satisfies _strong monotonicity_.

Proof. We show that for every $\alpha\in\mathbb{R}^{o}$, $A_{\alpha}$
satisfies _strong monotonicity_ , i.e., $\forall
p,q\in\hbox{\bf$\mathbb{Q}$}{}(p<q\wedge\neg\neg q\in A_{\alpha}\rightarrow
p\in A_{\alpha})$. Since $q\in A_{\alpha}\leftrightarrow\exists
k\in\mathbb{N}(q<\alpha(k))$, we have $\neg\neg q\in
A_{\alpha}\leftrightarrow\neg\neg\exists k\in\mathbb{N}(q<\alpha(k))$. By MP ,
$\neg\neg\exists k\in\mathbb{N}(q<\alpha(k))\leftrightarrow\exists
k\in\mathbb{N}(q<\alpha(k))$, (assume
$\beta(k)=\\{^{0~{}~{}~{}q<\alpha(k)}_{1~{}~{}~{}otherwise}$). So $\neg\neg
q\in A_{\alpha}\leftrightarrow q\in A_{\alpha}$, for any arbitrary
$q\in\hbox{\bf$\mathbb{Q}$}{}$. Then $p<q\wedge\neg\neg q\in
A_{\alpha}\leftrightarrow p<q\wedge q\in A_{\alpha}$, and since $A_{\alpha}$
satisfies _monotonicity_ , we derive $p\in A_{\alpha}$. $\dashv$

For choice sequences, if $\Phi$ is a total function from the collection of
choice sequences to natural numbers, the value of $\Phi$ for a sequence
$\alpha$ just depends on a finite segment of $\alpha$. Now this question seems
natural in constructive mathematics:

_how can we construct a total mapping
$\Phi:\mathbb{R}^{o}\rightarrow\mathbb{N}$?_

Assume we have a strategy to construct a total mapping
$\Phi:\mathbb{R}^{o}\rightarrow\mathbb{N}$. Then for any arbitrary sequence
$\alpha\in\mathbb{R}^{o}$, we must be able to construct a witness for
$\alpha$. Since $\Phi$ is well-defined, for any sequence $\beta=^{o}\alpha$,
we will have $\Phi(\beta)=\Phi(\alpha)$. Therefore, our strategy cannot depend
on any finite segment of $\alpha$. Because of this obstacle, one may expect
that it is not possible construct $\Phi$ unless $\Phi$ happens to be constant.
We fulfill this intention, using Brouwer’s weak continuity principle ([17],
page 209):

$\hbox{\bf WCN}{}\ \ \ \ \forall\alpha\in\hbox{\bf T}{}\exists
y(\Phi(\alpha)=y)\rightarrow\forall\alpha\in\hbox{\bf T}{}\exists
x\forall\beta\in\hbox{\bf
T}{}(\bar{\beta}x=\bar{\alpha}x\rightarrow\Phi\beta=\Phi\alpha)$,

where T is a spread and for each sequence $\alpha$,
$\bar{\alpha}x=\langle\alpha(0),\alpha(1),...,\alpha(x-1)\rangle$.

###### Proposition 2.8

$(\hbox{\bf WCN}{})$ Any (constructive) total function
$\Phi:\mathbb{R}^{o}\rightarrow\mathbb{N}$ is constant.

Proof. Let $\alpha,\beta\in\mathbb{R}^{o}$ be arbitrary, $\alpha\leq\beta$,
$M\in\hbox{\bf$\mathbb{Q}$}{}$ be an upper-bound for $\beta$. The set
$\mathbb{R}^{o}_{M}=\\{\gamma\in\mathbb{R}^{o}\mid\gamma<M\\}$ is a spread.
Since $\Phi$ is a total, by $\hbox{\bf WCN}{},$ there exists $t$ such that,
for all $\gamma\in\mathbb{R}^{o}_{M}$, if $\bar{\gamma}t=\bar{\alpha}t$ then
$\Phi(\gamma)=\Phi(\alpha)$. Find $\gamma\in\mathbb{R}^{o}_{M}$ passing
through $\bar{\alpha}t$ such that $\gamma=^{o}\beta$. By well-definedness of
$\Phi$ we conclude $\Phi(\beta)=\Phi(\alpha)$. So, for all
$\alpha,\beta\in\mathbb{R}^{o}$, if $\alpha\leq\beta$ then
$\Phi(\beta)=\Phi(\alpha)$. Now, for arbitrary
$\alpha,\beta\in\mathbb{R}^{o}$, define $\gamma(n)=min(\alpha(n),\beta(n))$.
Here, $\gamma\leq\alpha$ and $\gamma\leq\beta$. Then
$\Phi(\gamma)=\Phi(\alpha)$ and $\Phi(\gamma)=\Phi(\beta)$. Consequently
$\Phi(\beta)=\Phi(\alpha)$. $\dashv$

We showed that any total function from $\mathbb{R}^{o}$ to natural numbers is
constant in the presence of the Weak Continuity Principle, that is., if one
has a constructive method that for each oriented cut is able to introduce a
witness, a natural number, then the witness is unique. To avoid this, as a
replacement for natural numbers, we consider another category of objects
called almost natural numbers, and construct witnesses for oriented reals from
this category.

###### Definition 2.9

We let $\mathbb{N}^{\ast}$ be the set of all functions $\xi$ from $\mathbb{N}$
to $\mathbb{N}$ such that, for some $k$, for all $n$, $\xi(n)\leq\xi(n+1)\leq
k$. For all $\xi,\nu$ in $\mathbb{N}^{\ast}$ we define:

  * $\xi\leq\nu$ if and only if $\forall m\exists n(\xi(m)\leq\nu(n))$, and

  * $\xi=^{\ast}\nu$ if and only if $(\xi\leq\nu)\wedge(\nu\leq\xi)$.

We call $\mathbb{N}^{\ast}$, regarding equality $=^{\ast}$, the set of _almost
natural numbers_.

It is clear that $\mathbb{N}^{\ast}$ is classically isomorphic to $\mathbb{N}$
as a set. In fact, classically, elements in $\mathbb{N}^{\ast}$ are increasing
sequences that converge.

###### Lemma 2.10

For every $\xi\in\mathbb{N}^{\ast}$,

  1. 1.

$\neg\neg\exists n\forall m>n[\xi(m)=\xi(n)]$,

  2. 2.

$\neg\forall n\exists m>n~{}(\xi(m)\neq\xi(m+1))$.

Proof.

  1. 1.

To show $\forall\xi\in\mathbb{N}^{\ast}\neg\neg\exists n\forall
m>n[\xi(m)=\xi(n)]$, it is enough to show
$\neg\exists\xi\in\mathbb{N}^{\ast}\neg\exists n\forall m>n[\xi(m)=\xi(n)]$,
by the intuitionistic valid statement $\neg\exists xA(x)\leftrightarrow\forall
x\neg A(x)$. So assume for some $\xi_{0}\in\mathbb{N}^{\ast}$, $\neg\exists
n\forall m>n[\xi_{0}(m)=\xi_{0}(n)]$. Then, $\forall n\neg\forall
m>n[\xi_{0}(m)=\xi_{0}(n)]$.

Since $\xi_{0}\in\mathbb{N}^{\ast}$, there exists $k_{0}\in\mathbb{N}$ such
that $\xi_{0}(n)\leq\xi_{0}(n+1)\leq k_{0}$, for all $n\in\mathbb{N}$.

We prove that for every $k\in\mathbb{N}$, $(\forall n(\xi_{0}(n)\leq
k))\rightarrow(\forall n(\xi_{0}(n)<k))$ (1).

Assume there exists $t\in\mathbb{N}$ such that $\xi_{0}(t)=k$. Since $\xi_{0}$
is nondecreasing and $\forall n(\xi_{0}(n)\leq k))$, we have $\forall
n>t[\xi_{0}(n)=k]$. It contradicts with $\forall n\neg\forall
m>n[\xi_{0}(m)=\xi_{0}(n)]$. Hence $\forall n(\xi_{0}(n)<k)$.

Now let $k=k_{0}$. By (1), we derive $\forall n(\xi_{0}(n)\leq k_{0}-1)$.
Repeating using (1), we have got $\forall n(\xi_{0}(n)=0)$. It contradicts
with $\forall n\neg\forall m>n[\xi_{0}(m)=\xi_{0}(n)]$.

  2. 2.

Let $\xi\in\mathbb{N}^{\ast}$, there exists $k\in\mathbb{N}$ such that for all
$n$, $\xi(n)<k$. Assume

$\forall n\exists m>n~{}(\xi(m)\neq\xi(m+1))$. (2)

Define $f:\mathbb{N}\rightarrow\mathbb{N}$ as

$f(n)=\min\\{m\mid(m>n\wedge\xi(m)\neq\xi(m+1))\\}+1$.

Due to the assumption (2), the function $f$ is a constructive well defined
function. It is easy to see that $n<f(n)$ and $\xi(n)<\xi(f(n))$. Therefore,

$\xi(1)<\xi(f(1))<\xi(f(f(1)))<\xi(f(f(f(1))))<...$.

It contradicts with the fact that for all $n$, $\xi(n)<k$.

$\dashv$

It is worth mentioning that if $\gamma$ is an almost natural number then the
set $I=\\{n\in\mathbb{N}\mid\exists k~{}(n\leq\gamma(k))\\}$ is not necessary
finite, but it is _quasi-finite_ in the sense of [18], i.e., it is a subset of
a finite set. Moreover, its complement is _almost full_ [18], meaning that for
every strictly increasing sequence $\alpha$, one may find a natural number $n$
such that $\alpha(n)$ belongs to the complement of $I$.

Until now, the two notions of oriented reals and almost natural numbers have
been defined. The following proposition (accompanied by its proof) shows that
how these two notions are related to our intuition of orientation. In contrast
with proposition 2.8, we have

###### Proposition 2.11

There exists a non-constant total function
$\Phi:\mathbb{R}^{o}\rightarrow\mathbb{N}^{\ast}$.

Proof. Choose $d_{1}<d_{2}<...<d_{j}$ from $\mathbb{Q}$ for some fixed
$j\in\mathbb{N}$. For any $\beta\in\mathbb{R}^{o}$, we define

  * $\Phi(\beta)(0)=0$, and

  * $\Phi(\beta)(n)=max(\\{i\mid i\leq j\wedge(d_{i}\leq\beta(n))\\}\cup\\{0\\})$ for all $n\geq 1$.

One may easily check that the followings hold true for $\Phi$:

  1. 1.

$\Phi(\beta)\in\mathbb{N}^{\ast}$.

  2. 2.

$\Phi$ is well-defined, i.e., if $\alpha=^{o}\beta$ then
$\Phi(\alpha)=^{\ast}\Phi(\beta)$.

  3. 3.

$\Phi$ is non-constant. Assume two different oriented cuts $\alpha,\beta$,
such that $d_{1}\not\in A_{\alpha}$ and $d_{1}\in A_{\beta},d_{2}\not\in
A_{\beta}$. Then $\forall n\in\mathbb{N}$ $\Phi(\alpha)(n)=0$, whereas,
$\exists k\forall n>k$ $\Phi(\beta)(n)=1$.

The function $\Phi$ can be computed by the following algorithm as well:

  * 1.

Put $\Phi(\beta)(0):=0$;

  * 2.

put $t=1$;

  * 3.

For i=1 to j do:

    * $\\{$

    * while $(\beta(t)<d_{i})$ do:

      * $\\{$

      * define $\Phi(\beta)(t):=i-1$;

      * put $t=t+1$;

      * $\\}$

    * $\\}$

  * 4.

For k=t+1 to $\infty$ do:

    * $\\{$

    * Put $\Phi(\beta)(k):=j$;

    * $\\}$

The above algorithm checks the membership status of $d_{i}$’s in $A_{\beta}$
respectively, in an ordered manner. As soon as, the algorithm detects that
$d_{i}$ is in $A_{\beta}$, the future value of $\Phi(\beta)$ changes. We
defined $\Phi$ using the above algorithm, in order to evoke our sense of the
orientation explained in the introduction; the oriented real number $\beta$ is
assumed to be a moment in future. Rational numbers $d_{1}<d_{2}<...<d_{j}$ are
assumed as future moments which we examine the occurrence of $\beta$ at them,
as time passing. Using the information attained by the examination, the value
of $\Phi(\beta)\in\mathbb{N}^{\ast}$ is determined. $\dashv$

Therefore there exists a total function from $\mathbb{R}^{o}$ to
$\mathbb{N}^{\ast}$ which is not constant. _How a constructive total function
form $\mathbb{R}^{o}$ to $\mathbb{N}^{\ast}$ can be? or in other word, how can
we construct a total mapping
$\Phi:\mathbb{R}^{o}\rightarrow\mathbb{N}^{\ast}$?_ We answer this question
soon in section 2.2.

In the following, we illustrate that, although the oriented reals can not be
approximated by rationals, they can be approximated by almost rational
numbers, defined below.

###### Definition 2.12

Let $\mathbf{Q}$ be the set of all rational sequences
$\zeta:\mathbb{N}\rightarrow\mathbb{Q}$ in which

  * 1.

$\zeta$ is increasing, and

  * 2.

the image of $\zeta$ is a subset of a finite set.

For all $\zeta,\zeta^{\prime}\in\mathbf{Q}$, we define:

  * $\zeta\leq\zeta^{\prime}$ if and only if $\forall m\exists n(\zeta(m)\leq\zeta^{\prime}(n))$, and

  * $\zeta=\zeta^{\prime}$ if and only if $(\zeta\leq\zeta^{\prime})\wedge(\zeta^{\prime}\leq\zeta)$.

We call $\mathbf{Q}$ the set of _almost rational numbers_.

Note that $\mathbf{Q}$ is classically isomorphic to $\mathbb{Q}$ as a set. The
set of almost rational numbers $\mathbf{Q}$ is embedded into $\mathbb{R}^{o}$,
by $\zeta\mapsto\hat{\zeta}$ where $\hat{\zeta}(n)=\zeta(n)-\frac{1}{n+1}$,
for $n\in\mathbb{N}$, is an oriented real specifying the cut
$\\{q\in\mathbb{Q}\mid\exists k(q<\zeta(k))\\}$.

###### Definition 2.13

For every $\alpha\in\mathbb{R}^{o}$, and $r\in\mathbb{Q}$, define
$(\alpha+r)\in\mathbb{R}^{o}$, by $(\alpha+r)(n):=\alpha(n)+r$, for every $n$.

###### Definition 2.14

We say that an oriented real $\alpha$ can be approximated by rationals, if for
each $n$, there exists $q\in\mathbb{Q}$ such that
$(q\leq\alpha\wedge\alpha\leq q+2^{-n})$.

As we explained in the introduction, since moments of time sink back into the
past, it is not possible to approximate them using rationals. The following
proposition shows that oriented reals cannot be approximated by rational
numbers either.

###### Proposition 2.15

$(\neg\exists$-PEM$)$ It is false that every oriented real can be approximated
by rationals.

Proof. For an arbitrary bounded increasing sequence $\alpha$ of rationals, let
$\beta_{\alpha}$ be an oriented real defined by
$\beta_{\alpha}(n)=\alpha(n)-\frac{1}{n+1}$, for $n\in\mathbb{N}$, which
specifies the cut $A=\\{q\in\hbox{\bf$\mathbb{Q}$}{}\mid\exists
k(q<\alpha(k))\\}$. If $\beta_{\alpha}$ can be approximated by rationals, then
$\beta_{\alpha}$ is a Cauchy sequence. Consequently, the sequence $\alpha$ is
also Cauchy and so it converges. Since $\alpha$ was arbitrary, it would imply
that every bounded monotone sequence has a limit in Brouwerian real line. That
contradicts with $\neg\exists$-PEM ([17], page 268). $\dashv$

###### Definition 2.16

We say that an oriented real $\alpha$ can be approximated by almost rationals,
if $\forall
n\exists\zeta\in\mathbf{Q}(\hat{\zeta}\leq\alpha\wedge\alpha\leq(\hat{\zeta}+2^{-n}))$.

###### Proposition 2.17

Every oriented real can be approximated by almost rationals numbers.

Proof. Let $\beta\in\mathbb{R}^{o}$, and $M\in\mathbb{Q}$ be an upper bound
for $\beta$. We define $\zeta$ by induction. Let $\zeta(0)=\beta(0)$, and
assume we have defined $\zeta(k)$. We define

$\zeta(k+1)=\zeta(k)+2^{-n}t$, where $t\in\mathbb{N}$ and
$\zeta(k)+2^{-n}t\leq\beta(k+1)<\zeta(k)+2^{-n}(t+1)$.

Note that $\zeta\in\mathbf{Q}$, since
$image(\zeta)\subseteq\\{\beta(0)+2^{-n}t\mid\beta(0)+2^{-n}t<M,t\in\mathbb{N}\\}$
and the latter is not infinite. We claim that
$\hat{\zeta}\leq\beta\wedge\beta\leq\hat{\zeta}+2^{-n}$. For
$\hat{\zeta}\leq\beta$, we have for each $k\in\mathbb{N}$ there exists $m$
such that $\zeta(k)<\beta(m)$. For the second clause, i.e.,
$\beta\leq\hat{\zeta}+2^{-n}$, we have $\beta(k)<\zeta(k)+2^{-n}$.$\dashv$

A subset $B$ of $\mathbb{N}$ is called (intuitionistically) enumerable if it
is the image of a function on natural numbers (see [1]). Assuming the Kripke
schema ([17], page 236):

$\hbox{\bf KS}{}\ \ \ \ \forall X\exists\alpha\forall n[n\in
X\leftrightarrow\exists m(\alpha(n,m)=0)]$.

every inhabited 444A set $B$ is inhabited if $\exists x(x\in B)$. We indicate
that by $B\\#\emptyset$. subset of natural numbers is (intuitionistically)
enumerable [1].

###### Lemma 2.18

If $B\subseteq\mathbb{Q}$ is (intuitionistically) enumerable and bounded from
above, then there exists an oriented real number $\alpha$ such that
$A_{\alpha}=C(B)=\\{q\in\hbox{\bf$\mathbb{Q}$}{}\mid\exists p(p\in B\wedge
q<p)\\}$.

Proof. As $B\subseteq\mathbb{Q}$ is (intuitionistically) enumerable, there
exists a sequence $\gamma:\mathbb{N}\rightarrow\mathbb{Q}$ such that $\forall
q\in\hbox{\bf$\mathbb{Q}$}{}[q\in B\leftrightarrow\exists m(\gamma(m)=q)]$.
Since $B$ is bounded from above, the sequence $\gamma$ has an upper bound. By
lemma 2.6, there exists an oriented number $\alpha$ such that
$A_{\alpha}=C(B)$. $\dashv$

Assume $B\subseteq\mathbb{Q}$ is (intuitionistically) enumerable and upper
bounded. We define the supremum of $B$, $sup(B)$, to be the oriented real
number $\alpha$ that $A_{\alpha}=C(B)$. Assuming the Kripke schema, KS, every
upper bounded subset of $\mathbb{Q}$ has supremum.

###### Proposition 2.19

$(\hbox{\bf KS}{}).$ Assume $B\subseteq\mathbb{Q}$ is upper bounded. Then for
$\alpha=sup(B)$, the following hold.

  * 1.

$\forall q\in\mathbb{Q}(q\in B\rightarrow q\leq\alpha)$,

  * 2.

$\forall p\in\mathbb{Q}(p<\alpha\rightarrow\exists q\in\mathbb{Q}(q\in B\wedge
p<q))$.

Proof. (1.) Assume $p\in B$. For any $q<p$, we have $q\in C(B)=A_{\alpha}$.
Then there exists $n$ such that $q<\alpha(n)$. By items $1$ and $4$ of
proposition 2.3, we have $p\leq\alpha$.

(2.) Assume $p<\alpha$. Then $p\in A_{\alpha}$. As $A_{\alpha}=C(B)$, there
exists $q\in B$ such that $p<q$. $\dashv$

###### Definition 2.20

Let $D\subseteq\mathbb{Q}$ is lower bounded, and
$B=\\{p\in\mathbb{Q}\mid\forall q(q\in D\rightarrow q\geq p)\\}$. We define
the infimum of $D$, $inf(D)$, to be the supremum of $B$.

###### Proposition 2.21

$(\hbox{\bf KS}{}).$ Assume $D\subseteq\mathbb{Q}$ is lower bounded. Then for
$\alpha=inf(D)$, the following hold.

  * 1.

$\forall q\in\mathbb{Q}(q\in D\rightarrow\alpha\leq q)$,

  * 2.

$\forall p\in\mathbb{Q}[(\forall q\in\mathbb{Q}(q\in D\rightarrow p\leq
q))\rightarrow p\leq\alpha]$.

Proof. (1.) $\alpha=inf(D)$ is an oriented real such that
$A_{\alpha}=C(B)=\\{q\in\hbox{\bf$\mathbb{Q}$}{}\mid\exists p(p\in B\wedge
q<p)\\}$, where $B=\\{p\in\mathbb{Q}\mid\forall q(q\in D\rightarrow q\geq
p)\\}$. Let $q\in D$. Since
$A_{\alpha}=\\{q\in\hbox{\bf$\mathbb{Q}$}{}\mid\exists n(q<\alpha(n))\\}$ and
$\alpha$ is strictly increasing, we have $\alpha(n)\in A_{\alpha}$, for every
$n$. By the equality $A_{\alpha}=C(B)$, it is derived that for each $n$, there
exists $p\in B$ such that $\alpha(n)<p$. Then by definition of $B$,
$\alpha(n)<q$.

(2.) Assume $p\in\mathbb{Q}$ is such that for all $q\in D$, $p\leq q$. Then
$p\in B$, and thus for all $p_{0}<p$, $p_{0}\in C(B)=A_{\alpha}$. According to
definition of $A_{\alpha}$, there exists $n$ such that $p_{0}<\alpha(n)$.
Then, $(\forall p_{0}<p)\exists n(p_{0}<\alpha(n))$. By items $1$ and $4$ of
proposition 2.3, we have $p\leq\alpha$.$\dashv$

###### Theorem 2.22

$(\hbox{\bf KS}{})$. _The monotone convergence theorem_. For every upper
bounded nondecreasing sequence $(\alpha_{n})_{n\in\mathbb{N}}$ of oriented
reals, there exists an oriented real $\alpha$ such that

  * $1.$

$\forall n(\alpha_{n}\leq\alpha)$,

  * $2.$

$(\forall p\in\mathbb{Q})[(p<\alpha)\rightarrow(\exists m\forall n(n\geq
m\rightarrow p<\alpha_{n}))]$.

Proof. Let $B=\\{p\in\mathbb{Q}\mid\exists n(p<\alpha_{n})\\}$. The set $B$ is
upper bounded, so let $\alpha=sup(B)$. For all $n,m\in\mathbb{N}$, we have
$\alpha_{n}(m)\in B$, since $\alpha_{n}$ is a strictly decreasing sequence of
rationals. By proposition 2.19, we have for each $n$, for all $m$,
$\alpha_{n}(m)<\alpha_{n}(m+1)\leq\alpha$. So, by items (1) and (4) of
proposition 2.3, for each $n$, $\alpha_{n}\leq\alpha$.

If $p<\alpha$, then by proposition 2.19, there exists $q\in B$ such that
$p<q$. By definition of $B$, there exists $m\in\mathbb{N}$, $p<q<\alpha_{m}$.
As $(\alpha_{n})_{n\in\mathbb{N}}$ is a non-decreasing sequence, we have for
all $n\geq m$, $p<\alpha_{n}$. $\dashv$

Among the Dedekind lines $\mathbb{R}^{e}$, $\mathbb{R}^{be}$, and
$\mathbb{R}^{d}$ introduced in [17], the line $\mathbb{R}^{be}$ is much
similar to $\mathbb{R}^{o}$. The only difference between the cuts of
$\mathbb{R}^{be}$ and the cuts of $\mathbb{R}^{o}$ is that the first one
satisfies _strong monotonicity_ , whereas the second one just fulfils
monotonicity. On the other hand, as we have already noted, the reals in
$\mathbb{R}^{be}$ cannot be approximated by rationals, as desired as a
requirement for the temporal line. Hence, the line $\mathbb{R}^{be}$ could be
assumed as an appropriate mathematical model for the temporal line if there
was a _continuity principle_ for it, like Brouwer’s continuity principle for
_Cauchy_ reals $\mathbb{R}$. We note that the line $\mathbb{R}$ is the
_Cauchy_ completion of $\mathbb{Q}$ and the line $\mathbb{R}^{be}$ is the
_order completion_ of $\mathbb{Q}$.

Let us define a mapping $\Psi:\mathbb{R}^{o}\rightarrow\mathbb{R}^{be}$ as
follows:

$\Psi(\alpha):=\\{r\in\mathbb{Q}\mid(\exists
s\in\mathbb{Q})[r<s\wedge\neg\neg\exists n(s<\alpha(n))]\\}$.

One can easily check that $\Psi$ is well-defined and for all
$\alpha\in\mathbb{R}^{o}$, $\Psi(\alpha)\in\mathbb{R}^{be}$.

###### Proposition 2.23

$(\hbox{\bf KS}{})$. The mapping
$\Psi:\mathbb{R}^{o}\rightarrow\mathbb{R}^{be}$ is surjective.

Proof. Assuming the Kripke schema, it can be shown that every inhabited subset
of natural numbers is (intuitionistically) enumerable [1].

Let $S\in\mathbb{R}^{be}$, and $\gamma:\mathbb{N}\rightarrow S$ enumerates
$S$, i.e., $\forall q\in\mathbb{Q}[q\in S\leftrightarrow\exists
m(\gamma(m)=q)]$. Define

$\alpha(n)=max\\{\gamma(0),\gamma(1),...,\gamma(n)\\}-\frac{1}{n+1}.$

The sequence $\alpha$ is in $\mathbb{R}^{o}$. We claim $\Psi(\alpha)=S$.

Assume $r\in\Psi(\alpha)$. Then there exists $s\in\mathbb{Q}$ such that $r<s$
and $\neg\neg\exists n(s<\alpha(n))$. For the sake of argument, assume
$\exists n(s<\alpha(n))$. Note that for all $m$, $\alpha(m)\in S$. By the
_monotonicity_ of $S$, we have $s\in S$. Thus $\neg\neg s\in S$. By the
_strong monotonicity_ , we have $r\in S$.

For the converse, assume $r\in S$. Let $k$ be such that $\gamma(k)=r$. By
openness, there exists $r^{\prime}\in S$ such that $r<r^{\prime}$, and let
$\gamma(m)=r^{\prime}$ for some $m$. Choose $n>m$ such that
$r^{\prime}-r<\frac{1}{n+1}$. Then we have $r<\alpha(n)$ and thus
$r\in\Psi(\alpha)$. $\dashv$

###### Proposition 2.24

For all $\alpha,\beta\in\mathbb{R}^{o}$

$\alpha<\beta$ implies $\Psi(\alpha)<\Psi(\beta)$.

Proof. The relation “$<$” for cuts $S,T$ is defined as follows: $S<T:=\exists
r>0(S+r\subset T)$ (see definition 5.4 of [17]). Suppose $\alpha<\beta$. Then
for some $n$, we have $\forall m(\alpha(m)<\beta(n))$. Let
$r=\beta(n+1)-\beta(n)$. It is easy to check that
$\Psi(\alpha)+r\subset\Psi(\beta)$. $\dashv$

###  2.1 Arithmetic in $\mathbb{R}^{o}$

As explained in the introduction, we propose $\mathbb{R}^{o}$ as a model for
the temporal line. Each oriented real illustrates a moment. Then what does it
mean to add or multiply two moments? What is a proper arithmetic of the
temporal line?

The real line $\mathbb{R}^{be}$ is a _field_ due to the _strong monotonicity_
property of its elements. A field $\langle F,+,\cdot\rangle$ is an algebraic
structure with two _functions_ $+$ and $\cdot$ from $F\times F$ to $F$, where
$F$ equipped with the functions $+$ or $\cdot$, is a group, and $\cdot$ is
distributed over $+$. In our model, $\mathbb{R}^{o}$, the operations $+$ and
$\cdot$ are _relations_ instead of being functions, i.e, it is an algebraic
structure known as a _hyperstructure_ 555For similar definitions and
applications of hyper-algebraic structures as arithmetic, see [13, 8, 9]..

###### Definition 2.25

A _h-field_ is a tuple $\langle F,+,\ast,0,1\rangle$ where $+\subseteq F\times
F\times F$ and $\ast\subseteq F\times F\times F$ satisfy the following axioms:

  * -

_Inhabitance_.

$\begin{array}[]{l}\begin{array}[]{ccccc}(\forall x,y)(\exists
z)+(x,y,z)&&(\forall x,y)(\exists z)\ast(x,y,z).\end{array}\end{array}$

  * -

_Identity_.

$\begin{array}[]{l}\begin{array}[]{ccccc}(\forall x)+(x,0,x)&&(\forall
x)\ast(x,1,x).\end{array}\end{array}$

  * -

_Inverse_.

$\begin{array}[]{l}\begin{array}[]{ccccc}(\forall x\exists
y)+(x,y,0)&&(\forall x\exists y)\ast(x,y,1).\end{array}\end{array}$

  * -

_Commutativity_.

$\begin{array}[]{l}\begin{array}[]{ccccc}(\forall
x,y,z)(+(x,y,z)\leftrightarrow+(y,x,z))&&(\forall
x,y,z)(\ast(x,y,z)\leftrightarrow\ast(y,x,z)).\end{array}\end{array}$

  * -

_Associativity_.

$\begin{array}[]{l}\begin{array}[]{ccccc}(\forall
x,y,z,w,v,u)((+(x,y,w)\wedge+(w,z,v)\wedge+(y,z,u))\rightarrow+(x,u,v))\\\
(\forall
x,y,z,w,v,u)((+(x,y,w)\wedge+(x,u,v)\wedge+(y,z,u))\rightarrow+(w,z,v))\end{array}\end{array}$

$\begin{array}[]{l}\begin{array}[]{ccccc}(\forall
x,y,z,w,v,u)((\ast(x,y,w)\wedge\ast(w,z,v)\wedge\ast(y,z,u))\rightarrow\ast(x,u,v))\\\
(\forall
x,y,z,w,v,u)((\ast(x,y,w)\wedge\ast(x,u,v)\wedge\ast(y,z,u))\rightarrow\ast(w,z,v)).\end{array}\end{array}$

  * -

_Distributivity_.

$\begin{array}[]{l}\begin{array}[]{ccccc}(\forall
x,y,z,w,v,u,r)((\ast(x,v,w)\wedge+(y,z,v)\wedge\ast(x,y,u)\wedge\ast(x,z,r))\rightarrow+(u,r,w))\\\
(\forall
x,y,z,w,v,u,r)((+(u,r,w)\wedge+(y,z,v)\wedge\ast(x,y,u)\wedge\ast(x,z,r))\rightarrow\ast(x,v,w)).\end{array}\end{array}$

We define an _addition relation_ $+$, and a _multiplication relation_ $\ast$
on $\mathbb{R}^{o}$ as follows:

For $\alpha,\beta,\gamma\in\mathbb{R}^{o}$, we let

  * 1.

$+(\alpha,\beta,\gamma)$ if and only if
$\Psi(\alpha)+\Psi(\beta)=\Psi(\gamma)$, and

  * 2.

$\ast(\alpha,\beta,\gamma)$ if and only if
$\Psi(\alpha)\cdot\Psi(\beta)=\Psi(\gamma)$.

###### Proposition 2.26

$(\hbox{\bf KS}{})$. $\langle\mathbb{R}^{o},+,\ast,\hat{0},\hat{1}\rangle$ is
a h-field.

Proof. The proof is straightforward by using the fact that $\mathbb{R}^{be}$
is a field. $\dashv$

###  2.2 The Oriented Continuity Principle

In this part, we propose the oriented continuity principle which expresses
formally our sense of the notion of orientation. To do this, we study total
functions $\Phi$ from $(0,1]^{o}$ to $\mathbb{N}^{\ast}$, where
$(0,1]^{o}=\\{\alpha\in\mathbb{R}^{o}\mid\hat{0}<\alpha\leq\hat{1}\\}$.
Similarly, we let
$[0,1]^{o}=\\{\alpha\in\mathbb{R}^{o}\mid\hat{0}\leq\alpha\leq\hat{1}\\}$. We
assume such function $\Phi$ is well-defined, i.e., $\alpha=^{o}\beta$ implies
$\Phi(\alpha)=^{\ast}\Phi(\beta)$.

###### Lemma 2.27

$(\hbox{\bf WCN}{})$. Let $\Phi\in(0,1]^{o}\rightarrow\mathbb{N}^{\ast}$ be
total. Then
$\forall\alpha,\beta\in(0,1]^{o}(\alpha\leq\beta\rightarrow\Phi(\alpha)\leq\Phi(\beta))$.

Proof. For $\theta\in\mathbb{N}^{\ast}$ define
$N_{\theta}=\\{m\in\mathbb{N}\mid\exists n(m\leq\theta(n))\\}$. We need to
show that $\alpha\leq\beta$ implies $N_{\Phi(\alpha)}\subseteq
N_{\Phi(\beta)}$. Assume $\Phi(\alpha)(n)=k$ for some $n,k\in\mathbb{N}$. The
set $(0,1]^{o}$ is a spread and $\Phi$ is total, so by WCN we can find a $t$
such that for each $\delta\in(0,1]^{o}$, $\bar{\delta}t=\bar{\alpha}t$ implies
$\Phi(\delta)(n)=\Phi(\alpha)(n)$. We have, for each $\delta\in(0,1]^{o}$, if
$\bar{\delta}t=\bar{\alpha}t$ then $k\in N_{\Phi(\delta)}$. Since
$\alpha\leq\beta$ there exists $\lambda\in(0,1]^{o}$ such that
$\bar{\lambda}t=\bar{\alpha}t$ and $\lambda=^{o}\beta$. Note that $\Phi$ is
well-defined, therefore $k\in N_{\Phi(\beta)}$. Since we assumed
$k\in\mathbb{N}$ to be arbitrary, we have $N_{\Phi(\alpha)}\subseteq
N_{\Phi(\beta)}$. $\dashv$

The following theorem is by W. Veldman from our correspondence with him.

###### Theorem 2.28

$(\hbox{\bf WCN}{})$. Let $\Phi$ be a total well-defined function from
$(0,1]^{o}$ to $\mathbb{N}^{\ast}$. Then

$\forall\alpha\in(0,1]^{o}\neg\neg\exists
q\in\mathbb{Q}[q<\alpha\wedge\forall\beta\in(0,1]^{o}[(q<\beta\leq\alpha\rightarrow\Phi(\alpha)=^{\ast}\Phi(\beta)]]$.

Proof. Let $\alpha\in(0,1]^{o}$. Since $\Phi(\alpha)\in\mathbb{N}^{\ast}$, by
lemma 2.10, $\neg\neg\exists n\forall m>n[\Phi(\alpha)(m)=\Phi(\alpha)(n)]$.
For the sake of the argument, assume $\exists n\forall
m>n[\Phi(\alpha)(m)=\Phi(\alpha)(n)]$, and let $n_{0}$ be such that $\forall
m>n_{0}[\Phi(\alpha)(m)=\Phi(\alpha)(n_{0})]$. By WCN, there is a $t$ such
that for all $\beta$ in the spread $(0,1]^{o}$, if
$\bar{\beta}t=\bar{\alpha}t$ then $\Phi(\beta)(n_{0})=\Phi(\alpha)(n_{0})$. It
follows that, for all $\beta\in(0,1]^{o}$ passing through
$\bar{\alpha}t=\langle\alpha 0,\alpha 1,...,\alpha(t-1)\rangle$,
$\Phi(\alpha)\leq\Phi(\beta)$.

Define $q:=\alpha(t-1)$. Assume that $\beta\in(0,1]^{o}$ and
$q<\beta\leq\alpha$. Find $\lambda$ passing through $\bar{\alpha}t$ such that
$\lambda=^{o}\beta$, and conclude
$\Phi(\alpha)\leq\Phi(\lambda)=^{\ast}\Phi(\beta)$. On the other hand, by
lemma 2.27, since $\beta\leq\alpha$ we have $\Phi(\beta)\leq\Phi(\alpha)$.
Hence, for every $\beta$ satisfying $q<\beta\leq\alpha$,
$\Phi(\beta)=^{\ast}\Phi(\alpha)$. This conclusion is obtained from the
assumption: $\exists n\forall m>n[\Phi(\alpha)(m)=\Phi(\alpha)(n)]$. As we
know $\neg\neg\exists n\forall m>n[\Phi(\alpha)(m)=\Phi(\alpha)(n)]$, we may
conclude $\forall\alpha\in(0,1]^{o}\neg\neg\exists
q\in\mathbb{Q}[q<\alpha\wedge\forall\beta\in(0,1]^{o}[(q<\beta\leq\alpha\rightarrow\Phi(\alpha)=^{\ast}\Phi(\beta)]]$.
$\dashv$

Note that for any $\Phi:(0,1]^{o}\rightarrow\mathbb{N}^{\ast}$, there exists
$k\in\mathbb{N}$ such that for all $n$, $\Phi(\hat{1})(n)\leq k$, by
definition of almost natural numbers.

###### Proposition 2.29

$(\hbox{\bf WCN}{})$. Let $\Phi:(0,1]^{o}\rightarrow\mathbb{N}^{\ast}$ be
total, $k\in\mathbb{N}$ be such that for all $n$, $\Phi(\hat{1})(n)\leq k$,
and $T_{i}=\\{q\in\mathbb{Q}\mid\Phi(\hat{q})=^{\ast}\underline{i}\\}$, for
$0\leq i\leq k$, where $\underline{i}=\langle
i,i,i,...\rangle\in\mathbb{N}^{\ast}$. Then

  * (a).

if $i<j$, $q\in T_{i}$ and $p\in T_{j}$, then $q<p$,

  * (b).

for each $\alpha\in(0,1]^{o}$, if $A_{\alpha}\cap T_{i}\\#\emptyset$, then
$\Phi(\alpha)\geq\underline{i}$,

  * (c).

for each $\alpha\in(0,1]^{o}$, if $\Phi(\alpha)=^{\ast}\underline{i}$, then
$\neg(A_{\alpha}\cap T_{i}=\emptyset)$,

  * (d).

for each $\alpha\in(0,1]^{o}$, $\neg\neg\exists i[(0\leq i\leq
k)\wedge(\Phi(\alpha)=^{\ast}\underline{i})]$.

Proof. (a) and (b) are derived by lemma 2.27. (c) follows from theorem 2.28,
and (d) is a consequence of the definition of almost natural numbers, and the
fact
$\forall\alpha\in(0,1]^{\ast}(\Phi(\alpha)\leq\Phi(\hat{1})\leq\underline{k})$.
$\dashv$

###### Theorem 2.30

$(\hbox{\bf WCN}{}+\hbox{\bf KS}{})$. Assume
$\Phi:(0,1]^{o}\rightarrow\mathbb{N}^{\ast}$. Let $\delta_{i}=inf(T_{i})$, for
$T_{i}$s $0\leq i\leq k$, defined above, and $E=\\{\delta_{i}\mid 0\leq i\leq
k\\}$. For each $\alpha\in(0,1]^{o}$, define
$L_{\alpha}=\\{\gamma\in(0,1]^{o}\mid\gamma<\alpha\\}$. Then

$\forall\alpha,\beta\in(0,1]^{o}[(L_{\alpha}\cap E=L_{\beta}\cap
E)\rightarrow\neg\neg(\Phi(\alpha)=^{\ast}\Phi(\beta))]$.

Proof. First, observe that for $\alpha\in(0,1]^{o}$ and $0\leq i\leq k$,

  * (a)

$\delta_{i}\in L_{\alpha}\rightarrow\neg(A_{\alpha}\cap T_{i}=\emptyset)$, and

  * (b)

$A_{\alpha}\cap T_{i}\\#\emptyset\rightarrow\delta_{i}\in L_{\alpha}$

To prove the theorem, let $\alpha,\beta\in(0,1]^{o}$ such that $L_{\alpha}\cap
E=L_{\beta}\cap E$ and $\neg(\Phi(\alpha)=^{\ast}\Phi(\beta))$ (1). we want to
derive a contradiction. By proposition 2.29(d), we have $\neg\neg\exists
i[(0\leq i\leq k)\wedge(\Phi(\alpha)=^{\ast}\underline{i})]$ (2), and
$\neg\neg\exists i[(0\leq i\leq k)\wedge(\Phi(\beta)=^{\ast}\underline{i})]$
(3). Applying the intuitionistic valid statement
$\neg\neg(\varphi\wedge\psi)\leftrightarrow\neg\neg\varphi\wedge\neg\neg\psi$
to (1), (2) and (3), gives $\neg\neg(\exists i,j[(0\leq i,j\leq
k)\wedge(\Phi(\alpha)=^{\ast}\underline{i})\wedge(\Phi(\beta)=^{\ast}\underline{j})\wedge\neg(\Phi(\alpha)=^{\ast}\Phi(\beta))])$,
which implies

$\neg\neg(\exists i,j[(0\leq i,j\leq
k)\wedge(\Phi(\alpha)=^{\ast}\underline{i})\wedge(\Phi(\beta)=^{\ast}\underline{j})\wedge\neg(\underline{i}=^{\ast}\underline{j})])$.

Now, for the sake of argument, assume

$\psi\equiv\exists i,j[(0\leq i,j\leq
k)\wedge(\Phi(\alpha)=^{\ast}\underline{i})\wedge(\Phi(\beta)=^{\ast}\underline{j})\wedge\neg(\underline{i}=^{\ast}\underline{j})]$.

Either $i<j$ or $j<i$, and assume the first case. By proposition 2.29(b),
$\neg(A_{\alpha}\cap T_{j}\\#\emptyset)$, i.e, $A_{\alpha}\cap
T_{j}=\emptyset$. So by $(a)$, $\neg(\delta_{j}\in L_{\alpha})$, and by the
assumption $L_{\alpha}\cap E=L_{\beta}\cap E$, we have $\neg(\delta_{j}\in
L_{\beta})$. By $(b)$, $\neg(A_{\beta}\cap T_{j}\\#\emptyset)$, that is,
$A_{\beta}\cap T_{j}=\emptyset$. But it contradicts with proposition 2.29(c),
and thus $\neg\neg(\Phi(\alpha)=^{\ast}\Phi(\beta))$. By assuming $\psi$, we
derived a contradiction. By
$(\psi\rightarrow\varphi)\rightarrow(\neg\neg\psi\rightarrow\neg\neg\varphi)$,
assuming $\neg\neg\psi$ yield also a contraction. So
$\neg\neg(\Phi(\alpha)=^{\ast}\Phi(\beta))$.$\dashv$

The theorem says that if $\Phi$ is a total constructive function from the
temporal interval $(0,1]^{o}$ to $\mathbb{N}^{\ast}$ then there exists a _non-
infinite_ subset $E$ of $[0,1]^{o}$ such that

$\forall\alpha,\beta\in(0,1]^{o}[(L_{\alpha}\cap E=L_{\beta}\cap
E)\rightarrow\neg\neg(\Phi(\alpha)=^{\ast}\Phi(\beta))]$.

As mentioned in the introduction, a moment $\alpha$ on the temporal continuum
is a _flow_ from the past, i.e., the moment $\alpha$ is adhered to the
collection of all moments that happened before it. We think up of this flow as
$L_{\alpha}$. To construct a witness for a moment $\alpha$, the occurrence of
$\alpha$ is compared with the occurrence of _non-infinitely_ many moments in
$E$. If the results of the comparison are the same for two moments $\alpha$
and $\beta$, then it is not false that the witness constructed for $\alpha$ is
also a witness for $\beta$. In this way, the theorem 2.30 formalizes our sense
of the orientation discussed in the introduction.

We believe that the following form of theorem 2.30 is plausible to be accepted
as a principle, which we name the _oriented continuity principle_ , $\hbox{\bf
OCP}{}:$

Oriented Continuity Principle:

$\begin{array}[]{l}$For~{}every~{}$\Phi:(0,1]^{o}\rightarrow\mathbb{N}^{\ast},\\\
$if$~{}\forall\alpha\in(0,1]^{o}~{}\exists\xi\in\mathbb{N}^{\ast}~{}[\Phi(\alpha)=^{\ast}\xi]~{}$then~{}there~{}exists~{}a~{}\emph{non-
infinite}~{}subset$~{}E\subseteq[0,1]^{o}~{}$such~{}that$\\\
\forall\alpha,\beta\in(0,1]^{o}~{}[(L_{\alpha}\cap E=L_{\beta}\cap
E)\rightarrow(\Phi(\alpha)=^{\ast}\Phi(\beta))].\\\ \end{array}$

##  3 Topologies For Continuum

What is the proper topology of the intuitive temporal continuum? Clearly, the
topology must display the notion of orientation. As is emphasized in the
introduction, since $t$ sinks back to the past, the moment $t$ is not
distinguishable from moments in $(10:40,11:30]$. Therefore, if we have a
constructive method to introduce witnesses for all moments on the temporal
line then the witness for the moment $t$ is also the witness for all moments
in $(10:40,11:30]$. Thus, for a suitable topology of the temporal continuum,
it seems that every total function is continuous.

As Brouwer distinguishes between the intuitive continuum and the “full
continuum” of the unfinished elements of the unit segment (intuitionistic real
line) ([14], page 74), we do not claim that our oriented line, equipped with
topologies defined below is exactly the intuitive temporal continuum based on
our intuition. We believe that the only difference between the intuitive
continuum and the intuitive temporal continuum is taking into account the
notion of _orientation_ as a new aspect of the continuum besides
_inexhaustibility_ and _non-discreteness_. Our oriented line and the following
topologies are our suggestions for modeling the temporal continuum.

###  3.1 The Oriented Topology

###### Definition 3.1

The Oriented Topology on the temporal interval $(0,1]^{o}$. A subset $U$ of
$(0,1]^{o}$ is called open if and only if for every $\alpha$ in $U$ there
exists a _non-infinite_ set $E\subseteq[0,1]^{o}$ such that
$S_{E}(\alpha)\subseteq U$, where $S_{E}(\alpha)=\\{\beta\in(0,1]^{o}\mid
L_{\alpha}\cap E=L_{\beta}\cap E\\}$. We indicate the set of all open subsets
by $\mathcal{T}_{1}$, and refer to $((0,1]^{o},\mathcal{T}_{1})$ as the
oriented topological space.

###### Example 3.2

The interval
$(\frac{1}{4},\frac{1}{2}]^{o}=\\{\alpha\in(0,1]^{o}\mid\hat{\frac{1}{4}}<\alpha\leq\hat{\frac{1}{2}}\\}$
is open. For $\alpha\in(\frac{1}{4},\frac{1}{2}]^{o}$, let
$E=\\{\hat{\frac{1}{4}},\hat{\frac{1}{2}}\\}$. Then $L_{\alpha}\cap
E=\\{\hat{\frac{1}{4}}\\}$, and
$S_{E}(\alpha)\subseteq(\frac{1}{4},\frac{1}{2}]$.

We must show that the class of all open subsets of $(0,1]^{o}$ is a topology.
It is obvious that the empty set and $(0,1]^{o}$ belong to $\mathcal{T}_{1}$.
One may easily see that the topology $\mathcal{T}_{1}$ is closed under
arbitrary union. The next proposition shows that it is also closed under
finite intersection.

###### Proposition 3.3

The class of open subsets of $(0,1]^{o}$ is closed under finite intersection.

Proof. Let $U_{1}$ and $U_{2}$ be open, and $\alpha\in U_{1}\cap U_{2}$. Let
the non-infinite sets $E_{1},E_{2}$ be such that $S_{E_{1}}(\alpha)\subseteq
U_{1}$, and $S_{E_{2}}(\alpha)\subseteq U_{2}$. It is easily seen that
$S_{E_{1}\cup E_{2}}(\alpha)\subseteq U_{1}\cap U_{2}$. $\dashv$

###  3.2 The Ordinary Topology

We first define the notion of semi-metric spaces, and then introduce the
_ordinary_ topology based on this notion.

###### Definition 3.4

Assume $\mathbb{Q}^{+}=\\{q\in\mathbb{Q}\mid q>0\\}$. A semi-metric space is a
couple $(I,\hbox{\bf d}{})$, where $I$ is a set and d is a relation $\hbox{\bf
d}{}\subseteq I\times I\times\mathbb{Q}^{+}$ satisfying the following
properties

  * P1.

$\forall x\in I\forall q\in\mathbb{Q}^{+}\hbox{\bf d}{}(x,x,q)$,

  * P2.

$\forall x,y\in I\exists q\in\mathbb{Q}^{+}\hbox{\bf d}{}(x,y,q)$,

  * P3.

$\forall x,y\in I\forall q,p\in\mathbb{Q}^{+}(\hbox{\bf d}{}(x,y,q)\wedge
q<p\rightarrow\hbox{\bf d}{}(x,y,p))$,

  * P4.

$\forall x,y\in I\forall q\in\mathbb{Q}^{+}(\hbox{\bf
d}{}(x,y,q)\leftrightarrow\hbox{\bf d}{}(y,x,q))$,

  * P5.

$\forall x,y,z\in I\forall q,p\in\mathbb{Q}^{+}(\hbox{\bf
d}{}(x,y,q)\wedge\hbox{\bf d}{}(y,z,p)\rightarrow\hbox{\bf d}{}(x,z,p+q))$
_(triangle inequality)_ ,

The intended meaning of $\hbox{\bf d}{}(x,y,q)$ is ‘the distance of $x$ from
$y$ is less than $q$’. Then by this intention, all properties P1-P5 are
understood clearly. For $x\in I$ and $p\in\mathbb{Q}^{+}$, let
$S_{p}(x)=\\{y\in I\mid\hbox{\bf d}{}(x,y,p)\\}$.

###### Remark 3.5

Note that two notions of metric space and semi-metric space are classically
the same. Assume $\mathbb{R}$ is the classical real line. For a semi-metric
space $(I,\hbox{\bf d}{})$, we can define a metric function $dist:I\times
I\rightarrow\mathbb{R}$, such that $dist(x,y)=inf\\{p\mid\hbox{\bf
d}{}(x,y,p)\\}$. One may easily verify that $dist$ is a (classical) metric.
Also if $(I,dist)$ is a (classical) metric space, defining $\hbox{\bf
d}{}\subseteq I\times I\times\mathbb{Q}^{+}$ by $\hbox{\bf
d}{}(x,y,q)\leftrightarrow dist(x,y)<q$, would make $(I,\hbox{\bf d}{})$ a
semi-metric space.

###### Proposition 3.6

If $(I,\hbox{\bf d}{})$ is semi-metric space, then the collection of subsets
$U\subseteq I$ satisfying the following property is a topology:

$\forall x\in U\exists p\in\mathbb{Q}^{+}(S_{p}(x)\subseteq U)~{}~{}(\ast)$.

Proof. It is clear that both sets $I$ and the empty set satisfy $(\ast)$. We
only need to show that the collection is closed under arbitrary union and
finite intersection. Being closed under arbitrary union is trivial. We prove
that it is closed under finite intersection. Assume $U_{1}$ and $U_{2}$
satisfy the property and let $x\in U_{1}\cap U_{2}$. Since both $U_{1}$ and
$U_{2}$ satisfy the property, there exist $p_{1},p_{2}\in\mathbb{Q}^{+}$ such
that $S_{p_{1}}(x)\subseteq U_{1}$ and $S_{p_{2}}(x)\subseteq U_{2}$. Let
$q=min(p_{1},p_{2})$. Then $S_{q}(x)\subseteq U_{1}\cap U_{2}$. To show this,
assume $y\in S_{q}(x)$. Then $\hbox{\bf d}{}(x,y,q)$ holds, and by (P3), we
have $\hbox{\bf d}{}(x,y,p_{1})$ and $\hbox{\bf d}{}(x,y,p_{2})$, which
implies $y\in S_{p_{1}}(x)\cap S_{p_{2}}(x)\subseteq U_{1}\cap U_{2}$. Hence
$U_{1}\cap U_{2}$ satisfies the property. $\dashv$

###### Proposition 3.7

Let $\hbox{\bf
d}{}\subseteq\mathbb{R}^{o}\times\mathbb{R}^{o}\times\mathbb{Q}^{+}$, defined
by

$\hbox{\bf d}{}(\alpha,\beta,q):=\exists\zeta\in\mathbf{Q}\exists
p\in\mathbb{Q}^{+}(p\leq
q\wedge(\hat{\zeta}\leq\alpha,\beta\leq\hat{\zeta}+p))$ 666We use expression
$a\leq x,y\leq b$ as a short abbreviation of $a\leq x\leq b\wedge a\leq y\leq
b$..

Then $(\mathbb{R}^{o},\hbox{\bf d}{})$ is a semi-metric space.

Proof. We must show that d satisfies properties P1-P5.

  * •

P1. It follows from proposition 2.17.

  * •

P2. Let $\alpha,\beta$ be two oriented reals. There exist rational numbers
$p_{1},p_{2},q_{1},q_{2}$ such that $p_{1}<\alpha<p_{2},q_{1}<\beta<q_{2}$.
Let $\zeta\in\mathbf{Q}$ defined by $\zeta(n)=min(p_{1},q_{1})$ for each
$n\in\mathbb{N}$. Then

$\hat{\zeta}\leq\alpha,\beta\leq(\hat{\zeta}+|min(p_{1},q_{1})|+max(p_{2},q_{2}))$.

  * •

P3. Trivial.

  * •

P4. Trivial.

  * •

P5. Assume $\alpha,\beta,\delta\in\mathbb{R}^{o}$ and for some
$p,q\in\mathbb{Q}$, $\hbox{\bf d}{}(\alpha,\beta,q)$ and $\hbox{\bf
d}{}(\beta,\delta,p)$ hold. Then there exist
$\zeta_{1},\zeta_{2}\in\mathbf{Q}$, $q_{1}\leq q$ and $p_{1}\leq p$, such that

$\hat{\zeta_{1}}\leq\alpha,\beta\leq(\hat{\zeta}_{1}+q_{1})$

and

$\hat{\zeta_{2}}\leq\beta,\delta\leq(\hat{\zeta_{2}}+p_{1})$.

Let $\zeta(n)=min(\zeta_{1}(n),\zeta_{2}(n))$, for each $n$. We claim that

$\hat{\zeta}\leq\alpha,\delta\leq(\hat{\zeta}+(p_{1}+q_{1}))$.

From $\hat{\zeta_{1}}\leq\alpha$ and $\hat{\zeta_{2}}\leq\alpha$, we derive
$\hat{\zeta}\leq\alpha$. We also derive $\hat{\zeta}\leq\delta$ similarly.
Assume an arbitrary $t\in\mathbb{N}$. The fact
$\alpha\leq\hat{\zeta_{1}}+q_{1}$ implies that there exists $k\in\mathbb{N}$
such that $\alpha(t)<\hat{\zeta_{1}}(k)+q_{1}$. On the other hand, since
$\hat{\zeta_{1}}\leq\beta$ and $\beta\leq\hat{\zeta_{2}}+p_{1}$, there exists
$k^{\prime}\in\mathbb{N}$ such that
$\hat{\zeta_{1}}(k)<\hat{\zeta_{2}}(k^{\prime})+p_{1}$, and consequently
$\alpha(t)<\hat{\zeta_{2}}(k^{\prime})+p_{1}+q_{1}$. Let
$k^{\prime\prime}=max(k,k^{\prime})$. Since both of
$\hat{\zeta_{1}},\hat{\zeta_{2}}$ are strictly increasing, we have
$\hat{\zeta_{1}}(k)<\hat{\zeta_{1}}(k^{\prime\prime}),~{}\hat{\zeta_{2}}(k^{\prime})<\hat{\zeta_{2}}(k^{\prime\prime})$,
and hence $\alpha(t)<\hat{\zeta_{1}}(k^{\prime\prime})+q_{1}+p_{1}$ and
$\alpha(t)<\hat{\zeta_{2}}(k^{\prime\prime})+p_{1}+q_{1}$. Then
$\alpha(t)<(min(\zeta_{1}(k^{\prime\prime}),\zeta_{2}(k^{\prime\prime}))-\frac{1}{k^{\prime\prime}+1})+p_{1}+q_{1}$.
It is shown that $\alpha\leq\hat{\zeta}+p_{1}+q_{1}$. Similar argument works
for $\beta\leq\hat{\zeta}+p_{1}+q_{1}$.

$\dashv$

###### Definition 3.8

The Ordinary Topology on $\mathbb{R}^{o}$. The _ordinary_ topology on
$\mathbb{R}^{o}$ is the topology induced by the semi-metric
$(\mathbb{R}^{o},\hbox{\bf d}{})$, where d is defined above. We show the
_ordinary_ topological space by $(\mathbb{R}^{o},\mathcal{T}_{2})$

###  3.3 A consequence of OCP

We use OCP to prove that:

###### Theorem 3.9

Every total function from $((0,1]^{o},\mathcal{T}_{1})$ to
$(\mathbb{R}^{o},\mathcal{T}_{2})$ is continuous.

Proof. Let
$f:((0,1]^{o},\mathcal{T}_{1})\rightarrow(\mathbb{R}^{o},\mathcal{T}_{2})$ be
total. We prove that for each $n\in\mathbb{N}$, for every
$\alpha\in(0,1]^{o}$, there exists $S_{E_{n}}(\alpha)\in\mathcal{T}_{1}$ such
that for every $\beta\in S_{E_{n}}(\alpha)$, $\hbox{\bf
d}{}(f(\alpha),f(\beta),2^{-n})$. Consider rational numbers $q_{i}=i2^{-n}$,
$0\leq i\leq 2^{n}$. For each $\delta\in(0,1]^{o}$, define
$\phi(f(\delta))(0)=0$ and $\phi(f(\delta))(k)=i$ if and only if $q_{i}\leq
f(\delta)(k)<q_{i+1}$. The function $\Phi(\delta)=\varphi(f(\delta))$ from
$(0,1]^{o}$ to $\mathbb{N}^{\ast}$ is total and well-defined. By OCP , there
exists a non-infinite subset $E\subseteq[0,1]^{o}$, such that for every
$\alpha,\beta$, if $\beta\in S_{E}(\alpha)$ then
$\Phi(\alpha)=^{\ast}\Phi(\beta)$. It easily seen that for
$\zeta_{1}(i)=q_{\Phi(\alpha)(i)}\in\mathbf{Q}$ and
$\zeta_{2}(i)=q_{\Phi(\beta)(i)}\in\mathbf{Q}$, we have $\zeta_{1}=\zeta_{2}$
and

$\hat{\zeta_{1}}\leq f(\alpha),f(\beta)\leq(\hat{\zeta}_{1}+2^{-n})$,

i.e., $\hbox{\bf d}{}(f(\alpha),f(\beta),2^{-n})$. $\dashv$

##  4 Concluding Remarks

Our temporal continuum suggests a new framework for constructive analysis. We
need to investigate the following questions as further work:

  1. 1.

Is every total function on $\mathbb{R}^{o}$ with one topology (either the
oriented topology or the ordinary one) continuous, or do we need a third
topology?

  2. 2.

What happens to the _uniform continuity theorem_?

  3. 3.

What does the _intermediate value theorem_ looks like?

Acknowledgement. We are very thankful to Dirk van Dalen, Mark van Atten and
Wim Veldman for reading a draft of this paper and for their helpful comments
and suggestions. Moreover, our correspondence with Wim Veldman resulted in
modifications of some crucial claims in this paper.

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