Theoretical Physics : Infinity

Infinity

Potential versus actual infinities

Aristotle distinguished between two kinds of infinities:
-The potential infinity as a never-ending process in time which is actually finite at any given moment.
-The actual infinity which is a completed timeless infinity existing wholly at every instant of time or even outside time.
According to Aristotle, the second notion of actual or completed infinity is contradictory and incoherent and as such he was adamant in rejecting it.
However, this notion of actual infinity was actually used successfully by a minority of mathematicians and philosophers such as Archimedes and Leibniz since the time of Aristotle.

The great Leibniz for example envision reality as an actual infinity constituted of Monads (mind-like atoms or atom-like minds) and then used infinitesimals (which are infinitely small quantities) in his theory of calculus.

The notion of actual infinity was given a precise mathematical meaning (and thus accepted by the majority of mathematicians nowadays) starting with Cantor and his theory of set theory. In this framework potential infinities require for their definition the existence (at least platonic) of actual infinities. Indeed, a potentially infinite set is a growing finite subset of an actually infinite set and as such they can never become an actually infinite set. Said more differently, potential infinity is connected with ordering (and ordinal numbers) whereas actual infinity is connected with counting (and cardinal numbers).

The first actual infinity established by Cantor is the size (also called cardinality) $\aleph_0$ of the set of natural numbers and the second actual infinity is the size of the power set of the set of the natural numbers (that is $\aleph_1=2^{\aleph_0}$) which is also believed to be the cardinality of the set of the real numbers. In other words, it is believed that there are no other infinities between $\aleph_0$ and $\aleph_1$ a fact known as the continuum hypothesis.

The hierarchy of infinities continues undefinitely uninterpreted (there is no cardinal of the set of all cardinals which is known as Cantor's paradox and which was only resolved within Zermelo-Fraenkel’s set theory).

This state of affairs naturally outraged mathematicians and philosophers from the schools of constructivism, intuitionism, finitism and ultrafinitism which from their part continued their uninterpreted rejection of actual infinity.

Quine has a moderate stance. He accepted actual infinity in mathematics because it is indispensable to science but he only accepted the first three actual infinities because they are the ones which are actually needed in science. So Quine admits $\aleph_0$ (cardinality of the natural numbers), $\aleph_1$ (number of points on the continuum, i.e. the line or the plane or any higher dimensional space) and $\aleph_2$ (number of lines or curves in the plane for example) and he was certainly willing to admit more if they became indispensable and needed to/by science.

Achilles paradox

Let us see how we can solve Achilles paradox of Zeno of Elea along the lines proposed by Aristotle strengthen with more input from quantum gravity.

We will assume that the speed of Achilles is $v$ whereas the speed of the slower runner (imagined for obscure reasons as a tortoise by most writers who came after Zeno and Aristotle) is $\omega<v$ and that she is given a head start by the amount $L$.

Achilles needs to cover an initial distance $x_0=v t_0$ in order to reach the tortoise initial position while simultaneously the tortoise advances further away a distance $x_1= \omega t_0$ during the same time interval $t_0$. In this first leg of the journey Achilles spends therefore a time $t_0$ given by
\begin{eqnarray}
t_0=\frac{x_0}{v}=\frac{\omega}{v}\frac{L}{\omega}.
\end{eqnarray}
Next, Achilles needs to cover the distance $x_1$ which he will do in a shorther time interval $t_1$, i.e. $x_1= v t_1$ while simultaneously the tortoise advances even further away a distance $x_2= \omega t_1$. In this second leg of the journey Achilles spends therefore a time $t_1$ given by
\begin{eqnarray}
t_1=\frac{x_1}{v}=\frac{\omega t_0}{v}=(\frac{\omega}{v})^2\frac{L}{\omega}.
\end{eqnarray}
Achilles needs then to cover the further distance $x_2$ in an even shorter time interval $t_2$, viz $x_2=v t_2$ while simultaneously the tortoise advances even further away a distance $x_3= \omega t_2$. Achilles spends therefore an extra time $t_2$ in this leg given by

\begin{eqnarray}
t_2=\frac{x_2}{v}=\frac{\omega t_1}{v}=(\frac{\omega}{v})^3\frac{L}{\omega}.
\end{eqnarray}
This process continues undefinitely and thus the total time spent by Achilles in his journey is $T=t_0+t_1+t_2+...$ while the total distance covered by him before he can reach the tortoise is
\begin{eqnarray}
D=vT&=&\bigg[\frac{\omega}{v}+(\frac{\omega}{v})^2+(\frac{\omega}{v})^3+....\bigg]\frac{vL}{\omega}\nonumber\\
&=&\bigg[1+(\frac{\omega}{v})+(\frac{\omega}{v})^2+....\bigg]L
.\label{series}
\end{eqnarray}
From classical mechanics we know that $T$ and $D$ are given by
\begin{eqnarray}
T=\frac{L}{v-\omega}~,~D=\frac{vL}{v-\omega}.
\end{eqnarray}
Equation (\ref{series}) is a geometric series which involves an actual infinite number of terms and hence Zeno of Elea thought that one must go to all those locations (in other words complete the infinity) before Achilles can catch up with the tortoise.
Nowadays, we know how to calculate geometric series (Cauchy). We have for $|r|< 1$ the result
\begin{eqnarray}
(1+r+r^2+r^3+....)a=\frac{a}{1-r}
\end{eqnarray}
In our case the base $r=\omega/v$ is smaller than $1$ since the tortoise is slower than Achilles and $a=L$. We get immediately, the result

\begin{eqnarray}
\bigg[\frac{\omega}{v}+(\frac{\omega}{v})^2+(\frac{\omega}{v})^3+....\bigg]\frac{vL}{\omega}=\frac{L}{1-r}.
\end{eqnarray}
This agrees with the result from classical mechanics.

In the spirit of Aristotle let us recompute the geometric series with only a very large but finite number of terms $N+1$ (a potential infinity). The distance covered in $N+1$ steps is $D_N$ in a time $T_N$. We get immediately

\begin{eqnarray}
D_N=vT_N&=&\bigg[\frac{\omega}{v}+(\frac{\omega}{v})^2+(\frac{\omega}{v})^3+....+(\frac{\omega}{v})^{N+1}\bigg]\frac{vL}{\omega}\nonumber\\
&=&\bigg[1+(\frac{\omega}{v})+(\frac{\omega}{v})^2+....+(\frac{\omega}{v})^N\bigg]L\nonumber\\
&=&\frac{L}{1-r}(1-r^{N+1}).
\end{eqnarray}
The absolute error (or the distance left before Achilles catches up with the tortoise) is $D-D_N$ and thus the relative error is $(D-D_N)/D$. They are given explicitly by

\begin{eqnarray}
D-D_N&=&\frac{L}{1-r}r^{N+1}.
\end{eqnarray}
\begin{eqnarray}
\frac{D-D_N}{D}&=&r^{N+1}.
\end{eqnarray}
For simplicity, let us assume that the speed of the tortoise is half that of Achilles, i.e. $\omega=v/2$ or equivalently $r=1/2$. We get in this case

\begin{eqnarray}
D-D_N&=&\frac{L}{2^{N}}~,~\frac{D-D_N}{D}&=&\frac{1}{2^{N+1}}.
\end{eqnarray}
If we further assume that the original distance separating Achilles and the tortoise is around $1$ kilometers or more precisely $L=2^{10}m$ then Achilles needs to perform only $N+1=11$ steps in order to bring the relative distance between them to $1$ meter exactly with a relative error of $0.05$ per cent. Thus, from a practical side the partial sum $D_N$ will converge to the correct answer $D$ (and Achilles as a consequence will catch up with the tortoise) under all normal circumstances.

However, there is also a fundamental obstruction to going (or attempting to go) to an infinite number of locations (as required by the abstract thought of Zeno). The actual distance between Achilles and the tortoise after $N+1$ steps is given explicitly by
\begin{eqnarray}
d_N=(L+x_1+x_2+x_3+...)-D_N&=&L r^{N+1}=\frac{L}{2^{N+1}}.
\end{eqnarray}
This approaches $0$ when $N$ goes to $\infty$.

From a physical point of view it is well established that distances below the so-called Planck scale given by $l_P=\sqrt{\hbar G/c^3}$ (where $\hbar$ is the Planck constant, $c$ is the speed of light and $G$ is Newton's gravitational constant) can not be probed by any physical process and/or observer. Indeed, Heisenberg's uncertainty principle with Einstein’s theory of classical gravity leads to the conclusion that spacetime geometry breaks down at the Planck scale (because of the formation of black holes which traps therefore light) and it is then widely conjectured that the classical geometry of spacetime at this scale is replaced by a quantum structure (noncommutative geometry for example) where points, lines, etc have no clear operational meaning.

The distance $d_N$ between Achilles and the tortoise must therefore be bounded from below by the Planck scale, viz
\begin{eqnarray}
d_N\geq l_P.
\end{eqnarray}
The Planck scale corresponds therefore to a maximum number of locations $N_P+1$ which can be visited by Achilles given by (with $l_P=1.62\times 10^{-35}$)
\begin{eqnarray}
N_P+1=\frac{\ln l_P-\ln L}{\ln r}=\frac{-80.11-\ln L}{\ln r}.
\end{eqnarray}
In the example considered above where $r=1/2$ and $L=2^{10}$ we get $N_P+1=115.57+10\simeq 116$. This maximum integer number $N_P$ increases further if the distance $L$ increases (maximum is the size of the observable universe) or the ratio $r=\omega/v$ increases (maximum is $1$). For Aristotle (and quantum gravity) all natural infinities are really only potential ones.