数理解析研究所講究録
1011 巻 1997 年 42-57

42Rand.omness$\mathrm{R}\mathrm{e}.\mathrm{v}$ $\mathrm{i}\mathrm{s}\backslash \mathrm{i}\mathrm{t}\mathrm{e}\mathrm{d}$.

Physics and Beckman Institute, UIUC1
Y. OonoI

Introduction-Characteristics of Nonlinear Systems

Nonlinear systems are systems that are not linear. Hence, a ‘nonlinear system’ meansthat a system whose particular description [1] does not obey the superposition principle. ‘A system is nonlinear’ is a shorthand statement of this.

The word ‘nonlinear’ may not be, however, a fully satisfactory characterizationof ‘nonlinear’ systems, because the word does not positively characterize nonlinear
systems; it characterizes a system by the absence of linearity, just as infinity is charac-terized by the lack of finiteness. In this case Dedekind supplied a positive definitionof infinity with the aid of$\mathrm{o}\mathrm{n}\mathrm{e}- \mathrm{t}_{\mathrm{o}^{-}\mathrm{o}\mathrm{n}\mathrm{e}}$correspondence between a set and its proper

subset. Is there any positive characterization of nonlinear systems?
Why does the superposition fail in ‘non-linear’ systems? Because we cannot
separate different scales (length, time, etc.). Consider a nonlinear partial differen-tial equation such as the time-dependent Ginzburg-Landau equation. Decomposition
into different Fourier modes is useless to solve it because of ‘mode-coupling.’ Wecannot disentangle small and large scales in contrast to the case of, say, the lin-

ear diffusion equation. This suggests a positive definition of nonlinear systems as
scale-interfering systems [2]. Scale interference causes nonlinear systems to exhibit
various nontrivial and often unexpected phenomena such as phase transition, critical
1University of Illinois at Urbana-Campaign

43phenomenon, chaos, etc. ‘Nontriviality’ of nonlinear systems is solely due to this scaleinterference that promotes the unknowable relevant to what we can observe at our(space-time) scale. The effect of noise illustrates the point in a rather trivial fashion;

it is obvious that noise has no effect at all on the expectation values of the quantities
governed by linear evolution equations, but often has marked effects on nonlinear
systems as exemplified by critical phenomena.

In this report, I wish to consider a foundational aspect of randomness. Randomness becomes much more important to nonlinear systems than to linear systems
because of scale interference. Hence, much more serious attention should be paid tothe fundamental nature of randomness in Nature. As the reader can easily infer, thescale interference makes the world we observe-contained (open to the unknow-$\mathrm{n}\mathrm{o}\mathrm{t}-_{\mathrm{S}}\mathrm{e}\mathrm{l}\mathrm{f}$able universe). In the next section, the real significance of chaos is discussed fromthis point of view. In Section III, a fundamental nature of noise and randomness tonatural science is stressed. In Section IV characterization of randomness by van Lam-balgen is reviewed (with a theoretical physicist’s gross distortion), and its possible
serious consequences are discussed. Needless to say, the problem I wish to consider isso fundamental that I will not be able to say any positive comment. Rather, I wish
to point out that fundamental scientists (not mathematicians) should pay due atten-

tion to, e.g., the choice of axiomatic systems of set theory as a respectable empiricalquestion just as the question whether the world is Euclidean or not has been regardedso.
Before going into the main topic, it may be useful to recall the history of modern

mathematics. Most people agree that Fourier’s proposal to use Fourier series to solve
the diffusion equation made an epoch. Cauchy was spurred to put analysis on afirmer ground; Riemann had to invent his integral to compute Fourier coefficients

according to Fourier’s prescription. Along this line lies the fundamental work of44Lebesgue. On the other hand, the question of uniqueness of Fourier expansion drove

Cantor to think about the distinction between denumerable and non-denumerable
sets. To secure his theory of transfinite numbers, he had to invent set theory. To

secure his well-ordering proof, Zermelo had to formalize axiomatic set theory withthe axiom of choice$(\mathrm{A}\mathrm{C})$. We see that the (foundation of) mathematics we currently

use has largely been molded to solve various basic problems of Fourier expansion andtransformation. In other words, linearity molded our mathematics. It may be worth

reevaluating this fact before going into the study of nonlinear systems.

II

Chaos and Randomness

Perhaps the most dramatic effect of scale interference is ‘chaos.’ Intuitively, ‘chaos’is a ‘random’ behavior of a deterministic dynamical system. There have been many
characterizations of chaos. Here I choose one of the oldest definitions [3], because
it is naturally related to algorithmic randomness [4] (and also almost no one has

understood the significance of the definition; see the remark below).
If one wishes to proceed as intuitively as possible temporarily avoiding formaliza-tion of randomness, it is a sensible idea to use an intuitively clearly ‘random’ process
such as the coin-tossing process of a fair coin (the one-sided $B(1/2,1/2)$ process or
the full shift on $\{0,1\}^{N})$ to formalize ‘chaos.’ Roughly speaking, if we can find a reasonable correspondence between the sample histories of the coin-tossing process and(a subset of) trajectories of a given deterministic dynamical system, the dynamicalsystem may be said to exhibit chaos [5].

Definition [chaos]. Let$f\mathrm{b}\mathrm{e}\sim$

itself. We say that the mapthat$f^{n}$

restricted to

$A$

$f$

an

$\mathrm{e}\mathrm{n}\mathrm{d}_{0}\mathrm{m}\mathrm{o}\mathrm{r}\mathrm{p}\mathrm{h}\mathrm{i}_{\mathrm{S}\mathrm{m}}.$.

of a set (usua.lly a region)

exhibits chaos, if there exists $n\in N$ andis isomorphic to the shift dynamical system$\sigma$$\Gamma \mathrm{i}\mathrm{n}\mathrm{t}\mathrm{o}$$A\subset\Gamma$

,

such

on $\{0,1\}^{N}$ .

45That is, the following diagram is commutative [6]:$arrowarrow f^{\mathfrak{n}}\sigma$$\{0,1A\downarrow\varphi\}^{N}$$\{0,1A\downarrow\varphi\}^{N}$

The following theorem [8] summarizes nice features of the definition [9]:Theorem. Let $F$ : $Iarrow I$ be a$C^{0}$-endomorphism of an interval I.

Then, thefollowing (1) to (4) are equivalent [12]

(1).

$F$

exhibits chaos.

(2) $F$ has a periodic orbit of period not equal to an integer power of 2.
(3) There exists a positive integer $m$ such that

(.4)

$F$

has an$\dot{i}nvaria..nt$$F^{m}$

has a mixing invariant measure.

measure with a positive Kolmogorov-Sinai entropy.

$\square$

Notice that Ornstein and Weiss [13] identify chaos and positivity of entropy.
Remark. The equivalence of (1) and (2) is the well-known statement: Period$\neq 2^{n}$

implies chaos [3]. This may sound a straightforward generalization of the famous the-orem due to Li and Yorke [14]: Period three implies chaos, but there is an importantdistinction that is usually totally ignored [15]. That is, the definitions of chaos in theabove theorem and in the Li-Yorke theorem are distinct. Chaos implies the Li-Yorkechaos, but the converse is untrue, as can be seen easily from the equivalence to (4).
Obviously, the above theorem is not a corollary to the Li-Yorke theorem (even withthe aid of \v{S}arkovski’s theorem [18] . There is also a grave defect in the definition of$)$

the Li-Yorke chaos. The scrambled set carrying the Li-Yorke chaos is often measurezero [19], so that the Li-Yorke chaos is NOT observable. In contrast, the set$A$

is withpositive measure whenever there is an absolutely continuous invariant measure. The

Li-Yorke chaos never pays attention to randomness, which causes its defect.

$\square$

The above characterization may be stated that if the trajectories have natural

46

relation to algorithmic random sequences, then the system is chaotic. This obvious

restatement has a strong support of Brudno’s theorem [20]. This theorem implies thatthe loss of initial information (measured by the Kolmogorov-Sinai entropy) and impossibility of its short coding (measured by the Kolmogorov complexity) are equivalent.

Since the initial information is constantly being lost in chaotic systems, we must
provide infinitely precise initial information to specify a trajectory. Therefore, noinformation-compression is possible, and no prediction of the long future is possible.

Incidentally, Brudno’s theorem has some practical relevance, although I havenever seen the theorem quoted in physics literature except in a recent one [21] (except

ours [22] . Suppose we wish to compute a trajectory of a dynamical system whose$)$time one map is given by
$t$

$T$

. To code a trajectory starting from a point$x$

for a period, we need roughly $tK(x,T)$ of symbols, where $K(x, T)$ is the Kolmogorov complexityof the trajectory starting fromcost of the trajectory starting$x$. $tK(x, T)$ may be interpreted as a computational$i^{\mathrm{f}\mathrm{r}\mathrm{o}\mathrm{m}x}$

for time . That is, $K(x,T)$ may be understood$t$

as the necessary cost to compute one time unit into the future from the present data.
This idea has been successfully applied to the Sinai billiard system [23].

III

Significance of Randomness

As explained in the previous section, randomness is crucial to chaos. Where does its
randomness come from? It is from the initial condition, especially, from its details.
Because extremely small scales can interfere our scale (due to generally expansive

nature of chaotic dynamics), chaotic systems look random. Notice that such small
scales are unknowable. This is not due to our technological limitations but due toabsolute limitation. Chaos is conceptually important, because it reminds us of theexistence of the unknowable; Chaos magnifies what we can never observe, and becauseof this the unknowable can have significant effects on what we observe or experience.

47

To recognize the possibility of analyzing a set of trajectories of a mechanicalsystem into the equation of motion and initial conditions was a major ingredient of
the Newtonian revolution. To understand the equation of motion gives us a feeling ofunderstanding the system if it is simple, so that (perhaps reinforced by the elementarymechanics education) we thought the equation of motion was everything. Then, camechaos that exhibits counterexamples to our happy feeling of understanding in terms of
the equation of motion. In order to specify a state of a chaotic system we must knowmuch more than its equation of motion. One can say a chaotic system is a system

for which the information in the initial condition dominates the required information
to specify its behavior. Notice that the parallelism between chaos and algorithmic

random sequence is really good.
The significance of noise$=$

the unknowable has become easy to grasp even for

uncritical thinkers thanks to simple chaotic system examples. This is the most signif-

icant implication of chaos. This is, however, not a fundamentally important progressat all, because science is not a democratic business. It is too shallow a view point toadvocate that with chaos new era of science dawns.
$i^{\mathrm{F}\mathrm{r}\mathrm{o}\mathrm{m}}$the application point of view, the significance of chaos is that it is a mod-ifier of randomness given by Nature. The randomness in chaos comes from its initialconditions. One could use neatly modified noise by chaos (like chaotic itinerancy[24] to accelerate optimization processes. The problem of using chaos as a random$)$

number generator is how to minimize the distortion due to deterministic modulation.

In physics, there are other cases where randomness is significant. Everyone istaught the principle of equal probability as a foundational principle of statisticalmechanics. Although there have been many people attempting to reduce it to atheorem of mechanics, the attempt is intrinsically meaningless, because the principle

is a principle about the initial condition of the mechanical (quantum or classical)

48

system. Every sensible person knows that the initial condition is beyond the controlof mechanics. Hence, the principle, which specifies what we mean by random samplingin experiments, is a really fundamental principle, perhaps more so than the law of
mechanics.

Quantum mechanics also requires deep understanding of randomness. If the wave
contraction due to observation could be explained, for example, by the irreversibility

in the observation process, then again we are not free from the nature of initialconditions before the observation.

IV

van Lambalgen’s Axioms

We must conclude that randomness is a fundamental concept in natural science. In
the above the Solomonov-Kolmogorov-Chaitin algorithmic characterization of randomness was adopted. However, this characterization relies on the concept of com-putation, or the Church thesis [25].

Is the concept of computation or algorithm more fundamental than randomness?
There might be a viewpoint that computation or the Turing machine is a device toformulate the concept of randomness, so the aforementioned question is misplaced.
Let us consider the concept of continuity. The concept seems very fundamental, but

we now know that topology is a much more fundamental and simpler concept thancontinuity, so that we never regard topological space as a device to discuss continuity.
As can be seen in the definition of the Turing machine or the statement of Church’s
thesis, computation is not a simple concept (perhaps even ugly).

Also, one must not ignore that there is no guarantee that all the regularities are‘computable regularities’ detectable by a universal Turing machine; some inspiration(oracle) might tell us the existence of a different kind of regularity in a given sequence.

49

This is not an outrageous statement, if mathematical intelligence is, as supposed by

G\"odel, nonfinitary [26]. Furthermore, ‘random sampling’ may be done by Nature
Herself. In that case, why do we have to assume that Her capability is restricted tocomputation? I do not adopt a rather extreme view that Nature Herself is a computer,so I feel it unnaturaI to regard algorithm as a basic concept.
If we wish to regard randomness as fundamental, unless we can find something

intuitively more fundamental, we cannot define the concept ‘randomness.’ Hence,

only possible way to mathematize the concept is axiomatization. A very serious
attempt the author is aware of is that due to van Lambalgen [27].
The following is a rather degenerate version of his axiomatic effort interpreted

in a theoretical physicist’s way. Because random sequence is defined algorithmicallyby its incompressibility, and because our intuitive reason to feel a sequence to berandom is that we cannot find any regularity in it, we should axiomatize the conceptof ‘incompressibility.’

Van Lambalgen introduces the relation

as

$‘ x$

acle

$R$

such that $R(x;y)$ may be interpreted

cannot be information-compressed even with the extra information$y’$)’. Here,$x$

relation$R$(or ‘or-is interpreted as a 01 sequence (or some symbol sequence) andmay be a set of symbol sequences. Notice that

we already know

$y$

$x$

$x$

$y$

can be information-compressed if

itself (we have only to say ‘the same’), so $R(x;x)$ is untrue. Thisis specified by the following axioms, which are put very informally here [28]:Rl. There is a sequence that cannot be information-compressed without any ‘ex-ternal information (or oracle). [i.e.,$\exists xR(x,$$\emptyset).$]$(R(x, \emptyset)$is henceforth written as$R(x).)$

This asserts that there is a random sequence. $R(x)$ can be interpreted as the state-ment that$x$is random.

50

R2. If$x$

cannot be information-compressed with the information ofcannot be done so with the information of
and

$z$

$z$

$y$

and , then it$z$

only. [i.e., $R(x;yz)\Rightarrow R(x;z).$] (Heremay be understood as sets, and $yz$ their joint set, so the order of

$y$

and

$z$

$y$

does

not matter.)This asserts that there is a hierarchy in randomness, although $R(x;zy)$ does not necessarily imply that

R3. If$x$$x$

is more random than$x’$such that $R(x’; z)$ .

cannot be information-compressed by the information of , then
$y$

$x$

and

$y$

are

different. [i.e., $R(x;y)\Rightarrow x\neq y.$ ]These are very natural axioms.

R4. If a certain relation
then there is

$w$

$x$

$\phi$

of

$y$

to

$x$

does not allow one to information-compress ,$x$such that the -relation of

not information-compressHere$\phi$$\phi$$w$. [i.e.,

$y$

to

$w$

plus an extra information$\exists x(R(x, y)\wedge\phi(x, y))\Rightarrow\exists w(R(w, zy)$should not have any parameters other than listed insatisfying $R(x, y)$ is ‘more

implies that$x$$\mathrm{r}\mathrm{a},\mathrm{n}\mathrm{d}\mathrm{o}\mathrm{m}$(lawless)’ than

$w$

$y.$

A

$z$

does

$\phi(w, y))$.

] (I.n a certain sense,

satisfying $R(w)$ , because R2

satisfies $R(x)$ .

Consequently, this axiom demands that there always exists a ‘more’ random sequencethan a given one; Indeed, R4 implies, with no relation chosen as$\exists wR(w;zy)$$\exists xR(x;y)\Rightarrow$, or, without $y\exists xR(x)\Rightarrow\exists wR(w;z)$ for any . R4 may sound artificial.
$z$

However, if we negate this, then there is some set of sequences$x$$\phi,$

such that $R(x;A)$ . That is, if we know

$A$

$A$

, then any sequencesuch that there is no$x$

can be information

compressible. Hence, the axiom R4 implies the ‘non-existence of Almighty.’

There are more axioms, but I have mentioned only I need below.

51$\mathrm{v}$Consequences of Axiomatization

As briefly explained in the preceding section, the axioms of$R$

are intuitively under-

standable (appealing) natural requirements. The consistency of these axioms withthe usual Zermelo-Fraenkel set theoretical axioms (without
by van Lambalgen $[27, 29]$.

$\mathrm{A}\mathrm{C}$) (ZFR) is established$\cdot$However, the ZFR is contradictory to$\mathrm{A}\mathrm{C}$:

Theorem [van Lambalgen]. There is no choice function on the power set of reals.
$\square$[Outline of proof]. Take an arbitrary sequence . There is a sequence$y$be compressed with the knowledge of . Let
$y$

$g$

$x$

which cannot

be a choice function on the power set ofreals (this is expressed as a set). Even if we know , there must be an incompressible$g$sequence , so that $\{x|R(x;g)\}$ is a nonempty set. Hence the choice function
$x$

$g$

must

give an element of this set $y:y=g(\{x|R(x;g)\})$ . By definition, this implies $R(y;g)$ .
Intuitively speaking, this is strange, because$y$is determined by , so
$g$

$y$

should becompressible with the knowledge of . Hence, we arrive at a contradiction. Formally,$g$

we can proceed as(5.1)$\exists yR(y;g)\wedge y=g(\{x|R(x;g)\})$ .

With the aid of R4 (setting $z=g(\{x|R(x;g)\})$ )$\exists yR(y, g(\{x|R(x;g)\})g)\wedge y=g(\{x|R(x;g)\})$

.

(5.2)

However, R2 allows us to replace $yR(y, g(\{x|R(x;g)\})g)$ with $yR(y, g(\{x|R(x;g)\}))$

in this formula, so that this implies $y\neq g(\{x|R(x;g)\})$ , a contradiction.
Physics, or more generally, natural science may never require general AC as-

serting that any transcendental family has a choice function. The assertion may beinterpreted as the credo of ‘super-realism.’ If we admit only the countable axiom of

choice (CAC), we could keep all the theorems of analysis needed for physicists, and,

52

simultaneously, can get rid of non-measurable sets or Hausdorff-Banach-Tarski type
‘unpleasant’ theorems. Hence, physicists may even be delighted with the fall of thefull strengthLet$\mathcal{U}$$\mathrm{A}\mathrm{C}$. However, if CAC fails, we would be in big trouble.

be a countable family of sets and$\{x|R(x, g)\}$$g$

be a choice function on it. Let $B=$

. Now, make a countable family of sets joining$\mathcal{U}$and $B$ , and

$h$

be a

choice function on it. Of course, $R(h(B);g)$ , because $h(B)\in\{x|R(x, g)\}$ . In otherwords,$\exists yR(y, g)\wedge y=h(\{x|R(x, g)\})$

(5.3)

.

Hence, with the same logic employed in the proof above,$\exists yR(y, h(\{x|R(x, g.)\})g)\wedge y=h(\{x|R(x, g)\})$ ,which leads us to a similar contradiction. Hence,

VI
It is

$\dot{\mathrm{c}}\mathrm{A}\mathrm{C}$(5.4)

also fails.

What can we conclude?
$\mathrm{h}\dot{\mathrm{a}}\mathrm{r}\mathrm{d}$to conclude or propose soinething mathematically constructive. In thisconcluding section, however, I try to extract some lessons.

It seems that CAC and a natural axiomatic system ofra.n.

$\mathrm{d}\mathrm{o}.\mathrm{m}\mathrm{n}\mathrm{e},\mathrm{S}\mathrm{S}.\mathrm{a}.\mathrm{r}\mathrm{e}$not com-

patible. This is more than we physicists can swallow. If we accept that the hierarchyof randomness has an upper bound (so that we have an almighty oracle set), thencertainly we can avoid this ‘unpleasant’ conclusion. However, to avoid Almighty is aprerequisite of any serious science, so this boundedness by something or someone is
unacceptable.

Of course, the reader would say that to accept randomness as fundamental is
the source of problem. However, for empirical scientists I believe randomness is more

fundamental than some esoteric set theory axioms like large cardinal axioms. If one

53

wishes to stick to the idea that randomness is fundamental, then what are the possible

directions?
Van Lambalgen himself discussed various possibilities [27]. He considered thepossibility of replacing the extensionality axiom with a weaker version. Under thiscondition the ‘set’ $B$ above is not a set, so the demonstrations in the previous section

do not work. Besides, AC without extensionality is probably not powerful enough. Of
course, there must be a lot of room to modify the simplest axioms for ‘randomness.’
Perhaps, the current attempt assumes the transcendency of randomness excessively.

We could add many fine prints, but I do not wish to modify the simplicity of theabove mentioned axiomatic system for$R$. In any case all the proposals to try to

avoid contradiction between the current axiomatic system (ZFC) and randomness

seem to suggest modifications of extreme atomistic nature of the current mathematicalfoundation.
There is a drastic position to deny all the infinity: since we are made of atoms(or elementary particles), and the total number should be finite, because the universe

seems to be finite (or in any case we discuss the phenomena in front of us, so wehave only to think about bounded domains). Perhaps the claim is true and there

is no infinity in Nature. However, to understand is to idealize. Without limiting
procedures, it is impossible to idealize. If no infinity is allowed, even thermodynamic

phases cannot be defined unambiguously, because we cannot take the thermodynamic
limit. Hence, I totally reject any position denying idealization.
To consider what axiomatic system of geometry is adapted to the actual space

has been a respectable empirical question. To ask what axiomatic system of the set
theory is the most suitable to understand Nature can also be a respectable ques-tion. Of course, it has been a respectable question as can be seen in [30] among‘fundamentalists,’ but not yet among empirical scientists.

54

One intriguing point I wish to stress here is that the conceptual problem ofrandomness was never serious in the study of linear systems. Note (1) ZFC is almosta direct consequence of the quest for the foundation of Fourier expansion theory(i.e., a tool for linear systems), and (2) randomness is never important in linear
systems. However, everyone knows that linearity is only due to approximation (except
perhaps for the Schr\"odinger equation), so (1) scale interference implies that the linear
expansion tools are no more powerful, and (2) the effect of randomness becomes evenqualitatively significant. There might be a more natural foundation of mathematics

which can be a suitable tool to understand Nature full of nonlinearity. Perhaps, thecurrent mathematics is artificially constrained by simple minded idealization (called
linearization).

Acknowledgements. I acknowledge fruitful discussions with Shin-ichi Sasa. The
work is, in part, supported by the National Science Foundation Grant DMR93-14938.

55

Bibliography[1] Just as seen in the so-called Koopmanism, a nonlinear dynamical system can be

described by a linear equation such as the Liouville equation, so that there is not

an intrinsic linear or nonlinear system. Even if we believe that everything canbe described by the Schr\"odinger equation, we know there are many nonlinear

phenomena such as life.

[2] I ignore integrable systems, because they are pseudo-nonlinear systems or disguised linear systems.

[3] Y. Oono, Prog. Theor. Phys. 59, 1029 (1978).
[4] See, e.g., M. Li and P. Vit\’anyi, An Introduction to Kolmogorov Complexity and

Its Applications (Springer, 1993) as a convenient reference.
[5] Continuos dynamical systems can be discretized by one of many standard methods, so we need not consider them separately.

[6] Alekseev [7] defined quasirandom dynamical systems in terms of a Markov chain
with positive Kolmogorov-Sinai entropy. The idea is, in retrospect, the same.
[7] V. M. Alekseev, Math. USSR Sbornik, 5(1), 73-128 (1968).

[8] Y. Oono and M. Osikawa, Prog. Theor. Phys. 64, 54 (1980); M. Osikawa and Y.
Oono, Publ. RIMS, 17, 165 (1981).

56[9] See also [10]. Block and Coppel [11] define a ‘turbulent map’ f by the condition
that there are two intervals J and K sharing at most one point such that$f(J)\cap f(K)$ , and say that when$J\cup K\subset$

is turbulent for some positive integer n, f is

$f^{n}$

chaotic. This usage is exactly the one proposed in our definition. The existenceof such J and K is a lemma used in the proof of this theorem.
[10] L. Block, Proc. Amer. Math. Soc., 67, 357 (1978); ibid., 72, 576 (1978).
[11] L. Block and W. A. Coppel, Dynamics in One Dimension, Lecture Notes inMathematics 1513 (Springer, 1992).
[12] (1)$\Rightarrow(2),$(3), (4) are trivial.

[13] D. Ornstein and B. Weiss, Bull. Amer. Math. Soc. 24, 11 (1991).

[14] T.-Y. Li and J. A. Yorke, Am. Math. Month. 82, 985 (1975).
[15] For example, Hao [16] says on p24, “Unaware of the work of \v{S}arkovskii, manyequivalent formulations or corollaries of this theorem were suggested in the in-tervening years, among which the Li-Yorke theorem “Period 3 implies chaos”
and the Oono’s statement “Period$\neq 2^{n}$

implies chaos”, etc.” A similar misun-

derstanding can be found in Li [17].
[16] B.-L. Hao, Chaos (World Scientific, 1984).
[17] S. Li, Trans. Amer. Math. Soc. 339, 243 (1993).
[18] A. N. \v{S}arkovskii, Ukr. Mat. Z. 16, 61 (1964); P. \v{S}tefan, Commun. Math. Phys.
54, 237 (1977).

[19] Y. Baba, I. Kubo and Y. Takahashi, Nonlinear Analysis 26, 1611 (1996).

[20] A. A. Brudno, Trans. Mosc. Math. Soc. 44, 127 (1983).

57[21] A. Crisanti, M. Falcioni, G. Mantica, and A. Vulpiani, Phys. Rev. E50, 1959
(1994).
[22] For example, Y. Oono and C. Yeung, J. Stat. Phys. 48, 593 (1987).

[23] P. R. Baldwin, J. Phys. A 24, L941 (1991).

[24] I. Tsuda, World Futures 32, 167 (1991).
[25] R. I. Soare, Recursively Enumerable Sets and Degrees, (Springer, 1982).
[26] R. Gandy, “The Confluence of Ideas in 1936,” in The Universal Turing Machine,a half-century survey (Oxford UP, 1988).
[27] M. van Lambalgen, J. Symbolic Logic 55, 1143 (1990); J. Symbolic Logic, 57,

1274 (1992).
[28] There are slightly different versions of axioms given in van Lambalgen’s papers.

Also here the exposition is informal, so the numbering of the axioms is different
from the original versions.

[29] There must be details such as we use the first order logic, etc., but in this informal
exposition, I omit them.

[30] P. Maddy, Realism in Mathematics (Oxford, 1990).