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\date{April 20, 1999}
 
\title{Paradoxes in Mathematics and the Meaning of Truth\\
\mbox{ }\\
West Virginia University\\
Benedum Distinguished Scholar Award Lecture}
 
\author{
Krzysztof Ciesielski
\\
{\footnotesize Department of Mathematics,}
{\footnotesize West Virginia University,}\\
{\footnotesize Morgantown, WV 26506-6310, USA}\\
{\footnotesize e-mail: KCies@wvnvms.wvnet.edu}\\
{\footnotesize web page: {\tt http://www.math.wvu.edu/homepages/kcies}}
}


\begin{document}
 
\maketitle

First of all, in what sense the word {\it paradox\/} in the title is used? 
The Webster's Dictionary~\cite{Web}
gives the following relevant interpretation for it:

\begin{itemize}
\item[(1)] a statement that seems contradictory, unbelievable, or absurd
but that may be true in fact;

\item[(2)] a statement that is self-contradictory and, hence, false.
\end{itemize}
Most of the paradoxes mentioned in this talk will be of the first
kind, though some of the second kind will also be mentioned. 

One of the first paradoxes recorded in a written history 
is a Zeno's of Elea (495--435 B.C.) paradox known as
{\em Achilles\/} or {\em Achilles and the Turtle\/} Paradox:
\begin{quote}
Achilles running to overtake a crawling turtle can never overtake it
because he must first reach the place from which the turtle started; 
when Achilles reaches that place, the turtle has already departed, 
and so it is still ahead. 
Repeating the argument we easily see that the turtle will be always ahead. 
\end{quote}
Although the argument sounds convincing, we know from experience
that Achilles will easily overtake a turtle. So, what is wrong?

The problem with Zeno's argument is that he uses the word
{\em always\/} in a non-standard way.
He describes the ``race'' as follows. 
If Achilles starts at a point $P_1$ and turtle at a point $P_2$, by the 
time Achilles reaches point $P_2$ turtle will be already
at some point $P_3$;  by the time Achilles reaches point $P_3$,
turtle will move to a point $P_4$, and so we can repeat this process
to construct consecutive points $P_1,P_2,\ldots,P_n,\ldots$
for all natural numbers $n$. Also, it takes Achilles 
a positive time length $t_1$ to get from point $P_1$ to $P_2$,
a positive time length $t_2$ to get from point $P_2$ to $P_3$,
and, in general, 
a positive time length $t_n$ to get from point $P_n$ to $P_{n+1}$.
So, {\em always\/} in the Paradox refers to
{\em all\/} infinitely many time periods of the process, each 
of a positive length $t_n$. 
Thus, Paradox is true when {\em always\/} in its statements refers 
to the entire time period described which has 
length $t_1+t_2+t_3+\cdots+t_n+\cdots$.
It is very likely that Zeno was convinced that the infinite sum
$t_1+t_2+t_3+\cdots+t_n+\cdots$ of positive numbers 
must always be infinite, and in this case 
there would be no discrepancy in the meaning of {\em always}. 
However, as we learn in calculus, the infinite sum
$t=t_1+t_2+t_3+\cdots+t_n+\cdots$,
known as {\em series}, can be finite even 
if all the terms $t_n$ are positive. For example
\[
\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\cdots+\frac{1}{2^n}+\cdots=1
\]
In fact, this is the case 
in the situation described in the Paradox. 
In other words, if {\em always\/} in the Paradox is understood
in the standard way, as {\em forever,} 
the statement of the Paradox is simply false. 

\medskip

%Although Achilles Paradox is false, it is not self-contradictory.
%So, it is a paradox of the first kind (1) not of the second kind (2). 
Next we will consider other two statements paradoxical in the 
same {\em counterintuitive\/} sense.

The first is known as {\em Riemann's Theorem.} 
For this consider the infinite sum
\begin{eqnarray*}
\begin{array}{ccccccccccccccl}
S & = & a_1 & + & a_2 & + & a_3 & + & a_4 & + & a_5 & + & a_6 & + & \cdots\\
\\
& = &
  \frac{1}{1} & + & \left(-\frac{1}{2}\right) & + & 
  \frac{1}{3} & + & \left(-\frac{1}{4}\right) & + & 
  \frac{1}{5} & + & \left(-\frac{1}{6}\right) & + & \cdots
\end{array}
\end{eqnarray*}
and note that $S$ is finite. 
Riemann's Theorem tells that in this (or any other 
series converging {\em conditionally\/})
\begin{quotation}
\noindent 
by rearranging the order of the addition
one can arbitrary increase or decrease the value of the sum. 
\end{quotation}
For example, taking two terms with odd indexes, one term with even,
and so on, as in
$a_1+a_3+a_2+a_5+a_7+a_4+a_9+a_{11}+a_6+\cdots$
we get a sum
\[
T= 
  \frac{1}{1} + \frac{1}{3}  + \left(-\frac{1}{2}\right) +
  \frac{1}{5} + \frac{1}{7}  + \left(-\frac{1}{4}\right) +
  \frac{1}{9} + \frac{1}{11} + \left(-\frac{1}{6}\right) +\cdots
\]
which is strictly {\em greater\/} than $S$. 


This clearly contradicts the intuition of most of us,
which comes from the properties of finite sums, that 
the value of the sum does not depend of the order 
in which we add the terms. 

\bigskip
\bigskip

\noindent[However, most of the budget administrators had long a sense
that some kind of Riemann theorem must be true. This is the reason
for which 
\begin{center}
{\em they\/} usually {\em take an infinite time\/} 
\end{center}
to rearrange different items in the budget and, upon the completion,
they try to convince all other people that 
\begin{center}
{\em the total value of the budget has been increased\/} 
\end{center}
by all their rearrangements.]


\bigskip
\bigskip


%The remark above does not concern most of the budget people.
%They usually take an infinite time to
%rearrange different items in the budget, and upon the completion
%they try to convince other people that as a result our
%total budget has been increased by all the rearrangements.]


Even more counterintuitive is the next theorem known as
{\em Paradoxical Decomposition of the Ball\/} or 
{\em Banach-Tarski Paradox:}
\begin{quote}
A solid ball $B$ of radius one (so, of volume $\frac{4}{3}\pi$) 
can be split into five pieces and the pieces can be 
rearranged (by using only shifts and rotations)
in such a way that 
two of the rearranged pieces form one ball $B_1$ of radius one,
and the three other pieces form another identical ball
$B_2$ of radius one. 
As a result of this operation the volume of the 
ball $B$ has been doubled to $2\cdot\frac{4}{3}\pi$, 
the volume of two balls $B_1$ and $B_2$.
\end{quote}
This statement can induce some doubts in mathematics
even between the strongest believers in science. 
How it is possible to double the volume by 
simple operations of ``cutting'' and rearranging?
And, if it is really possible, why don't we 
double the US gold reserve by such a process? 

The answer for the second question becomes clearer 
when we take a closer look at the original purpose for proving the theorem. 
It was done to show that we cannot associate a ``reasonable''
volume to every abstract subset of the three dimensional space.
In particular, the pieces used in the decomposition
\begin{center}
{\em do not have a volume\/}
\end{center} 
in any reasonable sense of this term. 
Thus, there is no chance to cut any ball {\em in practice\/}
to such weird pieces. 
This also explain the paradox of duplicating the volume: 
since in the process we leave the realm of objects
that have a volume, all intuitive rules concerning 
the property of volume seize to bind. 
Thus Paradox is counterintuitive, but does not carry a contradiction. 

\medskip


These and other similar paradoxes convinced mathematicians that
we cannot confidently rely on our intuitions when deciding 
whether something is true or false. 
To overcome this ``luck of confidence'' 
we have decided to adopt the following {\em axiomatic approach\/}
to most mathematical theories:
\begin{itemize}
\item You choose some fundamental facts, called {\em axioms},
      that are basic enough to be accepted without any doubts.
      They are considered to be true. 
      
\item The other facts are considered as true in our theory 
      if they are logically deduced from the axioms. 
      The logical deduction of a fact is called its {\em proof}. 
      The properties that can be deduced that
      way are called {\em provable}. 
\end{itemize}

There is nothing new in this approach to build {\em theories.}
It was already used by Euclid (330--275 B.C.)
in his book {\em Elements,} which gave an account 
of elementary geometry. It has been also {\em attempted\/} by 
many philosophers since then. 
However, as soon as we accept the axiomatic approach as our tool for searching 
for truth, we should realize the first limit in reaching this goal:
\begin{center}
{\bf there is no absolute truth}
\end{center}
that can be discovered that way. All we can get that way is the
{\em conditional truth,} that is, the statements 
provable within a theory are true only as long as the axioms are true. 
It should be stressed here 
that as long as we {\em believe\/} that the axioms are true,
we should also {\em believe\/} that our conclusions are true.
But this requires at least some level of {\em faith.}

So, is this a weak point of axiomatic theories?
Should we abandon this approach because of it?

The fact is that in all our reasoning 
\begin{center}
we must have some starting point(s).
\end{center}
There is no way to deduce something from nothing.
In all human search some assumptions are made.
However, in most of the cases 
\begin{center}
the assumptions are made {\em implicitly}. 
\end{center}
This does not make the truth less conditional.
It only makes more difficult
to realize what part of the theory belongs 
to the base in which we need to believe, and which part 
is the consequence of these believes. 
Thus, if something goes wrong in
{\em na\"\i ve (i.e., non-axiomatic) theories\/} 
%in the case when something goes wrong
--- the experiments disagree with the observations ---
it is very difficult to sort the things out.
This happened in physics twice in the last century causing 
the births of the {\em theory of relativity\/}
and the {\em quantum theory\/} to be painful experiences. 
%have been painful because of that. 

\bigskip

[When Albert Einstein got his Noble prize in physics in 1921, well over a decade
after he formulated his relativity theory,
the Noble price committee 
stressed that the price was {\bf not} for his work on the 
relativity theory!]

\bigskip

Coming back to axiomatic theories ---
there are two fundamental questions that can be asked about
any such theory:
\begin{description}
\item[Not too big?:] How can we be sure that the axioms are really true?
           Can we at least be sure that the axioms do not contradict
           each other?
\item[Big enough?:] Based on the given set of axioms, can we deduce
           about each statement (concerning our theory)
           whether is it true or false?
\end{description}
The first question was already of interest to Euclid. 
He was not convinced whether the following fact, 
the fifth of his postulates (axioms) of geometry, was intuitive enough.
\begin{quote}
Given a straight line $L$ and a point $p$ not on $L$,
then in the plane determined by $L$ and $p$ it is possible to draw
{\em precisely one\/} line $L'$ through $p$ which never meets $L$. 
\end{quote}
Euclid apparently tried to deduce this fact from his other postulates.
Not being able to do so, he added it as an axiom. 
Nevertheless, throughout the text of his {\em Elements\/}
he was pointing out the theorems in which proofs the 
fifth postulate was used, creating an impression that they are
``less trustworthy.''

Euclid's fifth postulate is a perfect example how difficult 
sometimes may be to decide 
whether a property is convincing (true) enough 
to be accepted as an axiom. The postulate remained
unchallenged for 2200 years, until first half of 
nineteenth century, when Lobatchewsky
showed that Euclid was right in his reserve to this axiom
--- the postulate does not follow from the other axioms 
and it is legitimate to assume that it is false.
Lobatchewsky proved this last fact by showing that,
in a way, in the reality in which we live,
{\em on the surface of the Globe,}
the fifth postulate is false, while other axioms remain valid.
This is the case, since the straight lines on the Globe ---
the curves (known as {\em geodesics\/})
on which any two points are connected by a segment
with the shortest distance between them
--- are the big circles (the circles having the same radius that the Globe). 
And {\em every\/} two big circles on the Globe intersects. 

This part of discussion should convince us that 
the choice of the axioms is not simply dictated by 
some absolute truth. 
\begin{center}
What is true or false may depend on the reality which we describe!
\end{center}
Although the fifth postulate remains true (at least I believe so)
when we do not bound ourselves to the surface of the Earth
but stay within the realm of the classic geometry,
the same postulate is false when we adopt
Lobatchewsky point of view of the geometry
of the surface of Earth. 

Does it mean that all theories people are building 
tell us nothing about reality? 
Are they telling us anything about the truth?
The answer is that 
\begin{center}
all scientific facts are only {\em relatively\/} true. 
\end{center}
They remain true only in the reality 
for which they have been design, in which the basic
assumptions -- axioms -- remain valid. 
Thus, classical physics is true in a sense that their
predictions remains (reasonable) accurate
when we stay within the  realm of everyday sizes, distances, 
and speeds. 
The same classical physics becomes false 
(its predictions become far from the observations) 
when we become experiments concerning 
extreme speeds or subatomic particles. 

%At the same time \begin{center}
%there is no scientific theory which predictions are absolutely 
%(and verifiable) true. \end{center}

The above discussion should convince us 
not only that we should stick to the axiomatic theories
(out of lack of better choice). 
It also indicates that, in general, there is no good answer to the
question 
{\em How can we be sure that the axioms are really true?\/}
since the fifth postulate is neither ``really true'' 
not ``really false.''
(This depends on the reality which we would like to describe.)
So, lets concentrate on two remaining questions on how well we can choose
our axioms: 
\begin{itemize}
\item      Can we ensure that the axioms are {\em consistent}, that is,
           do not contradict each other?
\item      Based on the given set of axioms, can we deduce
           about each statement (concerning our theory)
           whether is it true or false?
\end{itemize}
Unfortunately, the answer to both questions is negative
in a sense described by the following two theorems of G\"{o}del.
\begin{quote}
Assume that the theory is rich enough 
to allow us to talk about natural numbers,
and that the axioms are chosen ``reasonable'' in a sense
that given a sentence we can {\em effectively\/}
decide whether it is an axiom or not. 
Assume also that the axioms do not contradict each other. 
Then
\begin{itemize}
\item We cannot prove the consistency of the theory within the theory itself.
      That is we can express the sentence $\psi$: {\em the theory is consistent\/}
      within the theory, however $\psi$  cannot be deduced from the axioms.
\item There is a sentence $\varphi$ (concerning our theory) which is {\em
independent\/}
      of the axioms, that is, we can deduce from the axioms
      neither $\varphi$ nor its negation. 
\end{itemize}
\end{quote}
Moreover, almost as interesting that the theorem itself is the fact that the 
sentence $\varphi$ has the following very easy intuitive interpretation:
\begin{center}
I (i.e., $\varphi$)  cannot be deduced (proved) from our axioms. 
\end{center}
Clearly this $\varphi$ cannot be proved from the axioms.
However its negation $\neg\varphi$:
\begin{center}
$\varphi$ can be deduced (proved) from our axioms 
\end{center}
cannot be proved as well, since otherwise both $\varphi$
and its negation $\neg\varphi$ would have the proofs
implying that the axioms are self-contradicting. 

G\"{o}del's main achievement was not the discovery of the sentence 
$\varphi$, since it is a version of an ancient Greek's paradox
stated below. His main work was in arguing that this
intuitive self-reference sentence, which is a paradox of kind (2), can be 
expressed (coded) as a property of the natural numbers. 
\begin{description}
\item[Liar Paradox:] Imagine a land in which some inhabitants, Cretans,
always lie and all other inhabitants, Athenians, always
tell the truth. In this land a person says:
\begin{center}
I am     a liar. 
%lying right now. 
\end{center}
Note that this person can be neither Cretan nor Athenian. 
\end{description}
Indeed, Cretan cannot say this phrase, since it is true, 
and he/she never tells the truth. Athenians cannot say it, since
he never lie and it would be a lie. 

\medskip

If you like to feel the depth of self-reference problem, as
in Liar Paradox, consider the following:
\begin{description}
\item[Brain Boggler:]
Imagine that somewhere deep in West Virginia countryside
there is a T-shape crossroad with roads going 
to Richmond,
Morgantown, and Charleston, respectively. 
At the crossroad live two brothers, identical twins. Both 
always answer all questions perfectly precise, but one 
{\em always tells the truth,} the other {\em always lie.}

Driving from Richmond, and being in a hurry for this lecture,
you approach to the crossroad and you realize
that you do not know which way is to Morgantown. 
Luckily, one of the twins comes out from 
the house ready to help you. But which one it is?
You do not know and you have a time to ask just {\bf one} question,
with an answer YES or NO, to find how to get to 
Morgantown on time. 

\medskip

What question should you ask?
\end{description}

%The answer will be provided at the end of the lecture. 


%You should point out at one of the possible roads and ask:
%What would your brother answer if I ask him whether this is the road to Morgantown?

The discussion above concentrated on limitations and difficulties 
%and weak points 
in searching for the ``real scientific truth''
in general, and through axiomatic approach in particular. 
Is there anything good that can be said about 
it after all?

The bright side of G\"{o}del's theorems is that
\begin{center}
there will be always room for the interpretation in science, 
\end{center}
even in highly structured axiomatic theories. 
There will be always a need for some ``human'' input. 
We will not be (easily) replaced by computers.
Our jobs, at least for a while, are safe!

There is even brighter side in seemingly very unpleasant fact that
{\em the truth of the theoretical predictions depends
on the truth of the axioms, and there is no such a thing as the 
``absolute truth'' of the axiomatic system.}  
The reason is that
\begin{quote}
any {\em consistent\/} axiomatic system, no matter how abstract and unbelievable,
may lead to the theory which, in fact, describes 
some portion of our reality. 
\end{quote}
For example, when Lobatchewsky (in the first half of the nineteenth 
century) was developing his 
``abstract geometries'' in different surfaces and
``deformed three-dimensional spaces'' 
(with ``straight lines'' identified with geodesics)
it was just a pure play of thoughts. 
It did not seem to have any ``real'' connection
with the reality. It was not until the 
general relativity theory was formulated when
people realized that, most likely, we in fact live in 
such a ``strange deformed three-dimensional space.''
Suddenly, purely speculative and abstract theory
become useful in describing the reality!

This happened to many abstract mathematical theories: 
being for years, sometimes centuries, 
unuseful and speculative they suddenly 
become practical tools for other sciences. 
%This is the power 





%We can summarize the above discussion in form of the following conclusions. 
%\begin{enumerate}
%\item {\bf All scientific truth is only of the conditional value.}
%       Certainly, if we believe (have a faith) in our assumptions, we 
%       can {\em treat\/} our conclusions as the ``real truth.''
%       But this always requires some level of faith. 
%\item {\bf There is no way to choose ``perfect'' set of axioms/postulates for 
%       any ``reasonable'' theory} in a sense given by G\"{o}del's theorems. 
%       
%       This makes even more difficult to judge what the ``real truth'' is. 
%\end{enumerate}

\bigskip 

My personal interests in this subject comes from 
the study of the theory of sets known as {\em set theory}.
This theory is the most fundamental for all mathematics
(with the exception of pure logic)
in a sense that all mathematical theories can be
treated as sub-theories of the set theory (can be {\em modeled\/} in it)
and use, to some extend, the axioms of set theory. 
In particular, in recent years I was mostly preoccupied with
studying the influence of the axioms of set theory
on the theory of real functions, known as {\em real analysis,}
which a theoretical base for {\em calculus}.
This branch of research has become known as {\em set theoretic analysis}. 

By G\"{o}del's theorems the axioms of set theory 
have a similar flaw as all other ``reasonable'' theories --- they cannot 
decide all statements that concern sets.
To help you appreciate the difficulty in choosing
``correct'' axioms for set theory let us consider the following
two principles: 

\begin{description}
\item[AC:] Let $\cal C$ be a collection (possibly infinite) of sets, 
           each of which has at least one element. Assume also that 
           no two different sets from $\cal C$ have a common element. 
%           (They are disjoint.)
           
           Then there exists a set $S$, called {\em selector}, 
           which has exactly one element in common with each set from $\cal C$.

\item[AD:] Let $I$ be the set of all numbers $x$ with $0\leq x\leq 1$
           which are identified with their decimal expansions:
           $x=0.\overline{x_1x_2x_3x_4\ldots}$. 
           For every set $A$ of numbers from $I$ consider the following infinite game
           $G(A)$: player I chooses the first digit $x_1$ from $0,1,2,3,4,5,6,7,8,9$
           and then player II chooses the second digit $x_2$; next player I chooses
           $x_3$ and player II picks $x_4$, and so on. 
           At the end of the game the players end up with a number
           $x=0.\overline{x_1x_2x_3x_4\ldots}$. 
           We declare that player I wins if $x$ belongs to $A$.
           Otherwise player II wins. 
           
           Then for every set $A$ of numbers from $I$ one of the players has a
           winning strategy in the game $G(A)$. 
\end{description}
Both statements are quite intuitive.

The first tells you that if you take some objects and distribute them
into different drawers --- sets from the collection $\cal C$ ---
then we can take a {\em sampler\/} $S$ containing exactly one 
example from each drawer. This seems pretty obvious.

The second principle tells you that in a game $G(A)$ 
in which both players have perfect information on the rules and 
all moves so far, one of the players should be able to guarantee
himself a victory, assuming that he will play ``perfect'' game. 
(This is true for all finite games, like chess, checkers, or go,
in which players have full information and control of their moves.
It does not mean, however, that it is easy to find 
such a wining strategy.)
This also should be easy to believe.

Do you believe that both these statements are true?
If not, which one seem to be more trustworthy? 

The first of this statement, AC, is known as the Axiom of Choice,
and it is commonly accepted as one of the axioms of set theory,
though it usually plays the role of the fifth Euclid principle
--- it is often singled out as the least believable. 

The second statement, AD, is known as the Axiom of Determinacy. 
\begin{center}
It {\bf contradicts} the Axiom of Choice! 
\end{center}
However it is sometimes considered as its alternative. 

I like to finish this lecture with 
%at least one example of a theorem 
three examples of theorems 
from the set theoretical analysis, my main line of recent
research.  For this consider the following fundamental fact 
which is taught at every course of multivariable calculus. 
\begin{quote}
Let $f(x,y)$ be a function with $0\leq x,y,f(x,y)\leq 1$.
If $f$ is continuous then the following integrals, 
known as the {\em iterated integrals},
exist and are equal
\[
I_1=\int_0^1\left(\int_0^1 f(x,y)\/ dx\right)\/dy\ \ \text{ and }\ \ 
I_2=\int_0^1\left(\int_0^1 f(x,y)\/ dy\right)\/dx.
\]
\end{quote}
If function $f$ is not continuous, the integrals may or may not exist.
However the following seem to be an intriguing question.
\begin{description}
\item[Question:] If for some function $f$ as above the iterated integrals
$I_1$ and $I_2$ exist, must they be equal?
\end{description}
Surprisingly, assuming only the standard axioms of set theory 
we cannot decide what is the answer to this question.
This means, and it has been proven by 
the methods of set theoretic analysis,
that by assuming either of the answers to this question
as an additional axiom of set theory,
the obtained theory will remain contradiction free. 

\medskip

A big part of my research concerned different generalizations
of continuities for the functions of one variable 
(from $\real$ to $\real$). 
For example recall the following basic property of 
of continuous functions  $f$ (from $\real$ to $\real$) known
as the {\em Intermediate Value Property\/} and taught at every 
calculus course. 
\begin{description}
\item[IVP:] For every $a<b$ and every number $y$ between 
            $f(a)$ and $f(b)$ there exists $c$ between $a$ and $b$
            such that $f(c)=y$. 
\end{description}
Thus, every continuous function has IVP. However, it is not difficult to 
find functions with IVP that are not continuous.  
(See e.g. $f(x)=\sin(\frac{1}{x})$ for $x\neq 0$ and $f(0)=0$.) 
But how much of continuity functions with IVP must have?
For example, is the following statement true?
\begin{description}
\item[($*$)] For every function $f$ from $\real$ to $\real$ which has
             IVP there exists a subset $A$ of $\real$ of the same size
             (cardinality) than $\real$ such that $f$ considered
             only on $A$ (i.e., restricted to $A$) is continuous. 
\end{description}

Once again, the answer is that neither property ($*$) nor its negation 
can be deduced from the usual axioms of set theory.
This has been proved in my 1997 paper~\cite{BCN}
(written with M.~Balcerzak and T.~Natkaniec).

\medskip 

%Not all set theoretic analysis theorems concerns 
%independence of the 
I like to finish this talk with citing 
one of my earlier set theoretical analysis result
related to the {\em Paradoxical Decomposition of the Ball\/} theorem.
To state this I have to specify what
we will understand by a three dimensional abstract volume.

\begin{quote}
An {\em abstract volume\/} $\mu$ on the  three dimensional
space $\real^3$ is a function associating 
to every subset $A$ of $\real^3$ from some fixed family
$\cal M$ of subsets of $\real^3$ a 
number $\mu(A)\geq 0$ (possible infinity $\infty$)
and such that
\begin{enumerate}
\item Every geometrically defined solid $S$ belongs to $\cal M$
      and $\mu(S)$ is equal to its standard volume.

\item If $A$ is obtained from $B$ by applying rigid motion
      and $A$ and $B$ belong to $\cal M$ then $\mu(A)=\mu(B)$. 
      
\item If $A_1,A_2,A_3,\ldots$ is a sequence of sets from $\cal M$
      no pair of which has common points
      and $A=A_1\cup A_2\cup A_3\cup \cdots$
      is a union of all sets $A_n$ then
      \[
      \mu(A)=\mu(A_1)+\mu(A_2)+\mu(A_3)+\cdots.
      \]
\end{enumerate}
\end{quote}
It is known that there exists an abstract volume 
(known as {\em Lebesgue measure\/}),
and it follows from the 
Paradoxical Decomposition of the Ball theorem
for any such abstract volume there are non-measurable sets, i.e., the
sets (pieces used in the Paradoxical Decomposition)
which do to belong to $\cal M$. 
So, there is no abstract volume  measuring everything.
But maybe at least there is the best abstract volume
measuring all it possibly can
(in a sense that no other sets can be added to $\cal M$)?
This question was asked by Sierpi{\' n}ski in 1935.
The answer was given in my 1985 paper~\cite{KC:Sierpinski1}, 
joint with Pelc, 
in which it is proved that
\begin{center}
there is no best abstract volume, 
\end{center}
that is, every abstract volume $\mu$ can be farther
extended to measure more sets. 
The full story on the discussion on
this topic can be found in my 1989 popular 
article~\cite{KC:HowGood} published in 
{\em Mathematical Intelligencer}.











\bigskip
\bigskip
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Are there any answers to Brain Boggler:? 
An answer can be found in my web page:\\

http://www.math.wvu.edu/homepages/kcies/


\begin{thebibliography}{22}

\bibitem{BCN} M.~Balcerzak, K.~Ciesielski, T.~Natkaniec,   {\it
Sierpi\'nski--Zygmund functions that are
Darboux, almost continuous, or have a perfect road}, Arch. Math. Logic
{\bf 37} (1997), 29--35.

\bibitem{KC:Sierpinski1} K.~Ciesielski, A.~Pelc, {\it Extensions of invariant  
measures on Euclidean spaces}, Fund. Math. {\bf 125} (1985), 1--10.

\bibitem{KC:HowGood} K.~Ciesielski,  {\it How good is the Lebesgue measure?},
Math. Intelligencer {\bf 11}(2) (1989), 54--58.



\bibitem{Web} {\it Webster's New Word Dictionary of American English},
third college edition, Webster's New Word, New York. 1988. 

\end{thebibliography}


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