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\cl {\bf A BRIEF HISTORY OF SET THEORY }
\noi
{\cl {\it A set is a collection, thought of as a whole}}
These are the words (sort of) of the mathematician Georg Cantor, circa late 19th century.
\noi
So what?
\noi
Why shouldn't we consider a collection of objects as a single unit?  
Certainly the collection of all kitchen utensils currently in your kitchen is innately a
 well defined concept, 
and saying "my kitchen utensils" is tantamount to
identifying them together. Pasting, so to speak.  
\noi
So what?
\noi
Well, for years, many philosophers and mathematicians refused to identify an 
infinite collection as a whole 
(I know, there are only SO many utensils in your kitchen). The natural numbers 
(or counting numbers, i.e. 1,2,3....) 
were merely thought of as "potentially
infinite", and no concept of a "completed infinity" was thought acceptable.  
\noi
Phooey! said Cantor. 
It is a small mind that will allow no such thoughts.  The natural
numbers were an entity unto themselves. Surely we must allow this. And thus with this seemingly 
simple idea was "Transfinite Set Theory" born.  I'm still surprised
at the debate Cantor's ideas instigated at the time. But as we shall see, Cantor's 
ideas led mathematics 
(dare I say science) down an utterly amazing path.
% and upon reflection I think these men as prescient.  
In the end, this fundamental  
debate led mathematicians to examine the very
foundations of their own "science".  And many surprising results have been found.  
\noi
This is that story.  
\noi 
First, Cantor was led to consider relative "sizes" of his
sets. It had always been common, and fruitful, to compare the sizes of finite sets. 
(I have more utensils than you! Why? Because we can match them up, one-for-one,
and after this is done, I still have some left over). In fact, Galileo had once had the 
following thought: 
Consider the set of natural numbers (call it $\fatn$) and the
proper subset of it consisting of merely the  natural numbers that are perfect squares 
(call it $\fatn^2$). 
Then $\fatn$ and $\fatn^2$ can be paired off, one-for-one, in an
utterly trivial manner. To each natural number $n$ there corresponds a unique 
perfect square, namely $n^2$. 
And conversely to each perfect square $m$ there
corresponds a unique natural number, namely $\sqrt m$.  So the sizes
of these two sets were comparable, and nothing was left over in the end. They had the same size! 
\noi
Maybe Cantor had read Galileo's work. Or maybe lightning struck
twice. But in the end, Cantor was hospitalized for mental depression, while Galileo 
merely dismissed the notion, 
thinking (as we STILL think) that clearly there
are many more natural numbers as there are perfect squares, for the perfect squares 
form merely a proper subset 
of the natural numbers. But Cantor
would freely extrapolate from the finite case, and was led to conclude that $\fatn$ and $\fatn^2$
 have the same size, which he called $\aleph_0$. (Aleph is the first
letter of the Hebrew alphabet). Using similar arguments, Cantor was able to establish that
 any infinite, 
proper subset of the natural numbers also had size
$\aleph_0$. He deemed such sets countable. But were there other size infinities? 
Something "bigger" than $\aleph_0$? Did uncountable sets exist?
\noi
 He considered the
rational numbers $ \ \fatq$ (i.e. the fractions). They were countable! 
\noi
He considered 
the real numbers $\fatr$ 
(i.e. the real number line, or the "continuum" as it was
then called). He supposed they were countable, and obtained a devious contradiction. 
So they couldn't be countable. Could they? 
They were (and still are, as far as
I know) uncountable!  He called their size c. 
\noi
More generally, Cantor considered the power set of an arbitrary set.  
This is just the set 
of all subsets of the given
set. In the finite case, if a set has $n$ elements, it turns out that its power set 
has $2^n$ elements.  Bigger. A lot bigger.  
\noi
And it turns out that the power set of
any infinite set will also always be "bigger". That is, after any one-for-one pairing, 
there will still be 
elements of the power set left over.  So Cantor took the
power set of $\fatn$ and dubbed its size $2^{\aleph_0}$.  
And the power set of this set was
 $2^{2^{\aleph_0}}$ big. And on and on. And $\aleph_0 < 2^{\aleph_0} < 2^{2^{\aleph_0}}$ 
(and so on). 
And cardinal arithmetic was born, where $\aleph_0 +  \aleph_0 = \aleph_0$. Huh? 
\noi
But how do these alephs compare with c? Cantor showed that $c$ was equal to
$2^{\aleph_0}$, and then hypothesized that there were no other sizes of infinity "between" 
$\aleph_0$ and $c$. 
This is what is known as the {\it Continuum
Hypothesis} or {\bf CH}. More (much) on that later. 
\noi
Cantor theorized that "Hey, if I can
consider  infinities as complete and not merely potential, then what is to keep me from
thinking about the big whopper: the set of all sets."
 Everything, taken together as a
whole.   And how big would it be? Surely the most big of them all. The largest
of the infinities. The last stop on the aleph train. But what about its power set? 
It had to be bigger. 
One more $2^?$.  So Cantor took the Galilean approach: Keep
it quiet! 
\noi
Thus began the paradoxes that would come to plague Cantor's theory. Russell's paradox 
is the most famous. 
It has many similarities to the so called "self
referential" statements like "I am lying" or "This sentence is false". 
And the mathematicians of the world took cover.  Their very science was threatened. At its
core.  
\noi
So at the beginning of the 20th century, mathematicians took on the task of 
investigating the 
foundations of their subject. The paradoxes of set theory had
shown them the necessity of reformulating Cantor's ideas. More carefully. 
Ernst Zermelo was likely the 
first to develop an axiom scheme for set theory. His axioms,
modified over the years by Adolf Fraenkel and others, are now known as the 
"Zermelo-Fraenkel (or {\bf ZF}) Axioms of Set Theory". 
They are still used today.  
\noi
Zermelo was led to
the foundations of set theory by his investigation of the {\it Well-Ordering Principle},
or {\bf WOP}.  This idea was first hypothesized by Cantor, and states that any set can be
well ordered. That is, an ordering can be found for which every nonempty subset has a 
"least" member. 
Cantor was sure this was true, but could not prove it. Why not?
Well, the natural ordering (less than or equal to) on the real numbers is not a 
well-order.  
Even the interval (0,1) has no least member under this ordering. But
surely some well order for $\fatr$ could be found. Well, to this day it has not!
 However, Zermelo claimed to have proved the {\bf WOP} (circa 1905) but his
proof did not construct a well-order, it merely claimed that one must exist. 
Should we believe 
him?
\noi
The catch here is that  he explicitly assumed what is called the 
{\it Axiom of Choice}, or {\bf AC}. An idea that
 once
again, like Cantor's infinite sets, caused many mathematicians discomfort. 
The {\bf AC} states that for any (possibly infinite) collection of sets, one
member can be chosen from each set to form a new set. Big deal, right? 
But the squeamishness of some mathematicians in extrapolating to the infinite case arose
again. And rightly so, as the paradoxes had shown.  
\noi
Russell pointed out the subtleties of the {\bf AC} this way: "If you have an 
infinite collection of
pairs of shoes, then the {\bf AC} is not needed to identify one from each pair, 
for certainly one 
can choose, for example, the left shoe from each pair.
However, the choice axiom must be invoked to choose one member from an infinite 
collection of pairs 
of socks. Since the socks are identical, no rule can be stated
as to how to choose one from each pair."
\noi
 That the best mathematicians of the day were considering such questions (shoes and socks?) 
would have the Greeks rolling
in their graves.  
\noi
Now the {\bf AC} was eventually shown to be equivalent to the {\bf WOP} 
(a nice little fact). 
So the fate of the {\bf WOP} was now directly tied to the fate of the {\bf AC}. But the
question still  remained, "Are these provable within the {\bf ZF} axiom system?"
It was thought that Zermelo's
axioms of sets could be used to prove them, and hence establish them as theorems of set theory. 
No such luck (we'll discuss this more later). Other axioms of set
theory were developed, and the attempts to prove the {\bf AC} continued. 
But most mathematicians came to accept some form of the axioms of set theory. The
main debate concerned the {\bf AC}. 
Many said if it could not be proved, then it should not be assumed. Many others took 
the easy route; if you can't prove
it then assume it. They merely added it as the last axiom of set theory.  
In fact, it might be worth mentioning that today the exact same debate rages. Why was it
not resolved? It was.
\noi
Sort of!
\noi
Not surprisingly, mathematicians at the turn of the century had seen 
(some 75 years earlier) 
a very similar situation as the one they were in now. 
The development of Non-Euclidean geometries 
throughout the 19th century had cast doubt on the previously
undoubtable Euclidean Axioms of Geometry, for over 2000 years the 
benchmark of mathematical 
thinking. 
 And Euclidean Geometry was found to be wanting, in some
respects. It wasn't as absolute as once thought. There are other, 
just as vital and knowable,
 geometries. 
And in one of these geometries did Einstein find his
theory of general relativity. So much for Euclid! So much for absolute certainty! 
But this is another story altogether. 
\noi
However, Non-Euclidean Geometries had given
mathematicians an insight into the important questions at the heart of any axiom system. 
Namely independency, completeness, and consistency. So armed with this
knowledge and with the desire to rid mathematics of the various paradoxes did the mathematicians 
of the early 20th century tackle the task of placing ALL of
mathematics (set theory, geometry, arithmetic) on a firm foundation.    
\noi
But what do we mean by "firm"? Well, any axiom system is just a list of undefined terms
together with a collection of statements (the axioms) taken to be truths. 
For example, point, line and incidence are undefined terms of Euclidean Geometry.
Euclid's first axiom then states that "Any two distinct points are incident to a unique 
line". 
This was taken as a truth, a priori. And using only his axioms and
the laws of logic did Euclid develop his geometry. The fact that there were undefined 
terms led the German mathematician David Hilbert to consider axiom systems as
stripped of any real-world interpretation, and merely to be considered as a formal system. 
In other words, as Hilbert said, "we should at all times be able to
substitute for point, line and circle the terms table, chair and beer mug".  
Making the appropriate translations would then convert the theorems of geometry into
theorems of German taverns! Oh my! 
\noi
So Hilbert stripped his axiom systems of any single interpretation and studied their 
validity via the following questions: 
\item {} 1)
Are the axioms independent? That is, if one of the axioms can be proven, as a theorem, 
from the others, 
then it was superfluous and need not be taken as an axiom.
This was a question of conciseness and simplicity. 
\item {} 2)	Is the axiom system complete? That is, is it possible to prove any 
�well-formulated� statement within the
system, or its negation. Hilbert hoped so. If completeness could be verified then 
mathematicians would know that any problem in mathematics could be solved. The
{\bf AC} and the {\bf CH} COULD be resolved. We just need to work harder! 
\item {} 3)	Is the axiom system consistent? That is, is it free of
contradiction.
\noi
Nearly every mathematician viewed this last question with utmost importance.  
The paradoxes had to be resolved, 
and more importantly, we need to know that we will never see them again!
So Zermelo, Fraenkel, Hilbert, Bernays, Von Neumann, Godel and others tackled the 
foundations of set theory. They were fairly successful.  They failed to resolve
the problem of the {\bf AC}, and couldn't prove consistency, but no paradoxes were 
discovered. 
They hoped that they were now free of the contradictions. And
Hilbert and Bernays were quite successful with geometry, laying the framework for various 
geometries and establishing, at least, their �relative� consistency.  And
arithmetic was attacked by Peano, Frege, Russell and others.  Peano developed axioms 
for the 
natural numbers. Frege and Russell attempted to found arithmetic on
logic alone, thinking that possibly the two subjects were the same. 
However, Frege had used Cantor's set theory and his work was undone by Russell's paradox. 
He
was shattered and gave up. Russell (and Whitehead) then attempted to fix things. 
 For 15 years they 
labored to produce the "Principia Mathematica", what they hoped
would become the indisputable foundation of arithmetic.  Using logic as their 
starting point,
 they produced three volumes of what sometimes looks like wallpaper, a
collection of symbols with no written text whatsoever. But they encountered many 
fundamental difficulties, and really didn't get very far (200 pages to prove that
$1+1=2$!). As Russell later said, "for 15 years I labored, and I finally came to the 
conclusion that there was nothing more I could do".  
\noi
So the struggles continued
throughout the early 20th century.  And there were modest successes. 
But the final resolution was yet to be obtained. Hilbert, in particular, 
was adamant that the
questions of completeness and consistency would be resolved to everyone's satisfaction.
 And some 30 years passed with many noble attempts to straighten things
out.  But then Kurt Godel, in a voice reeking with self-reference, stated something to the 
effect of: "Some things we just can't know, and one of these things that
we just can't know is whether or not mathematics can be freed of these paradoxes." 
And the mathematicians believed him! So much for certainty!
\noi
What Godel established was the following: that any consistent axiom 
system strong enough to contain arithmetic was inherently incomplete.  
That the price of consistency was uncertainty. That if the system was consistent, then it
would contain true statements that could not be proved. Moreover, Godel proved that 
the question of consistency itself was one such statement. That is, if the
system is consistent, then we can't prove it! Needless to say, Godel's results shocked 
the mathematical community.  Its not clear of their impact on Hilbert, who
by this time was getting on in age. But they had to destroy him.  It was Hilbert who had 
believed what Cantor had developed would certainly lead to
certainty. "No one can expel us from the paradise Cantor has created" he had claimed. 
Well, Russell's paradox had shown that maybe it wasn't quite a
paradise. And then Godel had said something to the effect of "it sure ain't 
no paradise". 
So it goes. 
\noi
So what could mathematicians do? This: pick your
favorite axioms. Be aware that there may be paradoxes lurking, and that you will never 
be able to establish that there aren't! Just hope you don't find one.  Also
be aware that there will be questions which can not be resolved.  True statements 
that you won't be able to prove. In fact in 1964, Paul Cohen, using some results
of Godel, showed that both the {\bf CH} and the {\bf AC} (along
with their negations) are independent 
of the other axioms of {\bf ZF} set theory, and thus they are undecidable
questions. In other words, they can neither be proven true or false. Either they are true
or false, some mathematicians would say, we just  will never 
know which. 
For over 60 years, many mathematicians had struggled to
decide these questions, and their efforts were indeed in vain!
\noi
 I would be remiss at this point in not mentioning the ideas of the so called 
Intuitionists, 
a group
led by the Dutch mathematician E.L. Brouwer in the early 20th century. The original 
outrage 
at Cantor's 
idea of a completed infinity was led by Leopold Kronecker,
who said "God made the natural numbers, all else is the work of man". And the 
Intuitionists 
radically 
stripped mathematics of the infinite. The great French
mathematician Henri Poincare is sometimes identified with this school of thought,
 as he
once said  "future generations of 
mathematicians will come to view Cantor's work as a
disease from which we have recovered".  Not quite the paradise espoused by Hilbert! 
And the Intuitionists attacked mainstream mathematics. The {\bf AC} was
simply invalid, they claimed. Brouwer was even led to question the validity of the law of 
excluded middle, a basic law of logic which states that every statement
is either true or false.  Brouwer reasoned that there may be a third possibility. 
And three-valued logic was born. 
(And today we have fuzzy logic). And in some
sense Godel and Cohen's work validated some of the Intuitionist philosophy, for 
they showed that indeed 
there were undecidable statements. Maybe not true. Maybe not
false.
\noi
Oh well, so it goes.

\vskip40pt
\hskip260pt D.Wick 

\hskip260pt March 2001
\bye