Georg CantorI. Georg CantorGeorg Cantor founded set theory and introduced the concept of infinite numberswith his discovery of cardinal numbers. He also advanced the study oftrigonometric series and was the first to prove the nondenumerability of thereal numbers. Georg Ferdinand Ludwig Philipp Cantor was born in St. Petersburg,Russia, on March 3, 1845. His family stayed in Russia for eleven years until thefather’s sickly health forced them to move to the more acceptable environment ofFrankfurt, Germany, the place where Georg would spend the rest of his life.

Georg excelled in mathematics. His father saw this gift and tried to push hisson into the more profitable but less challenging field of engineering. Georgwas not at all happy about this idea but he lacked the courage to stand up tohis father and relented. However, after several years of training, he became sofed up with the idea that he mustered up the courage to beg his father to becomea mathematician. Finally, just before entering college, his father let Georgstudy mathematics. In 1862, Georg Cantor entered the University of Zurich onlyto transfer the next year to the University of Berlin after his father’s death.

At Berlin he studied mathematics, philosophy and physics. There he studied undersome of the greatest mathematicians of the day including Kronecker andWeierstrass. After receiving his doctorate in 1867 from Berlin, he was unable tofind good employment and was forced to accept a position as an unpaid lecturerand later as an assistant professor at the University of Halle in1869. In 1874,he married and had six children. It was in that same year of 1874 that Cantorpublished his first paper on the theory of sets. While studying a problem inanalysis, he had dug deeply into its foundations, especially sets and infinitesets. What he found baffled him. In a series of papers from 1874 to 1897, he wasable to prove that the set of integers had an equal number of members as the setof even numbers, squares, cubes, and roots to equations; that the number ofpoints in a line segment is equal to the number of points in an infinite line, aplane and all mathematical space; and that the number of transcendental numbers,values such as pi(3.

14159) and e(2.71828) that can never be the solution to anyalgebraic equation, were much larger than the number of integers. Before inmathematics, infinity had been a sacred subject. Previously, Gauss had statedthat infinity should only be used as a way of speaking and not as a mathematicalvalue. Most mathematicians followed his advice and stayed away. However, Cantorwould not leave it alone. He considered infinite sets not as merely going onforever but as completed entities, that is having an actual though infinitenumber of members.

He called these actual infinite numbers transfinite numbers.By considering the infinite sets with a transfinite number of members, Cantorwas able to come up his amazing discoveries. For his work, he was promoted tofull professorship in 1879. However, his new ideas also gained him numerousenemies. Many mathematicians just would not accept his groundbreaking ideas thatshattered their safe world of mathematics. One of these critics was LeopoldKronecker. Kronecker was a firm believer that the only numbers were integers andthat negatives, fractions, imaginaries and especially irrational numbers had nobusiness in mathematics. He simply could not handle actual infinity.

Using hisprestige as a professor at the University of Berlin, he did all he could tosuppress Cantor’s ideas and ruin his life. Among other things, he delayed orsuppressed completely Cantor’s and his followers’ publications, belittled hisideas in front of his students and blocked Cantor’s life ambition of gaining aposition at the prestigious University of Berlin. Not all mathematicians werehostile to Cantor’s ideas. Some greats such as Karl Weierstrass, and long-timefriend Richard Dedekind supported his ideas and attacked Kronecker’s actions.However, it was not enough.

Cantor simply could not handle it. Stuck in a third-rate institution, stripped of well-deserved recognition for his work and underconstant attack by Kronecker, he suffered the first of many nervous breakdownsin 1884. In 1885 Cantor continued to extend his theory of cardinal numbers andof order types. He extended his theory of order types so that now his previouslydefined ordinal numbers became a special case.

In 1895 and 1897 Cantor publishedhis final double treatise on sets theory. Cantor proves that if A and B are setswith A equivalent to a subset of B and B equivalent to a subset of A then A andB are equivalent. This theorem was also proved by Felix Bernstein and by Schrder. The rest of his life was spent in and out of mental institutions and hiswork nearly ceased completely.

Much too late for him to really enjoy it, histheory finally began to gain recognition by the turn of the century. In 1904, hewas awarded a medal by the Royal Society of London and was made a member of boththe London Mathematical Society and the Society of Sciences in Gottingen. Hedied in a mental institution on January 6, 1918.

Today, Cantor’s work is widelyused in the many fields of mathematics. His theory on infinite sets reset thefoundation of nearly every mathematical field and brought mathematics to itsmodern form.II. InfinityMost everyone is familiar with the infinity symbol .

How many isinfinitely many? How far away is “from here to infinity”? How big is infinity?We can’t count to infinity. Yet we are comfortable with the idea that there areinfinitely many numbers to count with: no matter how big a number you might comeup with, someone else can come up with a bigger one: that number plus one–orplus two, or times two. There simply is no biggest number. Is infinity a number?Is there anything bigger than infinity? How about infinity plus one? What’sinfinity plus infinity? What about infinity times infinity? Children to whom theconcept of infinity is brand new, pose questions like this and don’t usually getvery satisfactory answers. For adults, these questions don’t seem to have verymuch bearing on daily life, so their unsatisfactory answers don’t seem to be amatter of concern. At the turn of the century Cantor applied the tools ofmathematical rigor and logical deduction to questions about infinity in searchof satisfactory answers. His conclusions are paradoxical to our everydayexperience, yet they are mathematically sound.

The world of our everydayexperience is finite. We can’t exactly say where the boundary line is, butbeyond the finite, in the realm of the transfinite, things are different.Sets and Set TheoryCantor is the founder of the branch of mathematics called Set Theory, which isat the foundation of much of 20th century mathematics. At the heart of SetTheory is a hall of mirrors–the paradoxical infinity.

Georg Cantor was known tohave said, “I see it, but I do not believe it,” about one of his proofs. The setis the mathematical object which Cantor scrutinized. He defined a set as anycollection of well-distinguished and well-defined objects considered as a singlewhole. A collection of matching dishes is a set, as well as a collection ofnumbers.

Even a collection of seemingly unrelated things like, television,aardvark, car, 6} is a set. They are well-defined and can be distinguished fromone another. Sets can be large or small. They can also be finite and infinite. Afinite set has a finite number of members. No matter how many there are, givenenough time, you can count them all. Cantor’s surprising results came when heconsidered sets that had an infinite number of members. Sets such as all of thecounting numbers, or all of the even numbers are infinite sets.

In order tostudy infinite sets, Cantor first formalized many of the things that areintuitive and obvious about finite sets. At first, it seems like theseformalizations are just a whole lot of trouble, a way of making simple thingscomplicated. Because the formalisms are clearly correct, however, they provide apowerful tool for examining things that are not so simple, intuitive or obvious.Cantor needed a way to compare the sizes of sets, some method for determiningwhether sets had the same number of members. If two sets didn’t have the samenumber of members, he needed a method for telling which one was larger. Ofcourse this is simple for finite sets. You count the members in both sets.

Ifthe number is the same, they are the same size. If the number of members in oneset is greater than the number of members in the other, then that set is larger.You can’t count the members in an infinite set, though, so this method won’twork for comparing their sizes. If there are two infinite sets, one must needsome other way to tell if one is larger.

The formal notion that Cantor used forcomparing sizes of sets is the idea of a one-to-one correspondence. A one-to-onecorrespondence pairs up the members of one set with the members of another. Setswhich can be matched to each other in this sense are said to have the samecardinality. We could pair up the elements of the imaginary set television,aardvark, car, 6} with the numbers 1,2,3,4}.

It is possible to do this so thatone member of each set is paired up with one member of the other, no member isleft out, and no member has more than one partner. Then we can be sure that theset1,2,3,4} has the same number of members as the set television, aardvark,car, 6}. one-to-one correspondence:television, aardvark, car, 6}1,2,3, 4}So, what is bigger? infinity+X? infinity+infinity ? Or infinity(infinity)? Tocalculate which is bigger cantor used sets and one-to-one correspondence.These one-to-one correspondence sets show that even though we add an unknownvariable, multiply by two, and square a set, the upper and lower sets stillremain equal. Since we will never run out of numbers any correspondence set withtwo infinite values will be equal.

All these sets clearly have the samecardinality since its members can be put in a one-to-one correspondence witheach other on and on forever. These sets are said to be countably infinite andtheir cardinality is denoted by the Hebrew letter aleph with a subscript nought,.OTHER INFINITIESCantor thought once you start dealing with infinities, everything is the samesize. This did not turn out to be the case. Cantor developed an entire theory oftransfinite arithmetic, the arithmetic of numbers beyond infinity. Although thesizes of the infinite sets of counting numbers, even numbers, odd numbers,square numbers, etc., are the same, there are other sets, the set of numbersthat can be expressed as decimals, for instance, that are larger.

Cantor’s workrevealed that there are hierarchies of ever-larger infinities. The largest oneis called the Continuum. Some mathematicians who lived at the end of the 19thcentury did not want to accept his work at all. The fact that his results wereso paradoxical was not the problem so much as the fact that he consideredinfinite sets at all. At that time, some mathematicians held that mathematicscould only consider objects that could be constructed directly from the countingnumbers. You can’t list all the elements in an infinite set, they said, soanything that you say about infinite sets is not mathematics. The most powerfulof these mathematicians was Leopold Kronecker who even developed a theory ofnumbers that did not include any negative numbers. Although Kronecker did notpersuade very many of his contemporaries to abandon all conclusions that reliedon the existence of negative numbers, Cantor’s work was so revolutionary thatKronecker’s argument that it “went too far” seemed plausible.

Kronecker was amember of the editorial boards of the important mathematical journals of his day,and he used his influence to prevent much of Cantor’s work from being publishedin his lifetime. Cantor did not know at the time of his death, that not onlywould his ideas prevail, but that they would shape the course of 20th centurymathematics.Science