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An Interview with Ann Varela – Georg Cantor

Apr 10, 2017 by

Michael F. Shaughnessy –

1) There is one name that stands out in my mind when I think about the concept of “set” in mathematics. That name is Georg Cantor. First of all, what do we know about where he was born and his early childhood?

Although Georg Cantor was born in St. Petersburg, Russia in 1845, by age eleven he fled to Germany with his family when his father became ill.  Prior to relocating to Germany, Cantor had a private tutor before attending primary school in St. Petersburg.  He was regarded as an excellent student who excelled in mathematics, especially trigonometry.  In fact, he graduated in 1860 with distinction from the Realschule in Darmstadt.

Two years later, Cantor was a student at the University of Zürich.  After his father’s death in 1863, he transferred to the University of Berlin, as his inheritance was a generous one, and concentrated his studies in the areas of physics, philosophy, and mathematics.  By 1867, Cantor submitted his dissertation on number theory at the University of Berlin.  Next, he taught for a while in a Berlin girl’s school before taking a teaching position at the University of Halle.  He spent the rest of his career there.

Cantor’s thesis qualified him to conduct self-contained university teaching and proved to be vital for being promoted to Extraordinary Professor in 1872, ultimately making full professor in 1879.

2) Now, when we talk about “set” in math, what was Cantor’s original ideas about a set? Different people probably have different conceptualizations of a set.

Simply put, a set is a collection of objects (elements) that have something in common or follow a rule.  Every object in the set is unique.  We may define a set by listing or describing its elements.  For example, E = {2, 4, 6, 8, 10, 12, 14} may describe the set of even numbers less than 15.

There are times when it is not reasonable to list all the elements of a set.  In that case, we can describe the set with a rule.  For example, if we are interested in a larger set of even numbers than what is listed in the previous example, we could rewrite the set as E = {even numbers less than 100}.

Cantor was mainly interested in comparing inherent differences in sets of numbers, which at first, may seem simple, but is actually quite complex.  His main interest was in comparing the sizes of sets.  To some, it may be obvious to simply count the elements in each set and compare those totals; however, one must have the ability to count.  To overcome the counting obstacle, one could match an element of one set with an element of another set and continue this process until there are no more matches to be made.  If there are no elements remaining from either set after this process, the sets are called equivalent (the same size).

This process of matching (or pairing) illustrates a one-to-one correspondence which permits us to compare sets without ever counting the number of elements in either set.  Cantor defined two sets as being equivalent if it is possible to put them together using some law, in such a way that each element from one set can only be paired with one and only one element of another set.  See Figure 1.

Figure 1

Eventually, Cantor compared the infinite set of rational numbers (numbers of the form a/b, where b cannot equal zero) with the infinite set of natural numbers by using a procedure of listing and counting all the rationals and then pairing each rational in this list with the consecutive natural numbers.  See Figure 2.

Figure 2

The figure demonstrates how rational numbers are countable and that the infinitude of rational numbers is the same size as the infinitude of natural numbers.  Notice the one-to-one correspondence between N and Q.

Prior to Cantor’s work, the concept of a set was a rather rudimentary one that had been used unreservedly since the beginning of mathematics, dating back to the philosophies of Aristotle.  During that period, only finite sets were studied, or even known about, for that matter.  Set theory was frequently misunderstood by other mathematicians of Cantor’s time, thus leading to resistance, ridicule, and tragically, Cantor’s manic depression and paranoia.

3) Now, is “set theory” different from the idea of a “set” or is it all intertwined?

Set theory is a field of mathematics that studies sets (collections of objects), so the two concepts are interrelated.  Set theory is based on the relationship between an object o and a set A.  If o is a member (or element) of A, we write o  A.  Since sets are considered to be objects, one set can relate to another unique set.

Some of the more common sets of significance include the empty set (the set containing no elements), the set of real numbers (the set of all rational numbers combined with the set of all irrational numbers), and the set of natural numbers (used for counting and ordering).  The following special symbols exist to define these sets of numbers:  Z represents the set of all integers, Q represents the set of all rational numbers, and R represents the set of all real numbers.

We may describe a set of interest by specifying some property that is shared by its elements but not by any other object as illustrated above in the example with the set of even numbers less than 15.  Other rules are more sophisticated, such as {x|x and }.  This notation stands for the set of all x such that x is an element of R and x is greater than zero.

Sets may be compared to each other by using the operations of Union (a set that contains each element of the two given sets and no others), Intersection (the set of all elements common to each of the two given sets), Relative Complement (the collection of elements that are in one of the two given sets by not in the other), Set Difference (denoted U \ A; the set of all members of U that are not members of A), Symmetric Difference (denoted A or ; the set of all elements that are in one of the sets, but not in both), Cartesian Product (denoted A x B; is the set whose members are all possible ordered pairs (a, b) where a is a member of A and b is a member of B), and Power Set (the set of all subsets of a given set).

4) How do mathematicians regard and work with set theory today, and how does the average person work with sets in their daily lives?

Anytime you classify objects, count them and perform operations on such collections, you apply set theory.  A good example of this is shopping on-line.  When you use a search engine, set theory is involved because items are stored in several separate sets, such as Women’s, Men’s, Children’s, Shoes, For the Home, etc., and our request makes the computer search its database and finds the intersection of the two sets:  Men’s and Shoes, Women’s and Sweaters, and For the Home and Window Treatments.

It may be helpful to use a pictorial representation, such as a Venn diagram, in the study of set properties.  Venn diagrams represent the elements of a set with points (elements) inside a circle.  Figures 3 and 4 illustrate examples of how set theory relates to applications in everyday life.   Figure 3 is used by Medical Researchers to show the overlapping genes associated with different brain diseases, while Figure 4 compares the features on different models of PlayStation 3 gaming consoles.

Figure 3

Figure 4

Mathematicians are not alone in applying set theory and logic to real-life situations.  For example, fuzzy logic has been used in computer science, engineering, social science, business, the automotive industry, and medical sciences for research and development of facial pattern recognition software, home appliances, antiskid braking systems, weather forecasting systems, robotics, and medical diagnosis and treatments plans, to name a few.

The average person may not even realize they are using set theory in their day-to-day lives.  As mentioned earlier, using a search engine is a common application of set theory.  For example, we may utilize numerous conditions when searching for a new car on the internet, such as make, model, estimated mpg, color, and special features.  In the end, we end up with our ideal car.

Students may unknowingly use set theory when scheduling college courses.  The course names could represent the sets, while the time of day could be the elements within each set.  Some courses would overlap, or intersect, if they are offered at the same time.  The following table illustrated this point.

Time Offered Course Name
MATH ENGLISH
9:00 a.m.
10:00 a.m. 10:00 a.m.
11:00 a.m. 11:00 a.m.
2:00 p.m.

 

Managing money is common to all of us and easily fits with the set theory model.  The set of bills to pay and the set of dollars owed can be classified, counted, and operations performed upon those sets.  Some bill payments may be the same dollar amount while others are unique.  Some bills may be paid, monthly, biannually, or annually, thus a third set is introduced.

5) Has anyone elaborated on his theory over the years or developed it further?

Cantor’s theory of sets and mathematics have been compared to true love—never running smooth.  Fortunately, this notion has not deterred people from further study of set theory.

There was a breakthrough with Cantor’s continuum hypothesis in 1940 when Kurt Gödel proved that the continuum hypothesis was soundly consistent with the other axioms of the theory; there was no way to disprove the continuum hypothesis.

There are other branches of set theory, besides fuzzy logic, that are currently being studied and utilized in the twenty-first century.  For example, Rough Set Theory deals with all things having something in common with at least one other thing.  This notion has made it possible to compare and draw conclusions regarding functioning components that may not be completely defined or measured.  There was a historical comparison of the Great Depression of the 1930s with the Credit Crisis of 2008.  These sets intersect and can reveal important confirmations to the effect of management operation prior to financial woes.

The hypothetical basis of fuzzy sets was explained and applied to the noise and visual effects associated with a “live” windfarm case study back in 2007.

As recent as 2014, set-theoretic geology has used set-forcing, a technique for proving consistency and independence results, to model mantle penetrates down through the inner mantles to the outer core.

Soft-set theory, a parameterized family of sets, has been used to develop a model for decision making problems based on soft-set methods.  Applications of this model may be used in computer science and game theory.

In the medical field, a set-theory based approach to identify patients with heart failure was used in 2016.  Researchers found that patients with a diagnosis of heart failure exhibited at least one of the three electronic health records characteristics (discharge diagnosis of heart failure, echocardiogram results, and loop diuretic administered).  The findings suggested that more complex algorithms were needed to improve heart failure phenotyping.  The significance of this study was disease management.

6) What have I neglected to ask?

There is a Cantor medal of the Deutsche Mathematiker-Vereinigung, a professional society of German mathematicians, awarded at the society’s biannual meetings.  Mathematicians associated with the German language are eligible prize winners.

In addition to brilliance in the field of mathematics, Cantor and his family were accomplished musicians.

Cantor was regarded as an exceptional violinist.

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