Archive for the ‘Set Theory’ Category

I read a review of the graphic novel Logicomix in the New York Times recently.

Logicomix is a graphic novel that dramatizes the work and lives of some of the most important mathematical logicians of the late 19th and early 20th century.

A review in the Financial Times explains:

First among them is Bertrand Russell, the English philosopher whose life story this is – at least as far as 1939. Also present are his fellow pioneers in the philosophy of mathematics: Alfred North Whitehead, with whom Russell sought, in the years before the first world war, to provide a logically rigorous, good-for-all-time foundation for mathematics; Ludwig Wittgenstein, the austere Austrian who argued that Russell’s project was misconceived; Kurt Gödel, Wittgenstein’s compatriot, who proved that it was; and assorted other pin-ups of higher mathematics – Cantor, Poincaré, Hilbert.

So, essentially, what was going on is that the British logician/mathematician Bertrand Russell, together with his collaborator Alfred North Whitehead, were trying to prove that mathematics was “self-contained.”  That is to say that they were trying to show that there is no “fudge factor” in mathematics.

It is understandable that someone might be sucked into believing that this is possible.  Math can be very reassuring in the apparently solid nature of its logical structure.  It sometimes appears as if there are no holes in the chains of reasoning upon which all of mathematics is built.

But, in reality, mathematics is and always has been a construct that has significant holes in its foundations.  Not being able to divide by zero may be the least of math’s logical dilemmas.

This is not to say that the lack of logical perfection makes mathematics useless.  Far from it, math has been and remains a very useful reasoning tool for humans.  In fact, one of the most amazing things to me about math is that Kurt Gödel used mathematical logic itself to demonstrate the holes that exist in the logical structure of mathematics.

Gödel, Escher and Bach

I was first introduced to these events in the history of mathematical logic through the book Gödel, Escher and Bach, which was written in 1979 by Douglas Hofstadter.

In the introductory chapter, Hofstadter explains the basic ideas of the work of Bertrand Russell and the Austrian mathematician/logician Kurt Gödel.  Russell and Alfred North Whitehead were working very hard in the early 20th century to establish a rock solid logical foundation for mathematics that would dispel all of the paradoxes and conundrums that both ancient and contemporary mathematicians had struggled with.  Russell and Whitehead’s work was eventually published as the three-volume Principia Mathematica.

The simplest example of the type of paradox that Russell and Whitehead were concerned with is the one that Hofstadter gives in Gödel, Escher and Bach:

That paradox is the so-called Epimenides paradox, or liar paradoxEpimenides was a Cretan who made one immortal statement: “All Cretans are liars.”  A sharper version of the statement is simply “I am lying”; or, “This statement is false”.  It is that last version which I will usually mean when I speak of the Epimenides paradox.  It is a statement which rudely violates the usually assumed dichotomy of statements into true and false, because if you tentatively think that it is true, then it immediately backfires on you and makes you think it is false.  But once you’ve decided it is false, a similar backfiring returns you to the idea that it must be true.  Try it!

This paradox is particularly vexing for mathematicians because mathematics relies extensively on the “Law of the Excluded Middle,” which assumes that a statement is either true or false.

In Gödel, Escher and Bach, Hofstadter explains that many mathematicians weren’t sure that Russell and Whitehead would be able to accomplish their task, so the mathematician David Hilbert set up a challenge to mathematicians to determine conclusively whether Russell and Whitehead were right or wrong.

In 1931, the Austrian mathematician Kurt Gödel demonstrated that Russell and Whitehead’s work was incorrect – or, if it was correct, it was incomplete.  This is the essence of Gödel’s Incompletness Theorem.  It says that no formal mathematical system can be both complete (“all-encompassing”) and logically consistent.

I was always curious as to why Russell felt that mathematics could be fully explained through logic, when it seems pretty clear that the world we live in is far more complex than the rules of logic seem to account for.

Logicomix explores Russell’s private life, both as a child and as an adult.  Russell’s family situation as a child was somewhat unsettled and he came from a family that had a history of madness.  It may be that Russell’s love of mathematics and craving for logical certainty was an understandable attempt to ground himself in a family situation that was somewhat unstable.

The scientific and mathematical results of this period of history are important milestones in our culture.  It was in the 1920’s that Werner Heisenberg, Niels Bohr and Albert Einstein were working through the implications of quantum physics and it was in 1935 that Schrödinger first conceived of his cat that was both alive and dead.

These ideas and conclusions put 20th century European and American science and philosophy on shaky ground, and mathematicians and physicists continue today to work through the implications of these ideas.

It is unclear to me whether or not it is possible to resolve the various paradoxes and seemingly nonsensical conclusions that arise out of this body of work.

I think that examining the scientists and mathematicians themselves as historical figures (as Logicomix does) can be helpful in creating a structure for inquiry into an area of knowledge that, almost by definition, lacks structure.


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The most basic form of mathematics is counting.

Most cultures count (although some don’t).  Thus, the simplest, most basic set of numbers is the counting numbers (1, 2, 3, ….) also known as the Natural Numbers, symbolized by a capital N.

An issue arises in algebra if we create an equation using only natural numbers whose solution is not a natural number –


This leads to the creation of the Integers, symbolized by a capital Z (for the German word for number – zahlen) which includes all the positive and negative whole numbers along with zero – (…, -3, -2, -1, 0, 1, 2, 3, …).

Again, a problem arises if we use integers to write an equation whose solution is not an integer.


This leads to the creation of the Rational Numbers (symbolized by a capital Q for quotient), which is the set of all numbers that can be represented as the ratio of two integers.  We can see a connection between the operations of multiplication and division and the set of Rational Numbers Q .

Again, we can create an equation using rational numbers whose solution is not a rational number.


This was actually a famous problem for the Pythagoreans, who believed that all numbers were rational and could be represented as the ratio of two whole numbers.  However, the diagonal of a square whose sides are length 1 will have a diagonal whose length is the square root of 2.  When the Pythagoreans realized that this number was not rational, it caused them great concern.

The existence of irrational numbers requires the existence of a set of numbers that includes both rational and irrational numbers – the Real Numbers.  This allows us to include such irrational numbers as square and cube roots as well old favorites like pi and relative newcomer e.

I think that the best way to think of the Real Numbers is on the number line.  Every Real Number corresponds to a point on the number line and every point on the number line corresponds to a real number.  In fact, even the technical mathematical definition of Real Numbers can be visualized on the number line.

Also, we can see a connection between the processes of finding roots and exponentiation to the need to broaden the conception of numbers to the Real Numbers.

The process of finding roots leads not only to the realm of the irrational numbers, but also to that of the Complex Numbers – symbolized with a capital C.

We’ll look at the complex numbers in the next post…

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