Ask Dr. Silverman 7 — Infinity: 1, 2, 3, 4… Until, You Can’t Count Anymore
Herb Silverman is the Founder of the Secular Coalition of America, the Founder of the Secular Humanists of the Lowcountry, and the Founder of the Atheist/Humanist Alliance student group at the College of Charleston. Here we talk about infinity.
Scott Douglas Jacobsen: What makes infinity, infinite? What makes the finite, finite? Does a middleground exist, at all, between the infinite and the finite? Common sense says, “No.” However, mathematical conclusions can produce anti- or non-intuitive results through basic logical implications of the formulations of the mathematics and the conceptualizations behind them.
Professor Herb Silverman: There is no middle ground between the finite and the infinite, but there are some strange, non-intuitive things we can say about infinity, some of which I’ll describe.
In mathematics, a set (of numbers) is said to be infinite if it can be put into a one-to-one correspondence with a proper subset (not all the numbers) of itself. For instance, the positive integers (1, 2, 3, 4, …) form an infinite set because there is a one-to-one correspondence between them and the proper subset of positive even integers (1~2, 2~ 4, 3~6, 4~8, 5~10, etc.).
A set of numbers that is not infinite is called finite. There is no one-to-one correspondence between the set of numbers {1, 3, 7, 13, and 18} and a proper subset of itself. We say that the cardinality of this set is 5 because it has 5 elements, the same as the cardinality of {10, 20, 30, 40, 50}.
Since there is a one-to-one correspondence between the positive integers and the positive even integers, we say there are as many even integers as integers (the same cardinality). The question is whether there are infinite sets with larger cardinality than the integers. What about the rational numbers (fractions)? As it turns out, there are no more rational numbers than integers. On the other hand, not all infinities are equal. There can be no one-to-one correspondence between the integers and the real numbers (rational and irrational). An irrational number, like pi and the square root of two, is a real number that is not rational.
Furthermore, we say that an infinite set is countable if there is a one-to-one correspondence between the set and a subset of the positive integers. If not, we say the infinite set is uncountable. So the rational numbers form a countably infinite set, while the real numbers are uncountable. (The numbers between 0 and 1 are uncountable, as well as the numbers between any other two numbers.) This means that the cardinality of the real numbers is larger than the cardinality of the integers. Are there larger uncountable sets than the real numbers? The German mathematician Georg Cantor showed that there are infinitely many cardinalities larger than the cardinality of the set of real numbers because the set of subsets of a set has a greater cardinality than the set. Stated another way, there are infinitely many different kinds of infinities.
There is also the concept of infinitely small, which involves limits (the basis of calculus). For instance, it’s easy to see that there is no smallest positive number because we can divide any number by 2 and get a smaller positive number. The sequence {1/n}, (n = 1, 2, 3, 4, …), comes arbitrarily close to 0 (and has 0 as the limit) but never reaches 0.
There is still a lot that is unknown in mathematics about infinity. For instance, a prime number is divisible only by 1 and itself. The first few are 2, 3, 5, 7, 11, and 13. Euclid proved that there are infinitely many prime numbers. What mathematicians have not yet been able to solve is whether there are infinitely many twin primes (two consecutive odd primes like (3,5), (5,7), (11,13), (17,19) …). If you can solve this, you will become very famous in the math world.
There are simpler non-intuitive examples that deal with the infinite. For example, did you know that 0.99999999… = 1, where the dots mean you continue the 9’s forever?
You learned to accept in elementary school that 0.33333333333 = 1/3, so just multiply this equation by 3 to obtain the previous equation.
This is all based on properties of infinite series, which I won’t get into here. On the other hand, I’ll let you know that the infinite series 1/2 + 1/4 + 1/8 + 1/16 + …. = 1, while
1/2 + 1/3 + 1/4 + 1/5 + 1/6 + …. = infinity.
Any other questions?
Jacobsen: Yes. Thank you for the opportunity and your time, Professor Silverman.
