Georg Cantor
Georg Cantor defined infinity in two steps. First he defined an infinite set and then "infinity":
- an infinite set is one that can be put in one-to-one correspondence with a proper subset of itself. E.g. natural numbers can be put in a one-to-one correspondence with the set of even numbers, which is a proper subset of the natural numbers. Here is the correspondence {0, 1, 2, 3, ... } maps to {0, 2, 4, 6, ...}.
- The cardinality of such a set is "infinity".
Cantor went a step ahead and defined a family of "infinities", each larger than the previous one.
- Cantor proved that even an infinite set cannot be put in one-to-one correspondence with its power set (i.e. the set of all its subsets), whose cardinality is 2^n (if the cardinality of the original set is n).
- This defines a family of "transfinite" cardinalities, each larger than the one before:
- aleph-0 (the cardinality of natural numbers),
- aleph-1 = 2^aleph-0,
- aleph-2 = 2^aleph-1
- and so on.
- Real numbers cannot be put in one-to-one correspondence with natural numbers. (See Cantor's diagonal argument). The cardinality of real numbers is called the cardinality of the continuum, c.
- Cantor hypothesized that c = aleph-1, which became known as the "Continuum Hypothesis". In other words, there is no transfinite cardinality between aleph-0 (cardinality of natural numbers) and c (cardinality of real numbers). But Cantor could not prove it. It turns out that there is a very good reason for that!
- By the way, Cantor called natural numbers or any subset thereof as countable, since they can be counted, i.e. put in one-to-one correspondence with 0, 1, 2, 3, ... . He called real numbers and higher transfinite cardinalities as uncountable, because they cannot be put in one-to-one correspondence with 0, 1, 2, 3, ... . (See Cantor's diagonal argument).
Kurt Godel
Kurt Godel came along and gave the world two "incompleteness" theorems.
- Informally, the first incompleteness theorem says that in any axiomatic system involving natural numbers, there are statements that cannot be proved or disproved within that system. In other words, there are some "undecidable" statements within the system.
- Stated differently, we can never come up with a finite set of axioms that can prove or disprove every statement in that system. Hence the system is always "incomplete".
- The intuition behind this is that there are only countably many provable statements, but uncountably many statements in the system. By the "pigeon hole" principle, some statements are unprovable. In fact, a vast majority of the statements are unprovable, since "uncountable" is much, much larger than "countable"!
- The second incompleteness theorem says that for some axiomatic systems, consistency cannot be proved within the system itself.
Axiom of Choice
- Simply stated, the axiom of choice says that given a bunch of non-empty sets, it is always possible to choose one element from each set to construct a new set.
- It seems logical and harmless. But complications arise when the original set has a large cardinality, e.g. aleph-1. How do we go about choosing one element from uncountably many sets? Where do we start - it cannot be put in one-to-one correspondence with natural numbers and hence cannot be labeled 1, 2, 3, ..., etc. So we wouldn't know which set is the first one, which is second and so on.
- One way to go about this complication is to just assume that there exists such a choice set and circumvent the above complexity.
- Not everyone agrees! So there are two different axiomatic set theories - one without the axiom of choice (ZF for Zermelo-Frankl, the formulators of set theory) and another with the axiom of choice (ZFC).
- Godel proved that the Axiom of Choice was consistent with ZF. In other words, ZFC is consistent.
The Finale
- Godel also proved that the continuum hypothesis was consistent with axiomatic set theory. In other words, it cannot be disproved within ZF.
- Paul Cohen comes along and proves that continuum hypothesis is independent of the other axioms in ZF. In other words, neither the continuum hypothesis nor its opposite can be proved within ZF. No wonder Cantor couldn't prove the "Continuum Hypothesis" from ZF axioms!
- Well, what does it all mean? Continuum hypothesis must either be true or be false, since aleph-1 must be either equal to c or not equal to c. The bottom line is that the result by Cohen and Godel say that neither the equality nor the inequality can be proved within the axiomatic system, no matter how smart you are or how hard you try!
- I, for one, believe that aleph-1 is indeed equal to c. In other words, there are no cardinalities in between that of the natural numbers and that of the real numbers.
References
- Leonard M. Wapner, "The Pea and the Sun", 2005.
