What Does He Mean?
Penrose's discussion of what he calls the "Dedekind Cut" in Chapter 2 of Road is tantalizing but unfortunately brief to the point of being almost impossible for me to understand. I was particularly confused by his reference to how irrational numbers such as the square root of 2 are defined by Dedekind cuts when "those to the left have no actual largest number and those to the right, no actual smallest one" (3.3). Huh?
But with some reference to various pages on Wikipedia (which I fear I'll be making much use of as I slog through this tome) I believe I have pieced together what Penrose is talking about. And it is quite interesting, and worth working through here.
Basically, what Penrose seems to be talking about is defining irrational numbers as the "gaps" between rational numbers on the number line. Let's define Q as the set of all rational (not real!) numbers. We then "cut" Q into subsets A and B. The trick is to define the point γ which divides A from B in such a way that γ belongs to neither subset. Since A and B are subsets of Q and hence also comprised of rational numbers, but γ is not a member of A or B, γ cannot be a rational number.
Suppose for example that you wanted to use such a cut to define the square root of 2. You would cut Q as follows:
The notation is stolen from Wikipedia: the vee-shaped thingy means logical OR, and the wedge-shaped thingy means logical AND. So A is the set of all rationals a in Q whose square is less than 2 OR all rationals less than or equal to 0. Similarly, B is the set of all rationals b in Q whose square is greater than or equal to 2 AND all rationals greater than 0.
Well what does this all add up to. It is pretty obvious that all a must be less than γ, since we are either talking about negative numbers or positive numbers (including 0) whose square is less than 2. It is not quite so obvious that all b must also be greater than γ. This is because the square root of 2 is not a rational number and hence cannot be a member of B. All members of B must therefore have squares greater than 2.
Looking at this we start to get some idea of what Penrose means by his mysterious references to "no actual largest/smallest numbers" in the two sides of the cut. However I think it really helps to get a bit more precise in this case so that we can understand what is and is not being discussed. Apparently Penrose is referring to the concepts of a supremum and infimum of a set. You can define the supremum S (my symbol) of a set T of real numbers to be the smallest real number greater than or equal to every member of T; similarly, you can define the infimum I (again my symbol) of T to be the largest real number less than or equal to ever member of T. Examples:
We can now state much more concisely (and precisely, and clearly) that while γ is not a member of A or B, γ = S{A} and γ = I{B}. The supremum and infimum for A and B respectively fall outside A and B: this is what Penrose is saying, I think (read: I hope).
Additional Thoughts
Once I managed to piece all of this together in a reasonably coherent way I skimmed a page in Wikipedia concerning the relationship of suprema and infima to the completeness or lack thereof of sets. I didn't really read closely but it was indeed interesting that it is possible to define subsets of rational numbers in a way that appears to violate the completeness requirements (since there are suprema and infima that fall outside these subsets, as we have seen). So the irrationals arise almost as a necessity (?) in order to complete the set of rationals and form the larger (complete?) set of all real numbers. Assuming that it even makes sense to talk about irrationals as a "necessary" complement to the rationals, one wonders whether there are other ways to cut the reals that "necessitates" complex numbers. I dunno. *shrug*
No comments:
Post a Comment