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"To Infinity and beyond!" is the motto of Buzz Lightyear, a cartoon character from the movie Toy Story. Itís humorous because it seems absurd. Infinity means endless and nobody can go beyond endless, because thereís no end to go beyond.
But sometimes in mathematics the seemingly absurd is actually real and thatís what many mathematicians have long thought about infinity. While he was working out the numerical consequences of set theory, Georg Cantor discovered the existence of infinities bigger than what we commonly think of as infinity, the size of the set of the natural numbers (the positive integers). He had discovered, so he thought, endlessnesses more endless than endlessness, which he designated with the Hebrew letter aleph (א). He used aleph-null (א0) to denote the common infinity, that of the natural numbers, which is the smallest of the infinities. He then offered a proof of the existence of a set with the size of aleph-one.
Mathematicians compare the relative sizes of sets by carrying out a one-to-one matching of the setsí respective elements: if one set has left-over elements when the match-up is finished, then it is larger than the other set. To prove and verify the existence of an infinity bigger than aleph-null, Cantor matched the set of the natural numbers with the set of the decimal fractions.
Cantor imagined making a double-entry list. On the left side he put all of the natural numbers and on the right side, in random order, he put all of the decimal fractions. He thus had each decimal fraction uniquely associated with a natural number, the basis for a one-to-one matching. But the match was not one-to-one, he asserted, because the natural numbers comprise a fixed set while the decimal fractions constitute an endlessly expandable set; that is, he could manufacture new decimal fractions, ones that were not originally in the set, endlessly, through a process called diagonalization.
To generate a new decimal fraction Cantor looked at the first decimal fraction on his list, noted which digit came first, took a different digit, and made it the first digit of his new fraction. Then he looked at the second digit in the second entry on his list, took a different digit, and made it the second digit of his new fraction. Going on down his list, Cantor took the n-th digit of the n-th fraction on the list, changed it, and made the new digit the n-th digit of his new fraction. Thus Cantor created, in concept, a new decimal fraction that was guaranteed to differ from every element in the original set in at least one place and, therefore, would exist as a new element on the list.
Because he had assumed a one-to-one match between the natural numbers and the original decimal fractions on his list, Cantor knew that his new fraction had no match. He could generate those unmatched fractions endlessly, so he knew that the full set of the decimal fractions had to be larger than the full set of the natural numbers, with a magnitude of aleph-one.
That proof fails in the light of a simple symmetry. If someone (letís call them Anyone Cansee) were to scramble the natural numbers on Cantorís list and then apply his diagonalization procedure to the result, they would obtain new positive integers that were not on the original list. They get essentially the same result that Cantor got with the decimal fractions. That fact seems to imply that the two sets have the same magnitude.
Anyone can prove and verify that implication easily. Pick a decimal fraction (say 0.52) and reflect it through the decimal point (obtaining 25.0). The relation between the fraction and the integer is perfectly unique. No other fraction reflected through the decimal point will yield the same natural number and there is no fraction to which we cannot apply that procedure. Likewise, Anyone can pick a natural number (say 1956) and reflect it through the decimal point (obtaining 0.6591). Again, no other integer will yield that fraction upon reflection and there is no natural number to which we cannot apply that procedure.
Those facts necessitate both sets having the same number of elements. If someone were to show us an unmatched element in one set, we need only reflect it through the decimal point to produce its match in the other set. Every one of the fractions that Cantor manufactures through his diagonalization procedure thus obtains its match in the natural numbers. Both sets, then, have precisely the same magnitude - aleph-null.
Now the real numbers come up. A real number consists of a natural number combined with a decimal fraction. We have an infinite set of decimal fractions that we can attach to each of an infinite set of natural numbers, so we calculate the magnitude of the set of the real numbers as infinity squared. Astoundingly enough, infinity squared equals infinity. Again, proof and verification of that statement comes easily.
Let Anyone pick a real number (such as 2550.3116) and reflect the fractional part through the decimal point, interleaving the digits among the digits of the integer part (obtaining 62151530 or 26515103, though we must be consistent in which pattern of interleaving we use). Only that particular natural number gets associated with the given real number through that procedure and there is no real number to which we cannot apply that procedure. Anyone can apply the procedure in reverse to a natural number (say 299792458) and get a real number (29948.5279 or 9725.84992) and there is no natural number to which Anyone cannot apply that procedure. Anyone must then conclude that the set of the real numbers has the same magnitude as does the set of the natural numbers - aleph-null.
Anyone conceives the geometric plane as an infinite set of points arrayed in two dimensions. That means that two real numbers identify each and every point, so Anyone has the plane represented as two infinite sets of real numbers combined. To determine whether that set has a larger magnitude than does the set of the natural numbers, Anyone takes the coordinates of one point (say x=324.175 and y=467.282) and interleaves the digits (obtaining 342647.127852). There is no pair of coordinates to which Anyone cannot apply that procedure and the results for the set contain no duplicates, so Anyone has, in concept, transformed the complete set of points in two dimensions into a set that contains all of the real numbers. As with the set of the real numbers, the magnitude of the set of the plane equals aleph-null.
Further, Anyone can apply that procedure repeatedly to the coordinates of the points comprising figures of any number of dimensions. Thus Anyone can transform any finite set of real numbers, representing multiple dimensions, into a single real number and can carry out that process endlessly. Anyone must conclude that it is not possible to construct a set of numbers with a magnitude of aleph-one. Only aleph-null exists.
The fact that only one infinity exists in mathematics might make Reality seem smaller, but even the small infinity of aleph-null is far vastly more enormous than our minds can comprehend. Mathematicians have barely begun to explore the implications of infinity and its properties. Consider one simple application to theoretical physics.
Physicists say that energy can be neither created nor annihilated. One way of rephrasing that statement says that in any isolated system the energy content of the system remains unchanged, regardless of any behavior of the system. The Universe is an isolated system, so its energy content must be unchanging and, therefore, definite.
Infinity is numerically indeterminate: it has no definite value. Therefore, a conserved quality, like energy, can only exist in a finite amount. But Anyone asks "finite relative to what standard?" Reality doesnít know joules or calories and yet it must have some basic unit of energy that would enable Anyone, in concept at least, to count up the total energy in the Universe and get a finite number. Consequently Anyone must infer the existence of a minimum, non-zero and non-infinitesimal, quantity of energy, the quantum that we associate with Max Planck. In that way Anyone can deduce the basis for the old quantum theory, which physicists used across the first quarter of the Twentieth Century.
Infinity has more to teach us. What else might mathematicians find when they explore its properties? Like infinity itself, the possibilities are endless.
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