<text><span class="style10">orrespondence, Counting and Infinity (6 of 6)</span><span class="style7"></span><span class="style10">Different infinities</span><span class="style7">One remarkable application of this definition of an infinite set shows that there are no more </span><span class="style26">rational numbers</span><span class="style7"> than there are natural numbers. Again, we look for a one-to-one correspondence between the two sets. There are many possible mappings, but the simplest is by first counting the fractions whose numerator and denominator add up to 1, then those for which the total is 2, then those totaling 3, and so on. Within each group, the fractions are put in order of size and any fraction that equals a previously listed fraction is omitted. This results in the rationals being listed in a unique order: Obviously, we can calculate both which natural numbers any rational corresponds to, and which rational corresponds to any given natural number, and this is sufficient to prove that there are no more rationals than there are natural numbers: It would be wrong, however, to conclude from all this that all infinite sets are equivalent. In fact, it is possible to prove that </span><span class="style26">real numbers</span><span class="style7"> (the rationals and irrationals together) cannot be counted off in the way that we counted off natural numbers and rationals. This gave rise, at the end of the 19th century, to new studies of what was called </span><span class="style26">transfinite arithmetic</span><span class="style7"> (</span><span class="style26">transfinite</span><span class="style7"> means 'extending beyond the finite'), and attempts were made to prove that the cardinality of the reals is the next transfinite cardinal after that of the integers. In 1963 it was finally shown that this conjecture can neither be proved nor disproved; this is a 'gap' in mathematics itself.</span><span class="style10">Intuitionism and infinity</span><span class="style7">The various paradoxes and discoveries about infinity caused much attention to be paid in the late 19th century to the philosophical and logical foundations of mathematics.One school of thought, called </span><span class="style26">intuitionism</span><span class="style7">, blamed the contradictions that be came apparent on the assumption that it is possible to complete infinite processes. Intuitionists resolved to ban infinity from mathematics, with the result that statements have to be allowed to be neither true nor false.As an example, let us consider . Suppose we define N to be 0 if the sequence 0123456789 occurs in the decimal part of , and to be 1 if this sequence never occurs. A </span><span class="style26">classical</span><span class="style7"> mathematician will be prepared to assert that 'N is either 0 or 1' is true, even though, at the time, he or she has no way of knowing it is. An intuitionist claims that, until we are able to prove one result or the other, the statement cannot be said to be either true or false. Arithmetic, however, requires that there is no largest integer. The intuitionist resolves this by looking at things in a different way from the classicist. For example, the classicist might think of a distance of 2 units as actually made up of infinitely many steps of length 1 unit, then 1/2 unit, then 1/4 unit, then 1/8 unit, etc. The intuitionist, however, would interpret the result that the infinite series of 1 + 1/2 + 1/4 + 1/8 . . . + 1/2n + . . . has a sum (in fact 2) to mean that the total of finitely many terms in order can be made as close as we like to 2 by taking sufficiently many - but still finitely many - terms of the series.EJB</span><span class="style10">HILBERT'S INFINITE HOTEL AND TRISTRAM SHANDY</span><span class="style7"> The German mathematician David Hilbert, who was probably the most eminent mathematician of his generation, dramatized the paradoxical property of infinite sets by an exercise of the imagination.Imagine a hotel with an infinite number of rooms; then it can be full and yet able to accommodate more guests. The manager simply moves the guest in room 1 into room 2, the guest in room 2 into room 3, and so on. Each guest is then in a room with a number higher than before, so that room 1 remains vacant for a late arrival.Unfortunately, not one latecomer, but an infinite busload of them arrive. Instead of the moves as before, the manager puts the guest in room 1 into room 2, the guest from room 2 into room 4, the guest from room 3 into room 6, and so on. The infinite number of rooms with odd numbers are now vacant for the infinite number of latecomers.The novelist Laurence Sterne constructed a similar paradox in his novel </span><span class="style26">Tristram Shandy</span><span class="style7">. The hero is trying to write his autobiography. Since, after two years, he has only described the first two days of his life, he concludes that his efforts are doomed to failure.However, Bertrand Russell remarked that this would not follow if Tristram Shandy were immortal. Then the infinite set of the days of his life would be the same size as the 365-times larger set of days needed to describe the 'smaller' set.</span><span class="style10"> CANTOR'S DIAGONAL THEOREM</span><span class="style7"> The German mathematician Georg Cantor was responsible for the development of the whole theory of cardinality outlined in the text.He proved that every set has a </span><span class="style26">power set </span><span class="style7">(set of all its subsets) that is strictly bigger than the given set; that is, the power set cannot be put in one-to-one correspondence with the given set - even in the case of an infinite set. The proof is easiest to understand in terms of the real numbers, by assuming that such a correspondence is possible and then showing that this assumption cannot be true since it leads to a contradiction.We consider the real numbers between 0 and 1, expressed as decimals (0.47936421 ... is an example), so that each number has an infinite number of digits after the decimal point. Where decimals terminate, we continue the number with zeros. Suppose real numbers can be listed in order, that is put into one-to-one correspondence with the natural numbers. We could then write all the real numbers in this form: 0. a1 a2 a3 a4 ...0. b1 b2 b3 b4 ...0. c1 c2 c3 c4 ...and so on.Now, we try to construct a new number. We make the first digit after the decimal point differ from a1, the second digit differ from b2, the third digit differ from c3, and so on. Thus, we have constructed a new real number between 0 and 1, but we have constructed it so that it differs from every member of the list of real numbers that we began by supposing was complete: we have derived our contradiction. It follows, therefore, that such a list of the real numbers is not possible.However, the construction required by Cantor's proof requires the completion of an infinite number of steps, and intuitionists therefore claim that the new number is ill-defined; they therefore reject 'Cantor's paradise' of different infinities.</span></text>
</content>
<content>
<layer>background</layer>
<id>23</id>
<text>ΓÇó NUMBER SYSTEMS AND ALGEBRAΓÇó SETS AND PARADOXESΓÇó FUNCTIONS, GRAPHS AND CHANGE</text>
