Paulo Bordoni
September 18, 2014
800

# Números reais: procurando entendender

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## Paulo Bordoni

September 18, 2014

## Transcript

1. ### History topic: The real numbers: Attempts to understand Epple writes

in [2]:- What was a real number at the end of the 19th century? An intuitive, geometrical or physical quantity, or a ratio of such quantities? An aggregate of things identical in thought? A creation of the human mind? An arbitrary sign subjected to certain rules? A purely logical concept? Nobody was able to decide this with certainty. Only one thing was beyond doubt: there was no consensus of any kind. Were the real numbers consistent? Would an inconsistency appear one day and much of the mathematical building come tumbling down? Some of the intuitive difficulties that began to be felt revolved around the fact that the real numbers were not countable, that is, they could not be put in 1-1 correspondence with the natural numbers. Cantor proved that the real numbers were not countable in 1874. He produced his famous "diagonal argument" in 1890 which gave a second, more striking, proof that the real numbers were not countable. To do this he assumed that the real numbers were countable, that is they could be listed in order. Suppose that this list is L = {n1 , n2 , n3 , n4 , ... } and let d(i, j) be the i-th digit of nj . Define the real number r to have k-th digit 1 if d(k, k) = 2 and r to have k-th digit 2 if d(k, k) ≠ 2. Then the real number r is not in the list L since if it were then it would be nt for some t. But the t-th digit of r differs from the t-th digit of nt by construction, so we have a contradiction. Hence the real numbers are not countable. We'll construct a certain real number which, although historically not one that was looked at, will let us understand some of the questions that arose. Let us start with the 100 two digit numbers. A simple code will let us translate these into letters, 00 become a, 01 become b, ... , 25 becomes z, 26 becomes A, 27 becomes B, ... , 51 becomes Z, then code all the punctuation marks, and then make all the remaining numbers up to 99 translate to an empty space. Now create a number, say c, starting from the 100 2-blocks. c = 0.01020304050607080910111213141516171819202122232425... Then continue with the 10000 pairs of 2-blocks 0000, 0001, 0002, ..., 0099, 0100, 0101, ...