Richard's Paradox // 197

Constants to 50 Places

p 3.141 592 653 589 793 238 462 643 383 279 502 884

197 169 399 375 11

e 2.718 281 828 459 045 235 360 287 471 352 662 497

757 247 093 699 96

p

2 1.414 213 562 373 095 048 801 688 724 209 698 078

569 671 875 376 95

p

3 1.732 050 807 568 877 293 527 446 341 505 872 366

942 805 253 810 38

log 2 0.693 147 180 559 945 309 417 232 121 458 176 568

075 500 134 360 26

f 1.618 033 988 749 894 848 204 586 834 365 638 117

720 309 179 805 76

g 0.577 215 664 901 532 860 606 512 090 082 402 431

042 159 335 939 94

d 4.669 201 609 102 990 671 853 203 820 466 201 617

258 185 577 475 76

Here f is the golden number (page 96), g is Euler's constant (page 96), and d is the Feigenbaum constant, which is important in chaos theory (page 117). See en.wikipedia.org/wiki/Logistic_map

........................................... mathworld.wolfram.com/FeigenbaumConstant.html

Richard's Paradox In 1905 Jules Richard, a French logician, invented a very curious paradox. In the English language, some sentences define positive integers and others do not. For example `The year of the Declaration of Independence' defines 1776, whereas `The historical significance of the Declaration of Independence' does not define a number. So what about this sentence: `The smallest number that cannot be defined by a sentence in the English language containing fewer than 20 words.' Observe that what- 198 // Richard's Paradox

ever this number may be, we have just defined it using a sentence in the English language containing only 19 words. Oops.

A plausible way out is to say that the proposed sentence does not actually define a specific number. However, it ought to. The English language contains a finite number of words, so the number of sentences with fewer than 20 words is itself finite. Of course, many of these sentences make no sense, and many of those that do make sense don't define a positive integer ­ but that just means that we have fewer sentences to consider. Between them, they define a finite set of positive integers, and it is a standard theorem of mathematics that in such circumstances there is a unique smallest positive integer that is not in the set. So on the face of it, the sentence does define a specific positive integer.

But logically, it can't.

Possible ambiguities of definition such as `A number which when multiplied by zero gives zero' don't let us off the logical hook. If a sentence is ambiguous, then we rule it out, because an ambiguous sentence doesn't define anything. Is the troublesome sentence ambiguous, then? Uniqueness is not the issue: there can't be two distinct smallest-numbers-not-definable-(etc.), because one must be smaller than the other.

One possible escape route involves how we decide which sentences do or do not define a positive integer. For instance, if we go through them in some kind of order, excluding bad ones in turn, then the sentences that survive depend on the order in which they are considered. Suppose that two consecutive sentences are:

(1) The number in the next valid sentence plus one.

(2) The number in the previous valid sentence plus two.

These sentences cannot both be valid ­ they would then contradict each other. But once we have excluded one of them, the other one is valid, because it now refers to a different sentence altogether.

two page view?

all devices