Repercussions from the Richard Paradox: Math Rules

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The Cantor Diagonal Originally “Counted” Real Numbers

Georg Cantor image via Wikimedia Commons

Georg Cantor image via Wikimedia Commons

In 1874, Georg Cantor had developed his diagonal method to demonstrate that the size of the set of Real numbers, its “cardinality”, was larger than the cardinality of the Natural numbers. While the sets of Natural or Rational numbers are “countably infinite”, the Real numbers are “uncountably infinite”.

Cantor’s work proved a claim made by Galileo about the number of points in a Real line segment, as previously reported.

Two Clever Tricks in Richard’s Paradox

Richard’s major contribution that he avoided ordering his subset of real numbers, ‘E’, by their numeric value. By using a different ordering based on the descriptions, he avoided an already “raging” argument about the continuity and ordering of the Real numbers.

Richard’s paradox also avoided the possibility of generating any new paradoxical number ‘f’ that would end in all zeroes or nines. The need for this is illustrated by the fact that 0.999… = 1.0 exactly.

The proof is: let “x = 0.999…”. so “10*x = 9.999…”. Then “10*x – x = 9*x = 9.999… – 0.999… = 9.0”. Since “9*x = 9.0”, it follows that “x = 9/9 = 1”.

Richardian Numbers are Special Paradoxical Positive Integers

A similar blend of natual language and mathematics confines itself to positive integers but results in the same paradox.

Let ‘M’ be the set of sets of positive integers and described by finite strings of English words. For example, the definition “one” would correspond to the number 1; but “divisible by five” would be the set {5, 10, 15…}.

As in Richard’s Paradox for Real numbers, this is an infinite set but it may be ordered by the descriptive strings. Therefore each of these descriptions has an index number, based on alphabetical order within the length of the string.

We classify each index number in ‘M’ as “Richardian” if that index number has the property that is being described. For example, if “divisible by five” is the 25th description, then it is indeed “Richardian”. Since “five” comes before “four” alphabetically, at least one of them is not Richardian.

Create the paradox by including the definition “all these positive natural numbers are not Richardian”. This definition will be number ‘m’ in the ordered set ‘M’. If we include ‘m’ in this set of non-Richardian numbers, then it has the property of being “Richardian”. Therefore we remove ‘m’ from this set. Now that ‘m’ is no longer listed in the mth set, it is not Richardian. Therefore it should be included.

Once again we have a vicious circle paradox.

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