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From: ins_ampm@jhunix.UUCP (Michael P McKenna)
Newsgroups: net.puzzle
Subject: Re: truth machine clarification**2
Message-ID: <2131@jhunix.UUCP>
Date: Sat, 8-Mar-86 22:25:26 EST
Article-I.D.: jhunix.2131
Posted: Sat Mar  8 22:25:26 1986
Date-Received: Thu, 13-Mar-86 07:46:41 EST
References: <423@watdragon.UUCP> <2664@pucc-h> <394@link.UUCP> <2109@jhunix.UUCP> <616@bentley.UUCP>
Reply-To: ins_ampm@jhunix.ARPA (Michael P McKenna)
Organization: Johns Hopkins Univ. Computing Ctr.
Lines: 111

In article <616@bentley.UUCP> mdr@bentley.UUCP writes:
>
>>You mean countable.  If the set is countable there exists a 1 to 1
>>correspondence with the set of positive integers.  Thus each true
>>statement can be associated with a unique integer.  We need only
>>specify that the machine produces the statements in this order to
>>guarantee that any true statement is eventually produced.
>
>
>And it is obvious that the set of true sentences is countabe.  I need
>only concatenate together the ASCII codes for each character in the
>sentence to come up with a unique positive integer for that sentence.
>If the sentences are identical, the integers will be identical; if the
>integers are identical, the sentences must have been the same.  Thus the
>number of sentences (not even necessarily true) is countably infinite,
>as is the number of computer programs, books, musical scores (using 
>Western notation), or any other printed matter.
>

This is obvious if we take the traditional view that sentences are
finite in length.  What if we extend the idea of a sentence to
include such things as:

    The sentence, "The sentence, "The sentence, ...   is true." is true."
      is true.

or worse yet

     The sentence, "The sentence, "The sentence, ... is false." is false."
      is false.

or from Martin Gardner's article on Mobius Bands in _Mathematical
Magic Show_

     Once upon a time there was a story that began, "Once upon a time
       was a story that began, "Once upon a time..."

and a LONG poem...

     One day
     A mad metapoet
     With little to say
     Wrote a mad metapoem 
     That started:
     "One day
      A mad metapoet
      With little to say
      Wrote a mad metapoem
      That started:
      "One day
               .
               .
               .
       Sort of a close,"
      Were the words that the poet
      Finally chose
      To bring his mad poem
      To some 
      Sort of a close,"
     Were the words that the poet
     Finally chose
     To bring his mad poem
     To some
     Sort of close.

Actually we can use a variation of the mapping above to show that the
number of sentences similar to these is countable.  We have two types
of sentences here, sentences like the third example and sentences like
the other three.  

Sentences like the third example consist of a phrase that infinitely
repeated.  Simply construct the real number formed by placing the
appending the ASCII numbers of each letter after a decimal point.  
This forms a repeating decimal, which is of course a rational number.

The other sentences consist of two blocks, the first block is 
infinitely repeated THEN the second block is infinitely repeated.
We shall form a repeating decimal in this manner.  Start with a 
decimal point.  Append the ASCII code of the first character in the
first block, then the first character in the second block, then
the second character in the first block, second in second block, etc.
If one block is longer than the other just append all the characters
left when the shorter block is exhausted, repeat until bored.  This
is clearly a repeating decimal.  This method clearly extends to 
any finite number of blocks.

Note that you can't mix the mappings, each mapping proves that a certain
set of sentences is countable, therefore the union of these sets (so far
a countable number of sets), is countable.

What would be more interesting would be to see a sentence that maps to
an irrational number using one of the above methods.  This can clearly
be done by stringing random words together, but we really want some
sort of meaning in the sentences (at least as much as in the examples
above).  Another method would be to string variations of the phrase
"Once upon a time there was a story that began" together at random.
Ideally we want some rule for telling what the next phrase is.
(For example the number .101001000100001000001... is irrational and
the generation method is obvious).  Of course generating such
sentences does not prove the uncountability of the set of extended
sentences, for there may be some other mapping that would map
them to rational numbers.

Anyway this is digressing from the original puzzle and is getting quite
long so I'll quit.  It would be nice to see some "irrational sentences"
posted though.

                                                     Dwight S. Wilson