Deep Thoughts # 17: Liar Paradox is a Formal Metamathematical Theorem

From: Charlie-Boo (chvol_at_aol.com)
Date: 10/05/04 

Date: 5 Oct 2004 05:21:02 -0700

The Liar Paradox is the following metamathematical theorem applied to
English:

Theorem 1 ("This is not true."): No system can enjoy all of the
following properties:
1. The provability predicate is expressible.
2. Negation is expressible.
3. Substitution is expressible. (S-M-N theorem)
4. The system (every wff) is consistent.
5. The system (every wff) is decidable.

This metamathematical theorem applies to any system, e.g. English,
Logic, Turing Machines and Set Theory. In English we include # 1, 2,
3 and 4. Therefore we forfeit # 5. "This is not true." is a wff that
if not decidable - it is neither true nor false.

1: The sentence "It is true." expresses the provability predicate,
which in English is the truth predicate.
2: Adding a trailing "not" maps a sentence that express a predicate to
a sentence that expresses its negation.
3: The word "of" expresses substitution.
4: We assume that if a sentence is true it cannot be false, and if it
is false it cannot be true (the equivalent contrapositive.)

where,

1. The inputs are represented by pronouns.
2. Constants are represented by noun phrases.
3. Predicates are represented by adjectives.
4. The verb to be joins predicates and their arguments.
5. The recursion theory combinator is substitution of "this" for "it".
 That is, for any English sentence that expresses a particular
predicate, substituting "this" for "it" creates a sentence N that
expresses that predicate applied to N.

(Applied to Turing Machines, Theorem 1 concludes that the set of TM
that don't halt yes on themselves is not recursively enumerable.
Applied to Set Theory: There is no set of those sets that don't
contain themselves. Applied to Logic: No wff expresses the assertion
that a given number is the Godel number of a wff that when applied to
its number produces an unprovable sentence.)

Theorem 2 ("This is false."): No system can enjoy all of the following
properties:

1. A particular set S is expressible.
2. S is the negation of the provability predicate.
3. The system (every wff) is consistent.
4. The system (every wff) is decidable.

The exception to # 4 generated by the English recursion theory
combinator is "This is false.".

1: S is the set of false sentences, which is expressed by "It is
false."
2: A sentence is defined to be false iff it is not true.
3: As before, we assume that if a sentence is true it cannot be false,
and if it is false it cannot be true (the equivalent contrapositive.)

(Applied to Turing Machines, Theorem 2 concludes that, since the set
of TM that halt no on themselves is recursively enumerable, then some
program must not halt.)

Applying these theorems to the other systems (and identifying the
resulting specific theorem if it has been published before) as well as
generating other theorems, is left as an exercise.

C-B

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