Re: Where is the paradox in liar?

Newberry wrote:

Let's consider liar's paradox:

This sentence is false. (1)

At this point we know that (1) is not true and (1) is not false. Where
is the contradiction?

There is one only if you assume every sentence is true or false. It is
a program in an infinite loop. Like Godel's wff that is neither
provable nor refutable. See:

http://www.cs.nyu.edu/pipermail/fom/2002-November/006031.html

Also note that all of the theorems of Godel, Rosser, Turing et. al. are
simple conclusions when you express each theorem as the logical
consequence of certain relations being representable or not in the
logic itself.

PR = the set of provable sentences
REF = the set of refutable sentences
PRIT = the relation "X is a proof of Y"
Px means P is representable in the Logic.
-Px means P is not representable in the Logic.

(Unfortunately established Mathematical formalisms do not include a
symbolism for Px or -Px.)

-~PRx => Godel's 1st. Incompleteness Theorem based on Soundness
~PRITx => Godel's 1st. Incompleteness Theorem based on w-consistency
-~PRx , PRx => Rosser's Extension to Godel's Theorems
REF , ~(PR v REF) => Turing's Unsolvability of the Halting Problem

C-B

.