Re: Contradiction or paradox

On May 7, 1:16 pm, Charlie-Boo <shymath...@xxxxxxxxx> wrote:
On Apr 19, 8:48 am, translogi <wilem...@xxxxxxxxxxxxxx> wrote:

Getting more and more confused.

What is the fine logical difference between a paradox and a
contradiction?

Can something be a contradiction without being a paradox?

Can something be a paradox without being a contradiction?

Or is a contradiction just a solvable "paradox" (there is a way out of
it)

Hope for a lot of discussion

Not much discussion is needed because the answer is simple. However,
surprisingly (paradoxically?) a lot of discussion ensues when the
solution is presented.

The Fundamental Proof of Paradox reaches an inconsistency (|-false)
from a system with 5 properties:

1. Consistency (CONS)
2. Completeness (COMP)
3. Self-Representation (SELF)
4. Negation (NEG)
5. Substitution (SUB)

Each of these often holds and is valuable. However, the "paradox" is
that we cannot achieve all 5. Thus Paradox is a set of conditions
such that:

1. Each is easy, useful and frequent.

2. Together they are impossible.

This is the fundamental definition of Paradox.

(I can show how to apply these 5 properties to any branch of Computer
Science: Set Theory, Theory of Computation, Recursion Theory,
Incompleteness in Logic (proof theory), Axiomatic Systems, Program
Systhesis, Paradoxes, etc.)

E.g. Program Synthesis is X # I / YES from which we derive the Program
Synthesis axioms and rules. Recursion Theory if X # X / YES, Set
Theory is X / SE etc.

C-B

How about prooftheory then and what do you mean (exactly ) with
1. Consistency (CONS)
2. Completeness (COMP)
3. Self-Representation (SELF)
4. Negation (NEG)
and
5. Substitution (SUB


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