Re: Contradiction or paradox

On May 9, 9:39 am, translogi <wilem...@xxxxxxxxxxxxxx> wrote:
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- Hide quoted text -

Good choice. As this is a metamathematical result we will need a
Computationally Based Logic to express and prove the necessary facts,
as opposed to Propositionally Based Logics (e.g. Predicate Calculus)
which are unsuitable for formalizing metamathematics. (Nonetheless,
mathematicians routinely make the mistake of using Predicate Calculus
in an attempt to formalize Set Theory, a branch of metamathematics,
creating a horrible mess like ZFC for a very simple concept, the
notion of a set.)

Have you used a Computationally Based Logic before?

I use CBL, which is somewhat of a de facto standard for
Computationally Based Logics. The primitive assertion "mathematical
object M in system Q calculates mathematical process P" is formalized
as the expression M#P/Q. In the simplest case, P is a one-place
relation and Q is a two-place relation. This means that P =Q(M) that
is for all (a): P(a) = Q(M,a). Also P/Q means that there is an M such
that M#P/Q.

For example, if Q(a,b) iff Program a halts yes on input b, then P/Q
means that P is recursively enumerable. Do you see how to define Q to
make the assertion that P is expressible in some particular Logic?
That it is representable? That there is a set corresponding to P?

(Already we see the power of CBL. These primitive, useful notions -
r.e., expressible, representable, etc. - are not even formally
represented and manipulated in the published literature, while in CBL
they are all very simple expressions in a general system that
formalizes all of them.)

C-B

- Show quoted text -


.

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