nonri.xyz // 논리 / paradox engine

§ 01 / AXIOMS

THE GROUND BREAKS.

We accept five things without proof. From those five, everything follows. From those five, contradictions also follow. The choice of axioms is arbitrary—the consequences are not.

A1Identity. x = x. Even when x lies.

A2Excluded middle. P ∨ ¬P. There is no third option, except when there is.

A3Non-contradiction. ¬(P ∧ ¬P). Sometimes violated for art.

A4Modus ponens. If P then Q. P. Therefore Q. Therefore everything.

A5Choice. Pick one from each set. The set may be empty.

§ 02 / GÖDEL

THIS SENTENCE IS UNPROVABLE.

Any system rich enough to count is rich enough to break itself. Gödel encoded a sentence that says only "I cannot be proved"—and was right. Every formal system has truths it cannot reach. Logic, looking in the mirror, finds itself incomplete.

G ≡ ¬Provable(⌜G⌝)

if G is provable      → contradiction
if G is unprovable    → G is true (and unprovable)

∴ truth ⊋ proof

§ 03 / HALTING

DOES THE PROGRAM STOP?

Suppose H decides if any program halts. Build D(x) = if H(x,x) then loop else stop. Now run D(D). If D halts, it loops. If it loops, it halts. The decider cannot exist. Some questions have no algorithm.

function D(x):

if H(x, x) then

while true: // loop forever

else

return 0 // stop

D(D) // what does this return?

// EXCEPTION: undecidable

§ 04 / SPECIMENS

SPECIMENS OF FAILURE.

A short collection of statements that should not exist, but do.

P-001

Liar

"This statement is false."

P-002

Russell

The set of all sets that don't contain themselves. Does it contain itself?

P-003

Berry

"The smallest number not nameable in fewer than fourteen words." (named in thirteen)

P-004

Sorites

One grain is not a heap. Add one grain. Still not a heap. Repeat. When?

P-005

Curry

"If this sentence is true, then everything is true." It is true.

P-006

Grelling

Heterological = does not describe itself. Is "heterological" heterological?

§ 05 / ORACLE

ASK THE PARADOX.

Type any yes-or-no question. The oracle returns one of: TRUE, FALSE, or BOTH. You will not be able to tell which is correct. That is the point.

>

// awaiting input

§ 06 / GEOMETRY

SHAPES THAT CANNOT EXIST.

Drawn in two dimensions, refusing to live in three.

PENROSE TRIANGLE

NECKER / IMPOSSIBLE

MÖBIUS LOOP

THIS PAGE IS FALSE.

THIS PAGE IS TRUE.