Undecidable

A statement in formal logic is called undecidable if there is no proof or disproof of the statement in formal logic. A common misconception is that undecidable statements have no truth value, but this statement is not true. For example, many set theorists now believe that the continuum hypothesis (which is known to be undecidable in Zermelo-Fraenkel set theory) is actually false.

As a more concrete example, suppose we had a "theory of colored shapes" Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} where the objects were colored shapes (red triangles, blue squares, etc.), and the possible atomic sentences were of the form "shape Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} is a square." Since we do not have the language to state "shape Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} is red" then any statement of this form is undecidable in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} , though it could be true or false in some larger theory Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T'} .

Undecidability and Faith

The existence of undecidable statements proves the illogic of atheistic attempts to demand proof for the existence of God. Because we know that there exist true statements that we can never logically prove, there are necessarily things that must be believed on the basis of faith, rather than logic. The atheistic mantra that "the burden of proof lies with the believer" ignores this classic result in mathematical logic, and exposes their ignorance. It is notable that Kurt Gödel who demonstrated the existence of undecidability in mathematical logic was a devoutly religious man.

Famous Undecidable Statements

• Axiom of Choice
• Banach-Tarski paradox
• Continuum hypothesis
• Existence of large cardinal numbers
• Halting problem
• König's lemma
• Lefschetz principle
• Liar's paradox
• Ramsey theory involving infinite sets
• Russell's paradox