Lambda the Ultimate

Incompleteness theorems prove that there are logically undecidable propositions, i.e., that there are propositions that are neither provable nor disprovable in certain classes of theories.

Incompleteness of Principia Mathematica was proved informally using proof by contradiction in a stratified metatheory by Gödel [1931] with restrictive conditions. Then Rosser [1936] informally proved incompleteness using proof by contradiction in a stratified metatheory assuming consistency of Principia Mathematica.

However, information on the modern Internet is pervasively inconsistent and restricting reasoning to use only stratified metatheories is impractical. Consequently Direct Logic has been developed which is inconsistency tolerant and does without stratified metatheories. And incompleteness has been formally self-proved for every theory of Direct Logic without requiring the hypothesis of consistency. Moreover, because incompleteness is self-proved, logically necessary inconsistency immediately follows.

Contents

• Gödel on Incompleteness of Principia Mathematica
• Wittgenstein versus Gödel
• Absolute incompleteness in inconsistency tolerant logic
• Absolute Inconsistency
• Logical Necessity of Inconsistency
• Incompleteness in set theory
• Bibliography