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 GÃ¶del [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.
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