[spam][crazy][fiction][random] Non-Canon MCBoss Spinoffs

[mailbombbin]

<blockquote>'''First Incompleteness Theorem''': "Any consistent formal
system {{mvar|F}} within which a certain amount of elementary
arithmetic can be carried out is incomplete; i.e., there are
statements of the language of {{mvar|F}} which can neither be proved
nor disproved in {{mvar|F}}." (Raatikainen 2020)<!-- this is a direct
quote from (Raatikainen 2020) --></blockquote>

== Boolos's short proof ==

[[George Boolos]] (1989) vastly simplified the proof of the First
Theorem, if one agrees that the [[theorem]] is equivalent to:
<blockquote>
"There is no [[algorithm]] {{math|''M''}} whose output contains all
true sentences of arithmetic and no false ones."
</blockquote>
"Arithmetic" refers to [[Peano axioms|Peano]] or [[Robinson
arithmetic]], but the proof invokes no specifics of either, tacitly
assuming that these systems allow '<' and '×' to have their usual
meanings. Boolos proves the theorem in about two pages. His proof
employs the language of [[first-order logic]], but invokes no facts
about the [[logical connective|connectives]] or [[Quantifier
(logic)|quantifier]]s. The [[domain of discourse]] is the [[natural
number]]s. The [[Gödel sentence]] builds on [[Berry's paradox]].

—
Boolos made a short proof, but it hinges in agreeing on a different
expression of the theorem.