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I was reading the Wikipedia post on Emil Post(no pun intended), I quote:

"In his doctoral thesis, later shortened and published as "Introduction to a General Theory of Elementary Propositions" (1921), Post proved, among other things, that the propositional calculus of Principia Mathematica was complete: all tautologies are theorems, given the Principia axioms and the rules of substitution and modus ponens. Post also devised truth tables independently of Ludwig Wittgenstein and C. S. Peirce and put them to good mathematical use. Jean van Heijenoort's well-known source book on mathematical logic (1966) reprinted Post's classic 1921 article setting out these results."

and then the next paragraph is:

"While at Princeton, Post came very close to discovering the incompleteness of Principia Mathematica, which Kurt Gödel proved in 1931."

Is Principia complete or not?

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  • $\begingroup$ It is incomplete. Any system complex enough to include arithmetic will be underdetermined--there will always be unprovable propositions, and proving such will require adding axioms. $\endgroup$
    – RobertTheTutor
    May 6, 2021 at 20:11
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    $\begingroup$ Principia contains more than the propositional calculus. Post proved the propositional calculus is complete, not all of Principia. $\endgroup$
    – Thomas Andrews
    May 6, 2021 at 20:16
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    $\begingroup$ Principia is propositional calculus + predicate logic (of any order) + type theory ("equivalent" to modern set theory). Post proved completeness of prop calculus; Gödel completeness of (first-order) predicate logic, Gödel proved incompleteness of the logic of Principia plus Peano axioms for arithmetic, Leon Henkin proved the completeness of theory of types. $\endgroup$
    – Mauro ALLEGRANZA
    May 7, 2021 at 5:58

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Principia contains a lot more than the propositional calculus.

Propositional calculus is the language of logic alone, with no meaning ascribed to elementary statements. In particular, there is nothing about numbers in propositional calculus.

In propositional calculus, we can say things like “If $p$ and $p\implies q$ are true, then $q$ is true.” It lets us deduce thing logically.

But substituting actual sentences about numbers for $p$ and $q$ is the rest of Principia.

Post proved the propositional calculus in Principia is complete, not all of Principia.

Gödel proved the whole Principia is incomplete.

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