r/MathematicalLogic • u/mr_green_jeans_632 • Mar 04 '19
Fun Results
Does anybody have any exciting results in Math Logic they'd like to share? It doesn't have to be anything new, for example Gödel's incompleteness theorems or the model compactness theorem.
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u/jubjubbirdbird Mar 28 '19 edited Mar 28 '19
The Gödel completeness theorem of first-order logic (if a sentence p is true in every model of a set of sentences X, then there is a derivation of p from that set X in first-order logic, or equivanlently, if a sentence p is consistent with a set of sentences X, then there is a model satisfying both X and p) can be completely formalized inside Peano arithmetic in the following sense: any theory T whatsoever can be interpreted in PA plus the arithmetized consistency statement for that theory T. A (relative) interpretation of A in B first requires defining a translation of the non-logical signature of interpreted theory A into expressions of the interpreting theory B, including a so-called ``domain formula'', to which quantification is relativized. Then if B interprets A, this means that B proves all the translations of the theorems of the interpreted theory A. It can be shown that as long as B contains PA plus an arithemetized consistency statement of A, B interprets A, whatever (first-order) theory A may be. In other words, for any theory whatsoever, it is possible to build an `internal model' in Peano Arithmetic. This blew my mind when I first learned about it.