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$$ \text{You gave a counterexample: true in N but not PA-provable.} \ \text{But there are so many examples with: true in N and PA-provable: } \ \text{I define: } \ \text{I_N} = {0,1,2, ...} \text{ is the set of intuitive natural numbers} \ \bar 0 ; and ; \bar 1 \text{ are the constants from PA} \ \oplus and \circ \text{ are the functions in PA} \

\text{when z is element from I_N, I define:} \ \bar z := \bar 1 \oplus ... \oplus + \bar 1 \quad \text{n-times} \

\text{I proofed the following statement "S1"} \ \text{When n is element from I_N then we have:} \ PA \vdash \forall x \forall y ; (\bar z = x \oplus y \quad .\rightarrow. \quad \bar x=\bar 0 \land y=\bar z \lor x=\bar 1 \land y=\overline{z-1} \ ;\lor ... \lor ; x=\bar z \land y=\bar 0) \ \text{I assume, that there are "many" of "similar" statements like S1.} \ \text{Is it possible to formulate a statement "S", so that "S1" follows from "S" ?} \ $$

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