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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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