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Your proof strategy via (1) and (2) is impossible. If $PA\cup\Sigma$ proves that Goodstein's theorem is false, then the proof will have finite length, and so there will be some finite $\Sigma_0\subset\Sigma$ such that $PA\cup\Sigma_0$ proves that Goodstein's theorem is false. This would imply by (1) that Goodstein's theorem is false in the standard model. But Goodstein's theorem holds in the standard model, as Goodstein proved.