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As an example of theorems like part 1 in the questioninteractions between number theoretic conjectures and model theoretic problems, Macintyre and Wilkie proved that if the Schanuel's conjecture is true then the theory of real field with exponentiation ($R_{exp}$) is decidable.

Macintyre, A. & Wilkie, A. J. (1996). "On the decidability of the real exponential field". In Odifreddi, Piergiorgio. Kreiseliana: About and Around Georg Kreisel. Wellesley: Peters. pp. 441–467.

As an example of theorems like part 1 in the question, Macintyre and Wilkie proved that if the Schanuel's conjecture is true then the theory of real field with exponentiation ($R_{exp}$) is decidable.

Macintyre, A. & Wilkie, A. J. (1996). "On the decidability of the real exponential field". In Odifreddi, Piergiorgio. Kreiseliana: About and Around Georg Kreisel. Wellesley: Peters. pp. 441–467.

As an example of interactions between number theoretic conjectures and model theoretic problems, Macintyre and Wilkie proved that if the Schanuel's conjecture is true then the theory of real field with exponentiation ($R_{exp}$) is decidable.

Macintyre, A. & Wilkie, A. J. (1996). "On the decidability of the real exponential field". In Odifreddi, Piergiorgio. Kreiseliana: About and Around Georg Kreisel. Wellesley: Peters. pp. 441–467.

Source Link
user42090
user42090

As an example of theorems like part 1 in the question, Macintyre and Wilkie proved that if the Schanuel's conjecture is true then the theory of real field with exponentiation ($R_{exp}$) is decidable.

Macintyre, A. & Wilkie, A. J. (1996). "On the decidability of the real exponential field". In Odifreddi, Piergiorgio. Kreiseliana: About and Around Georg Kreisel. Wellesley: Peters. pp. 441–467.