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Discussion on answer by David C. Ullrich: How to distinguish two groups $(\mathbb{Z},+)$ , $(\mathbb{Z} \times \mathbb{Z},+)$ using first order logic?

Imported from a comment discussion on math.stackexchange.com/questions/2483407/…

first-order-logic logic

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Discussion on answer by David C. Ullrich: How to distinguish two groups $(\mathbb{Z},+)$ , $(\mathbb{Z} \times \mathbb{Z},+)$ using first order logic?