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For each m > 1 let Zm = { 0 , 1 , 2 , . . . , m 1 } , and let

For each m >1 let Zm ={0,1,2,...,m 1}, and let Fm =(Zm,+,\times ) be the model whose universe is Zm and that has relations corresponding to the + and \times relations computed modulo m. For each m, the theory Th(Fm) is decidable because the universes are finite. Therefore, the quantifiers cannot be tested by brute-force.
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