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It is easy to show that there exist finite sets of sentences $\phi_p$ such that for any $p$ prime, the models of $\phi_p$ are exactly the fields with characteristic $p$. It is also easy to show that there exists an infinite set of sentences $\phi_0$ such that the models of $\phi_0$ are exactly the fields with characteristic 0.

Does there exist a finite set of sentences such that its models are exactly the fields with characteristic $0$?

I tried arriving at a contradiction using various constructions and the Completeness Theorem, but it seems like it is necessary to make an explicit choice of $\phi_0$ in order to produce one.

user35443
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1 Answers1

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This is not possible. Note that if there was such a finite set of sentences, then we could take their conjunction to get a single sentence characterizing fields of characteristic $0$, and taking its negation gives a formula $\psi$ characterizing fields of positive characteristic.

Since fields of arbitrary positive characteristic satisfy $\psi$, we get that the theory consisting of $\psi,2\neq 0,3\neq 0,5\neq 0,\dots,p\neq 0$ is consistent for all primes $p$. But then by compactness so is the theory consisting of $\psi$ and $p\neq 0$ for all primes $p$ at once. But a field satisfying all those statements must have characteristic $0$, contradicting our assumptions on $\psi$.

Wojowu
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