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Structural identifiability is a property of an ODE model with parameters that allows for the parameters to be determined from continuous noise-free data. This is a natural prerequisite for practical identifiability. Conducting multiple independent experiments could make more parameters or functions of parameters identifiable, which is a desirable property to have. How many experiments are sufficient? In the present paper, we provide an algorithm to determine the exact number of experiments for multi-experiment local identifiability and obtain an upper bound that is off at most by one for the number of experiments for multi-experiment global identifiability. Interestingly, the main theoretical ingredient of the algorithm has been discovered and proved using model theory (in the sense of mathematical logic). We hope that this unexpected connection will stimulate interactions between applied algebra and model theory, and we provide a short introduction to model theory in the context of parameter identifiability. As another related application of model theory in this area, we construct a nonlinear ODE system with one output such that single-experiment and multiple-experiment identifiability are different for the system. This contrasts with recent results about single-output linear systems. We also present a Monte Carlo randomized version of the algorithm with a polynomial arithmetic complexity. Implementation of the algorithm is provided and its performance is demonstrated on several examples. The source code is available at https://github.com/pogudingleb/ExperimentsBound.