Definition of the spectrum in first order logic

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I want to understand the definition of the spectrum and therefore I want to know, what it means that a model has n elements or that a model is of size n. What is said to be an element? Are these only the elements of the universe?

Furthermore, I am trying to find a sentence $\phi$ without any "="-relation, so that $sp(\phi) = \{n \in \mathbb{N}: n > 5\}$. Any idea on that?