By A.M. Fink
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Extra info for Almost Periodic Differential Equations
The Dirichlet principle of number theory reads: To all ε > 0 and x1 , . . , xn ∈ R there is an integer N ∈ N such that each of the products N x1 , . . , N xn diﬀers from some integers by at most ε. We are done on to apply this theorem with the chosen parameters. 15. It is worth observing that the inﬁnite proximity (as well as equivalence) of reals is not a subset of the cartesian square R × R. 9. Note also that the monad μ(R) is indivisible in the following implicit sense: n−1 μ(R) = μ(R) for every standard n.
Then we declare given a language with alphabet A and call the chosen expressions well-formed formulas. The next step consists in selecting some ﬁnite (or inﬁnite) families of formulas called axioms in company with explicit description of the admissible rules of inference which might be viewed as abstract relations on Φ(A). A theorem is a formula resulting from the axioms by successive application of ﬁnitely many rules of inference. Using common parlance, we express this in a freer and cozier fashion as follows: The theorems of a formal theory comprise the least set of formulas which contains all axioms and is closed under the rules of inference of the theory.
In the opposite case, x is unbound or free in ϕ. We also speak about free or bound occurrence of a variable in a formula. Intending to stress that only the variables x1 , . . , xn are unbound in the formula ϕ, we write ϕ = ϕ(x1 , . . , xn ), or simply ϕ(x1 , . . , xn ). ” A formula with no unbound variables is a sentence. Speaking about verity or falsity of ϕ, we imply the universal closure of ϕ which results from generalization of ϕ by every bound variable of ϕ. It is also worth observing that quantiﬁcation is admissible only by variables.