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By Robert C. Gunning, Hugo Rossi

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Apply transfer to the formula ∀X ⊂ R ∃r ∈ R (X ≤ r) ⇒ ∃s ∈ R (X ≤ s ∧ ∀x < s X ≤ x) , where X ≤ r abbreviates ∀x ∈ X (x ≤ r). 2 Overspill, underspill and limits The usage of the transfer often takes place in the following form. 4. Let A ⊂ ∗R be an internal subset. (i) (Overspill) Suppose A contains arbitrarily large positive finite numbers. Then A contains an infinite number. (ii) (Underspill) Suppose A contains arbitrarily small positive noninfinitesimal numbers. Then A contains a positive infinitesimal.

For example, groups, rings, fields belongs to such category. So ( ∗R, ∗+, 0) is still called a group. • On the other hand, if the structures with the name N consist of more than one sort of elements in a natural way, internal sets from ∗ F ∩ Vn (R) are called internal N or ∗N . For example, vector n∈ω spaces consist of two sorts of elements: vectors and scalars from a field, and falls into this category. Hence, if X ∈ V (R) is a vector space over C, then ∗X is an internal vector space (or a ∗vector space) over the field ∗ C.

We leave it as an exercise to prove using the κ-saturation of (V, ε) the above two fundamental principles of nonstandard analysis. In nonstandard analysis, by transfer, we can lift a standard problem, solve it in V ( ∗R), and push the solution back to V (R) and make it into a solution of the original problem. In the process, κ-saturation is needed to ensure enough set existence. t. certain properties, one obtains a new standard space. 6. As a consequence of the transfer, we have the following useful tool for building internal sets: Internal Definition Principle: Let A, C1 , .

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Analytic Functions of Several Complex Variables by Robert C. Gunning, Hugo Rossi

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