By Hrbacek K., Lessmann O., O'Donovan R.

ISBN-10: 149870266X

ISBN-13: 9781498702669

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**Extra info for Analysis with ultrasmall numbers**

**Example text**

Q and p is observable relative to q1 , . . , q , then x is observable relative to q1 , . . , q . In accordance with the idea that all levels of observability should have the same properties, we re-interpret the definitions and axioms given so far as applicable to every level. Definitions 1 and 2 and the definition of observable neighbor apply to any context. The Existence, Closure and Observable Neighbor Principles are valid relative to any context. Example. (1) The number 1 is standard (observable relative to every context); therefore every x observable relative to 1 is standard.

Xn ) could be used. C can be viewed as an operation, defined for those values of x1 , . . , xn for which such unique object exists. The above argument applies to concepts that depend on parameters: Any mathematical object uniquely defined from parameters x1 , . . , xn by a statement of traditional mathematics is observable provided x1 , . . , xn are observable. 18 Analysis with Ultrasmall Numbers In particular, if C is an operation defined in traditional mathematics and x1 , . . , xn are observable, then C(x1 , .

Q , then x is observable relative to q1 , . . , q . In accordance with the idea that all levels of observability should have the same properties, we re-interpret the definitions and axioms given so far as applicable to every level. Definitions 1 and 2 and the definition of observable neighbor apply to any context. The Existence, Closure and Observable Neighbor Principles are valid relative to any context. Example. (1) The number 1 is standard (observable relative to every context); therefore every x observable relative to 1 is standard.

### Analysis with ultrasmall numbers by Hrbacek K., Lessmann O., O'Donovan R.

by George

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