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The following principles are consequences of axioms added to the classical axioms of set theory. When teaching analysis one usually does not study axioms but their consequences on real numbers. These will thus be considered axiomatically together with other properties of real numbers.

Definition: The context of a property, function or set is the list of parameters used in its definition. The context can be the empty list since some definition have no parameters.


Numbers, sets and functions are observable relative to some context.

The word "observable" , by convention, refers to a context. Informally: the context is the parameters (numbers, sets and functions) the statement is about. Therefore to determine the context of a statement, one must be able to understand it and describe what it says and about what it says something.

Observability principle
Closure principle
Ultrasmall principle
Contextual notation
Stability principle
Observable neighbour principle


The problem of induction