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Internal Set Theory August 19, 2008

Posted by Alexandre Borovik in Uncategorized.

From Edward Nelson’s introduction to his book:

Ordinarily in mathematics, when one introduces a new concept one defines it. For example, if this were a book on “blobs” I would begin with a definition of this new predicate: x is a blob in case x is a topological space such that no uncountable subset is Hausdorff. Then we would be all set to study blobs. Fortunately, this is not a book about blobs, and I want to do something different. I want to begin by introducing a new predicate “standard” to ordinary mathematics without defining it.

The reason for not defining “standard” is that it plays a syntactical, rather than a semantic, role in the theory. It is similar to the use of “fixed” in informal mathematical discourse. One does not define this notion, nor consider the set of all fixed natural numbers. The statement “there is a natural number bigger than any fixed natural number” does not appear paradoxical. The predicate “standard” will be used in much the same way, so that we shall assert “there is a natural number bigger than any standard natural number.” But the predicate “standard”— unlike “fixed”—will be part of the formal language of our theory, and this will allow us to take the further step of saying, “call such a natural number, one that is bigger than any standard natural number, unlimited.”

We shall introduce axioms for handling this new predicate “standard” in a consistent way. In doing so, we do not enlarge the world of mathematical objects in any way, we merely construct a richer language to discuss the same objects as before. In this way we construct a theory extending ordinary mathematics, called Internal Set Theory that axiomatizes a portion of Abraham Robinson’s nonstandard analysis. In this construction, nothing in ordinary mathematics is changed.

Edward Nelson, “Internal set theory: A new approach to nonstandard analysis,” Bulletin American Mathematical Society 83 (1977), 1165–1198.

Alain Robert, “Analyse non standard,” Presses polytechniques romandes, EPFL Centre Midi, CH–1015 Lausanne, 1985; translated by the author as “Nonstandard Analysis,” Wiley, New York, 1988.


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