Transfer principle

From Academic Kids

In mathematics particularly in non-standard analysis, the transfer principle is a rule which transforms assertions about standard sets, mappings etc., into one about internal sets, mappings etc. For the precise context of this principle as discussed here, see the article non-standard analysis.

In its most general form, transfer has the properties of an elementary embedding between structures. However, the formulation at this level of generality is false for the superstructure approach to non-standard analysis, where it is replaced by formulas with bounded quantification.

Example

The principle of mathematical induction. This the formula (stated in mostly symbolic terms)

For any subset A of N if
1 is an element of A
for every element n of A, its successor n+1 is also an element of A
then
A= N

The outer quantification of the induction principle does not formally appear as a bounded quantification, but in fact

<math> A \subseteq \mathbb{N} \iff A \in 2^{\mathbb{N}}<math>

so it indeed it is bounded. Applying transfer to the induction principle gives us the formula

For any A such that A is an element of the internal powerset of *N, if
1 is an element of A
for every element n of A, its successor is also an element of A
then
A= *N

Finally note that the internal powerset of an internal set A is exactly the set of all internal subsets B of A. This is the principle of internal induction. The principle of external induction is the usual induction principle on the natural numbers, viz:

  • Any subset (internal or not) of *N containing 1 and closed under successor contains N.

However, N is in many precise ways, a very small subset of the nonstandard natural numbers *N.

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