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Encyclopedia > Transfer principle

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. Euclid, detail from The School of Athens by Raphael. ... Non-standard analysis is that branch of mathematics that formulates analysis using a rigorous notion of infinitesimal, where an element of an ordered field F is infinitesimal if and only if its absolute value is smaller than any element of F of the form 1/n, for n a natural... Analysis is the generic name given to any branch of mathematics which depends upon the concepts of limits and convergence, and studies closely related topics such as continuity, integration, differentiability and transcendental functions. ... In mathematical logic, in particular in model theory and non-standard analysis, an internal set is a set that is a member of a model. ... Non-standard analysis is that branch of mathematics that formulates analysis using a rigorous notion of infinitesimal, where an element of an ordered field F is infinitesimal if and only if its absolute value is smaller than any element of F of the form 1/n, for n a natural...


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. In mathematical logic, given models and in the same language , a function is called an elementary embedding if is an elementary substructure of . ...


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

so it is indeed 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.


  Results from FactBites:
 
Transfer principle - Wikipedia, the free encyclopedia (271 words)
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.
Hyperreal number - Wikipedia, the free encyclopedia (3067 words)
The application of hyperreal numbers and in particular the transfer principle to problems of analysis is called nonstandard analysis; some find it more intuitive than standard real analysis.
The transfer principle however doesn't mean that R and *R have identical behavior.
The essence of the axiomatic approach is to assert (1) the existence of at least one infinitesimal number, and (2) the validity of the transfer principle.
  More results at FactBites »

 
 

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