# The Transfer Principle

Real **R** and hyperreal numbers **R*** are two models of the same first order theory, meaning that any (first-order) statement true in one is true in the other.

Every natural number n > 1 has a predecessor: (∀n > 1)(∃m)(m + 1 = n). When interpreted in **N***, the statement asserts that every hypernatural number, except 1, has a predecessor, implying, in particular, that there is no smallest infinite number.

Not every statement that holds in **N** holds in **N***. One example is the principle of mathematical induction:

∀M⊂**N**[(1∈M ∧ ∀k(k∈M⇒(k+1)∈M)] ⇒ M = **N**.

The analog of that in **N*** would mean that

∀M⊂**N***[(1∈M ∧ ∀k(k∈M⇒(k+1)∈M)] ⇒ M = **N***

which clearly fails for M = **N** because **N** is a proper subset of **N***.

Even more obviously not every statement that holds in **N*** holds in **N**. For example,

∃ω∈**N***∀n∈**N** (ω > n),

which claims the existence of infinitely large numbers has no analog in **N**.

... to be continued ...

### References

- Leif Arkeryd,
__The Evolution of Nonstandard Analysis__,*The American Mathematical Monthly*, Vol. 112, No. 10 (Dec., 2005), pp. 926-928 - J. M. Henle, E. M. Kleinberg,
*Infinitesimal Calculus*, Dover, 2003 - K. Ito,
__Nonstandard Analysis__, in*Encyclopedic Dictionary of Mathematics*, v. 1, MIT Press, Press, 2000 (fourth printing), pp. 1100-1103 - A. Robinson,
*Non-standard Analysis*, Princeton University Press (Rev Sub edition), 1996

- Infinitesimals. Non-standard Analysis
- Formal Languages
- Theories and Proofs
- Models and Metamathematics
- Hyperintegers and Hyperreal Numbers
- Structure of Hyperreal Numbers
- The Transfer Principle
- Common Concepts - A Non-standard View
- Is .999... = 1? A Non-standard View

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