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 ...
- 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
Back to What Is Infinity?
|Contact| |Front page| |Contents| |Up| |Algebra|Copyright © 1996-2018 Alexander Bogomolny