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

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