Editing Transfer principle (section)

== Generalizations of the concept of number ==
Historically, the concept of [[number]] has been repeatedly generalized.  The addition of [[0 (number)|0]] to the natural numbers <math>\mathbb{N}</math> was a major intellectual accomplishment in its time.  The addition of negative integers to form <math>\mathbb{Z}</math> already constituted a departure from the realm of immediate experience to the realm of mathematical models.  The further extension, the rational numbers <math>\mathbb{Q}</math>, is more familiar to a layperson than their completion <math>\mathbb{R}</math>, partly because the reals do not correspond to any physical reality (in the sense of measurement and computation) different from that represented by <math>\mathbb{Q}</math>.  Thus, the notion of an irrational number is meaningless to even the most powerful floating-point computer.  The necessity for such an extension stems not from physical observation but rather from the internal requirements of mathematical coherence.  The infinitesimals entered mathematical discourse at a time when such a notion was required by mathematical developments at the time, namely the emergence of what became known as the [[calculus|infinitesimal calculus]].  As already mentioned above, the mathematical justification for this latest extension was delayed by three centuries.  [[Howard Jerome Keisler|Keisler]] wrote:
:"In discussing the real line we remarked that we have no way of knowing what a line in physical space is really like. It might be like the hyperreal line, the real line, or neither.  However, in applications of the calculus, it is helpful to imagine a line in physical space as a hyperreal line."

The [[consistency|self-consistent]] development of the hyperreals turned out to be possible if every true [[first-order logic]] statement that uses basic arithmetic (the [[natural number]]s, plus, times, comparison) and quantifies only over the real numbers was assumed to be true in a reinterpreted form if we presume that it quantifies over hyperreal numbers. For example, we can state that for every real number there is another number greater than it:

: <math> \forall x \in \mathbb{R} \quad \exists y \in\mathbb{R}\quad   x < y. </math>

The same will then also hold for hyperreals:

: <math> \forall x \in {}^\star\mathbb{R} \quad \exists y \in {}^\star\mathbb{R}\quad  x < y. </math>

Another example is the statement that if you add 1 to a number you get a bigger number:

: <math> \forall x \in \mathbb{R} \quad x < x+1 </math>

which will also hold for hyperreals:

: <math> \forall x \in {}^\star\mathbb{R} \quad  x < x+1. </math>

The correct general statement that formulates these equivalences is called the transfer principle.  Note that, in many formulas in analysis, quantification is over higher-order objects such as functions and sets, which makes the transfer principle somewhat more subtle than the above examples suggest.