Transfer Principle - Generalizations of The Concept of Number

Generalizations of The Concept of Number

Historically, the concept of number has been repeatedly generalized. The addition of 0 to the natural numbers was a major intellectual accomplishment in its time. The addition of negative integers to form already constituted a departure from the realm of immediate experience to the realm of mathematical models. The further extension, the rational numbers, is more familiar to a layperson than their completion, partly because the reals do not correspond to any physical reality (in the sense of measurement and computation) different from that represented by . 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 infinitesimal calculus. As already mentioned above, the mathematical justification for this latest extension was delayed by three centuries. 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 self-consistent development of the hyperreals turned out to be possible if every true first-order logic statement that uses basic arithmetic (the natural numbers, 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:

The same will then also hold for hyperreals:

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

which will also hold for hyperreals:

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.