Everything about Hyperreal Numbers totally explained
The system of
hyperreal numbers represents a rigorous method of treating the ideas about
infinite and
infinitesimal numbers that had been used casually by mathematicians, scientists, and engineers ever since the invention of
calculus by
Newton and
Leibniz. The hyperreals, or
nonstandard reals (usually denoted as *
R), denote an
ordered field which is a proper
extension of the ordered field of
real numbers
R and which satisfies the
transfer principle. This principle allows true
first order statements about
R to be reinterpreted as true first order statements about *
R.
An important property of *
R is that it has
infinitely large as well as
infinitesimal numbers, where an
infinitely large number is a number that's larger than all numbers representable in the form
»
The use of the definite article
the in the phrase
the hyperreal number is somewhat misleading in that there isn't a unique ordered field that's referred to in most treatments.
However, a
2003 paper by Kanovei and
Shelah shows that there's a definable, countably
saturated (meaning
ω-saturated, but not of course countable)
elementary extension of the reals, which therefore has a good claim to the title of
the hyperreal numbers.
The condition of being a hyperreal field is a stronger one than that of being a
real closed field strictly containing
R. It is also stronger than that of being a
superreal field in the sense of Dales and
Woodin.
The application of hyperreal numbers and in particular the transfer principle to problems of
analysis is called
nonstandard analysis; some find it more intuitive than standard
real analysis. When
Newton and (more explicitly)
Leibniz introduced differentials, they used infinitesimals and these were still regarded as useful by later mathematicians such as
Euler and
Cauchy. Nonetheless these concepts were from the beginning seen as suspect, notably by
Berkeley, and when in the
1800s calculus was put on a firm footing through the development of the epsilon-delta definition of a
limit by Cauchy,
Weierstrass and others, they were largely abandoned.
However, in the 1960s
Abraham Robinson showed how infinitely large and infinitesimal numbers can be rigorously defined and used to develop the field of nonstandard analysis. Robinson developed his theory nonconstructively, using
model theory; however it's possible to proceed using only
algebra and
topology, and proving the transfer principle as a consequence of the definitions. In other words hyperreal numbers
per se, aside from their use in nonstandard analysis, have no necessary relationship to model theory or first order logic.
The transfer principle
Historically, the concept of
number has been repeatedly generalized. At each step in this process of generalization, mathematicians wished to retain as many properties as possible from the earlier concepts of numbers. However, some properties always had to be given up. For example, in moving from rational numbers to real numbers, one loses the property that every number is a ratio of two integers. In moving from real numbers to
complex numbers,
order is abandoned, and in moving from complex numbers to
quaternions,
commutativity is abandoned (moreover, when moving from quaternions to
octonions, one is even forced to abandon
associativity).
In the case of the hyperreals, a long historical delay in their development was caused by uncertainty among mathematicians as to exactly which properties could be retained, and which would have to be given up. 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's another number greater than it:
» :
of functions from κ to
R. The hyperreal fields we obtain in this case are called
ultrapowers of
R and are identical to the ultrapowers constructed via free
ultrafilters in model theory.
Further Information
Get more info on 'Hyperreal Numbers'.
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