Hyperreal number

In mathematics, hyper- real numbers are a central issue to the non-standard analysis. The amount of the hyper- real numbers is usually written as; it extends the real numbers to infinitesimally adjacent numbers, as well as infinitely large ( infinite ) numbers.

When Newton and Leibniz their differential calculus with " fluxions " or " monads " conducted, they used infinitesimal numbers, and even Euler and Cauchy found it helpful. Nevertheless, these figures have been considered from the outset skeptical, and in the 19th century, the analysis was provided by the introduction of the limit epsilon- delta definition by Cauchy, Weierstrass and others on a rigorous basis. Infinitesimal numbers have not been used from then on.

Abraham Robinson showed then in the 1960s, how infinitely large and small numbers can be rigorously defined formally, and so opened the field of nonstandard analysis. The construction given here is a simplified but no less strict version that was first given by Lindstrom.

Due to the hyper- real numbers a formulation of differential and integral calculus is possible without the limit concept.

Elementary Properties

The hyper real numbers form an ordered field which contains as a subfield. Both are completed even real.

The signature determines which symbols you can use in the statements. The restriction to the first-order logic means that one can quantify over elements of the body, but not on subsets. The following statements apply, for example, both in and in:

• Any number that is greater than zero, has a square root.

This does not mean that and behave exactly the same; they are not isomorphic. For example, there are a member that is greater than all natural numbers. However, this can not be expressed by a statement of the above form, you need to infinitely many:

Such a figure there are not. A hyper- real number as it is called infinitely or indefinitely, the reciprocal of an infinitely large number is an infinitesimal number.

A further difference is that the real numbers are order complete, ie every non-empty, bounded above subset of has a supremum in. This requirement characterizes the real numbers as a parent body clearly, that is, up to isomorphism unique. however, is not order complete: The set of all finite numbers in does not have a supremum, but is, for example, limited by the above. The reason is that one has to quantify the formulation of the order completeness of all subsets; they therefore can not be formalized in first-order logic.

The hyper real numbers are equally powerful to the real numbers:

Construction

The set of all sequences of real numbers ( ) form an extension of the real numbers, if one identifies the real numbers with the constant consequences.

The prototypes for "infinite large " numbers are in this set sequences that are eventually greater than any real number, for example, the sequence:

On one can now define the term-wise addition and multiplication:

The equation

Although both are equal to zero. It must therefore have consequences on an equivalence relation can be identified. The idea is that sequences are equivalent if the set of all points where the consequences are different, a minor is. What is the set of all non-essential quantities? In particular, yes consequences to be equivalent if they behave the same at infinity, so if they are only a finite number of points are different. All finite sets are therefore not significant. And the example of p and q is, that for each subset of either portion or its complement, is immaterial. Among other is still needed, that the intersection of two insignificant quantities is immaterial, since the equivalence relation must be transitive. This leads to an ultrafilter:

A filter on the natural numbers is a set of subsets of the natural numbers, in which:

• The empty set is not.
• If it contains two sets, then their intersection.
• When it contains an amount, then the upper volumes.

A filter is free if:

• Contains no finite sets

He is an ultrafilter if the following holds:

• If a certain subset does not contain contains its complement.

The existence of a free ultrafilter follows from the lemma of Zorn. With the help of this ultrafilter can define an equivalence relation:

On the set of equivalence classes is designated, then the addition and multiplication of the equivalence classes may be defined by representatives. This is well defined, since a filter. There is even an ultra- filter, each element other than 0 has an inverse. For example, one of the two sequences, and is equivalent to zero, and the other to one.

The equivalence class of the sequence is larger than any real number, as is true for a real number

Then is shown later that the constructed body is actually equivalent to elementary. This is done by an inductive proof of the structure of formulas, wherein use is made of the ultra- filtering properties.

Comments

• Each filter on the natural numbers corresponds to an ideal of the ring. An ultrafilter corresponds to a maximal ideal, that's why the quotient is also a body. The choice of a non-free ultrafilter would mean that the body of the equivalence classes isomorphic to the output body.
• This design is a special case of the ultra potency. Among other things, this means that the embedding is in an elementary embedding and that is - saturated.
• From the axioms of set theory ( ZFC ) plus the continuum hypothesis follows that this construction does not depend on the choice of the ultrafilter. ( This means that different ultrafilter lead to isomorphic Ultra products. )

Infinitesimals and infinitely large numbers

A hyper real number is infinitesimal when it is less than any positive real number, and is greater than any negative real number. The number zero is the only infinitesimal real number, but there are other hyper real infinitesimal numbers, for example. It is greater than zero but less than any positive real number, because of the ultrafilter contains all complements of finite sets.

A hyper- real number is called finite if there is with a natural number, otherwise called infinite. The number is an infinite number. Note: The term " infinite " usually refers to a number that is greater than any natural number, "infinite", but also includes figures that are less than any integer, such as.

A number other than 0 if and only infinite if is infinitesimal. For example,.

It can be shown that every finite hyper- real number " very close " is at exactly one real number. If is a finite hyper- real number, then there is exactly one real number, so that infinitesimal: More. The number is called the standard part of the difference to the non-standard part. St The figure has some nice features: For all finite hyper- real numbers, then:

• If and only if is real

Where the means in particular that the term is defined on the left side, so that, for example, is finite, if both are finite. The set of finite numbers thus form a subring in the hyper- real numbers. It should also

• If not infinitesimal,

In addition:

• The mapping st is continuous with respect to the order topology on the set of finite hyper- real numbers, it is even locally constant.

The first two properties (and the implication of the third property) state that st is a ring homomorphism.

For example, the hyper- real number is term by term less than, that is. But it is larger than any real number less than 1 Therefore, it is infinitesimally adjacent to 1 and 1 is its standard part. Your non- standard part is ( the difference from 1)

Remember though that the real number is as the limit of sequence equal to 1.

Other properties

The hyper real numbers are equally powerful to the real numbers, because the thickness must be at least as large as that of the real numbers, since they contain the real numbers, and can be at most as large as the amount is equally powerful to the real numbers. The order structure of the hyper- real numbers is uncountable cofinality, that is, there exists no unbounded countable set, so no unlimited series of hyper- real numbers: Be given a series of hyper- real numbers by representatives. Then the hyper- real number with the representative,

An upper bound. It can therefore reach no consequence arbitrarily large hyper- real numbers. The order of the hyper- real numbers induces an order topology. By means of these can be the usual topological concepts of limits and continuity on the hyper- real numbers transmitted. As a parent body they have, with the addition on a compatible with the topology group structure, ie it is a topological group. This induces a uniform structure, so you can speak to the hyper- real numbers also of uniform continuity, Cauchyfiltern etc.. From the uncountable cofinality followed by consideration of inverses that there are several hyper- real numbers can be no sequence of 0 ( or according to any other hyper- real number ), which comes arbitrarily close to the 0. Therefore, the topology of the hyper- real numbers does not meet the two Abzählbarkeitsaxiome, so it is not particularly metrizable. From the uncountable cofinality also follows that they are not separable. It follows from the absence of Suprema numerous amounts that the space is totally disconnected and not locally compact.