Hyper real number

In mathematics , hyperreal numbers are a central subject of nonstandard analysis . The set of hyper-real numbers is mostly written as; it expands the real numbers by infinitesimally neighboring numbers as well as by infinitely large (infinite) numbers. ${\ displaystyle {} ^ {*} \ mathbb {R}}$

When Newton and Leibniz carried out their differential calculus with “fluxions” and “monads” respectively, they used infinitesimal numbers, and Euler and Cauchy found them useful. Nonetheless, these numbers were viewed with skepticism from the start, and in the 19th century analysis was put on a strict foundation without infinitesimal with the introduction of the epsilon-delta definition of the limit value and the definition of real numbers by Cauchy, Weierstrass, and others Sizes.

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

The hyper-real numbers make it possible to formulate differential and integral calculus without the concept of limit values.

Elementary properties

The hyper real numbers form an ordered body that contains as part of the body. Both are really closed . ${\ displaystyle {} ^ {*} \ mathbb {R}}$ ${\ displaystyle \ mathbb {R}}$

Infinitesimal (ε) and infinites (ω) on the hyper-real number line

The body is constructed in such a way that it is elementarily equivalent to . This means that every statement that applies in also applies in if the statement can be formulated in the first-order predicate logic above the signature . ${\ displaystyle {} ^ {*} \ mathbb {R}}$${\ displaystyle \ mathbb {R}}$${\ displaystyle \ mathbb {R}}$${\ displaystyle {} ^ {*} \ mathbb {R}}$ ${\ displaystyle \ {0,1, +, -, \ cdot, <\}}$

The signature determines which symbols can be used in the statements. The restriction to first-level predicate logic means that one can only quantify over elements of the body, but not over subsets. The following statements apply e.g. B. both in and in : ${\ displaystyle \ mathbb {R}}$${\ displaystyle {} ^ {*} \ mathbb {R}}$

• ${\ displaystyle \ forall x \ \ exists y \ x
• ${\ displaystyle \ forall x \ x
• Any number that is greater than or equal to zero has a square root. In formulas:${\ displaystyle \ forall x \ ((x> 0 \ lor x = 0) \ rightarrow (\ exists y \ x = y \ cdot y))}$

That doesn't mean that and behave exactly the same way; they are not isomorphic . For example, there is an element that is greater than all natural numbers. However, this cannot be expressed by a statement of the above form, one needs an infinite number: ${\ displaystyle \ mathbb {R}}$${\ displaystyle {} ^ {*} \ mathbb {R}}$${\ displaystyle {} ^ {*} \ mathbb {R}}$${\ displaystyle w}$

${\ displaystyle 1

There is no such number in . A hyper-real number as it is called infinitely or infinite, the reciprocal of an infinite number is an infinitesimal number . ${\ displaystyle \ mathbb {R}}$${\ displaystyle w}$

Another difference: The real numbers are order-complete , ie every non-empty, upwardly bounded subset of has a supremum in . This requirement uniquely characterizes the real numbers as ordered fields, i.e. H. except for clear isomorphism. is, however, not orderly complete: The set of all finite numbers in has no supremum, but is e.g. B. limited by the above . This is due to the fact that one has to quantify over all subsets to formulate the order completeness; therefore it cannot be formalized in the first-order predicate logic . ${\ displaystyle \ mathbb {R}}$${\ displaystyle \ mathbb {R}}$${\ displaystyle {} ^ {*} \ mathbb {R}}$${\ displaystyle {} ^ {*} \ mathbb {R}}$${\ displaystyle w}$

The hyper real numbers are equal to the real numbers:

${\ displaystyle | {} ^ {*} \ mathbb {R} | = | \ mathbb {R} | = 2 ^ {\ aleph _ {0}}}$

construction

The set of all sequences of real numbers ( ) form an extension of the real numbers if the real numbers are identified with the constant sequences. ${\ displaystyle \ textstyle \ mathbb {R} ^ {\ mathbb {N}} = \ prod _ {n = 1} ^ {\ infty} \ mathbb {R}}$

${\ displaystyle 1}$So is with the result ,${\ displaystyle (1; 1; 1; 1; 1; 1; \ dotsc)}$
${\ displaystyle 2}$is thus identified with the sequence .${\ displaystyle (2; 2; 2; 2; 2; 2; \ dotsc)}$

The prototypes for “infinitely large” numbers are sequences in this set that will eventually become larger than any real number, e.g. B. the consequence:

${\ displaystyle A = (1; 10; 100; 1000; 10000; \ dotsc)}$

At you can now define addition and multiplication term by term: ${\ displaystyle \ mathbb {R} ^ {\ mathbb {N}}}$

${\ displaystyle (1; 1; 1; 1; 1; 1; \ dotsc) + (2; 2; 2; 2; 2; 2; \ dotsc) = (3; 3; 3; 3; 3; 3; \ dotsc)}$

This results in a commutative unitary ring , but this has zero divisors and is therefore not a field. It applies z. B. for ${\ displaystyle \ mathbb {R} ^ {\ mathbb {N}}}$

${\ displaystyle p = (1; 0; 1; 0; 1; 0; \ dotsc)}$
${\ displaystyle q = (0; 1; 0; 1; 0; 1; \ dotsc)}$

the equation

${\ displaystyle p \ cdot q = 0}$,

although both and are non-zero. Sequences therefore still have to be identified using an equivalence relation . The idea is that sequences are equivalent if the set of all places where the sequences differ is an insignificant one . Now what is the set of all immaterial sets ? In particular, sequences should be equivalent if they behave the same in infinity , i.e. if they are only different in a finite number of places. All finite sets are therefore insignificant. And the example with and shows that for each subset either the subset or its complement is insignificant. Among other things, it is then necessary that the union of two insignificant sets is insignificant, since the equivalence relation must be transitive. This leads to an ultrafilter: ${\ displaystyle p}$${\ displaystyle q}$${\ displaystyle p}$${\ displaystyle q}$

A filter on the natural numbers is a set of subsets of the natural numbers for which the following applies: ${\ displaystyle U}$

• The empty set is not in .${\ displaystyle U}$
• If it contains two sets, so does their intersection.
• If it contains a set, then also its supersets.

A filter is free if: ${\ displaystyle U}$

• ${\ displaystyle U}$ does not contain finite sets

It is an ultrafilter if:

• If a particular subset does not contain, its complement contains .${\ displaystyle U}$${\ displaystyle U}$

The existence of a free ultrafilter follows from Zorn's lemma . With the help of this ultrafilter , an equivalence relation can be defined: ${\ displaystyle U}$

${\ displaystyle (x_ {0}, x_ {1}, x_ {2}, \ dotsc) \ sim (y_ {0}, y_ {1}, y_ {2}, \ dotsc)}$if .${\ displaystyle \ {n: x_ {n} = y_ {n} \} \ in U}$

The addition and multiplication of the equivalence classes can now be defined using representatives on the set of equivalence classes that is designated with . This is well defined as there is a filter. Since there is even an ultrafilter, every element except 0 has an inverse. For example, one of the two sequences is and equivalent to zero and the other to one. ${\ displaystyle {} ^ {*} \ mathbb {R}: = {} ^ {\ textstyle \ mathbb {R} ^ {\ mathbb {N}}} \! / \! {} _ {\ textstyle \ sim} }$${\ displaystyle U}$${\ displaystyle U}$${\ displaystyle {} ^ {*} \ mathbb {R}}$${\ displaystyle p}$${\ displaystyle q}$

Now we have to define one more order. This is done through ${\ displaystyle {} ^ {*} \ mathbb {R}}$

${\ displaystyle (x_ {0}, x_ {1}, x_ {2}, \ dotsc) <(y_ {0}, y_ {1}, y_ {2}, \ dotsc)}$if .${\ displaystyle \ {n: x_ {n}

It is easy to see that this defines a total order on (for totality it is important that there is an ultrafilter). ${\ displaystyle {} ^ {*} \ mathbb {R}}$${\ displaystyle U}$

The equivalence class of the result is greater than any real number, because for a real number applies ${\ displaystyle w}$${\ displaystyle r}$

${\ displaystyle \ {n: w_ {n}> r \} \ in U}$

Then it has to be shown that the constructed body is actually elementarily equivalent to . This is done through an induction proof of the structure of the formulas, whereby use is made of the ultrafilter properties. ${\ displaystyle \ mathbb {R}}$

Remarks

• Every filter on the natural numbers corresponds to an ideal of the ring (but not vice versa). An ultrafilter corresponds to a maximum ideal, so the quotient is a body. Choosing a non-free ultrafilter would result in the body of the equivalence classes being isomorphic to the starting body.${\ displaystyle \ mathbb {R} ^ {\ mathbb {N}}}$
• This construction is a special case of ultrapotence . Among other things, this means that the embedding of is in an elementary embedding and that - is saturated .${\ displaystyle \ mathbb {R}}$${\ displaystyle {} ^ {*} \ mathbb {R}}$${\ displaystyle {} ^ {*} \ mathbb {R} \ \ omega _ {1}}$
• From the axioms of set theory (ZFC) plus the continuum hypothesis it follows that this construction does not depend on the choice of the ultrafilter. (This means that different ultrafilters lead to isomorphic ultraproducts.)

Infinitesimal and infinitely large numbers

A hyper-real number is called infinitesimal if it is less than any positive real number and 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 smaller than any positive real number, because the ultrafilter contains all complements of finite sets. ${\ displaystyle a = (1; 0 {,} 1; 0 {,} 01; 0 {,} 001; \ dotsc)}$

A negative infinitesimal number is greater than any negative real number and less than any positive real number, e.g. B. . ${\ displaystyle -a = (- 1; -0 {,} 1; -0 {,} 01; -0 {,} 001; \ dotsc)}$

A hyper real number is called finite if there is a natural number with , otherwise it is called infinite . The number is an infinite number. Note: The term "infinitely large" usually denotes a number that is larger than any natural number, but "infinite" also includes numbers that are smaller than any whole number, such as . ${\ displaystyle x}$${\ displaystyle n}$${\ displaystyle -n ${\ displaystyle x}$ ${\ displaystyle A = (1; 10; 100; 1000; \ dotsc)}$${\ displaystyle -A = (- 1; -10; -100; \ dotsc)}$

A number other than 0 is infinite if and only if is infinitesimal. For example is . ${\ displaystyle x}$${\ displaystyle 1 / x}$${\ displaystyle A = {\ tfrac {1} {a}}}$

It can be shown that every finite hyper-real number is “very close” to exactly one real number. More precisely: if is a finite hyper-real number, then there is exactly one real number such that is infinitesimal. The number is called the standard part of , the difference to is the non-standard part . The mapping st has some pleasant properties: For all finite hyper-real numbers , the following applies: ${\ displaystyle x}$${\ displaystyle \ operatorname {st} (x)}$${\ displaystyle x- \ operatorname {st} (x)}$${\ displaystyle \ operatorname {st} (x)}$${\ displaystyle x}$${\ displaystyle x}$${\ displaystyle x}$${\ displaystyle y}$

• ${\ displaystyle \ operatorname {st} (x + y) = \ operatorname {st} (x) + \ operatorname {st} (y),}$
• ${\ displaystyle \ operatorname {st} (xy) = \ operatorname {st} (x) \ operatorname {st} (y),}$
• ${\ displaystyle \ operatorname {st} (x) = x}$if and only if is real${\ displaystyle x}$

Whereby this means in particular that the term is defined on the left-hand side, so that e.g. B. is finite if both and are finite. The set of finite numbers thus form a subring in the hyper-real numbers. Also is ${\ displaystyle x + y}$${\ displaystyle x}$${\ displaystyle y}$

• ${\ displaystyle \ operatorname {st} \ left ({\ frac {1} {x}} \ right) = {\ frac {1} {\ operatorname {st} (x)}}}$if is not infinitesimal,${\ displaystyle x}$

The following also applies:

• 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) say that st is a ring homomorphism . ${\ displaystyle \ operatorname {st} (0) = 0, \ operatorname {st} (1) = 1}$

For example, the hyper-real number is term-wise less than , so is . But it is greater than any real number less than 1. It is therefore infinitesimally adjacent to 1 and 1 is its standard part. Your non-standard part (the difference from 1) is ${\ displaystyle g = (0; 0 {,} 9; 0 {,} 99; 0 {,} 999; \ dotsc)}$${\ displaystyle (1; 1; 1; 1; \ dotsc)}$${\ displaystyle g <1}$

${\ displaystyle g-1 = (- 1; -0 {,} 1; -0 {,} 01; -0 {,} 001; \ dotsc) = - a}$.

Note, however, that the real number is 1 as the limit of the sequence . ${\ displaystyle 0 {,} 999 \ dotso}$${\ displaystyle g}$

Other properties

The hyper-real numbers are equal to the real numbers, because the cardinality must be at least as great as that of the real numbers, since they contain the real numbers, and can at most be as large as the set is equal to the real numbers. The order structure of the hyperreal numbers has uncountable confinality , i. H. there is no unlimited countable set, that is, no unlimited sequence of hyper-real numbers: Let a sequence of hyper-real numbers be given by representatives . Then the hyper real number with the representative , ${\ displaystyle \ mathbb {R} ^ {\ mathbb {N}}}$ ${\ displaystyle (a_ {n}) \ in {} ^ {*} \ mathbb {R} ^ {\ mathbb {N}}}$${\ displaystyle A \ in \ mathbb {R} ^ {\ mathbb {N}}}$

${\ displaystyle A_ {n} = \ max _ {i \ leq n} a_ {in}}$

an upper bound. So hyperreal numbers of any size cannot be achieved with any sequence. The order of the hyper real numbers induces an order topology . By means of this, the usual topological terms of limit values ​​and continuity can be transferred to the hyper-real numbers. As an ordered body, with the addition they show a group structure compatible with the topology , so it is a topological group . This induces a uniform structure , so that one can also speak of uniform continuity, Cauchy filters, etc. on the hyperreal numbers. From the uncountable confinality it follows by considering reciprocal values that there is also no sequence consisting of hyperreal numbers other than 0 (or any other hyperreal number) that comes arbitrarily close to 0. Therefore, the topology of the hyperreal numbers does not fulfill the two axioms of countability , so it is especially not metrizable . It also follows from the uncountable confinality that they are not separable . The absence of numerous sets of Suprema implies that space is totally disconnected and not locally compact .