non-standard analysis

non-standard analysis

Non-standard analysis

is a branch of mathematics that formulates

analysis

using a

rigorous

notion of

infinitesimal

, where an element of an

ordered field

F

is infinitesimal if and only if its

absolute value

is smaller than any element of

F

of the form

1

n

, for

n

a

natural number

. Ordered fields that have infinitesimal

elements

are also called non-Archimedean. More generally, non-standard analysis is any form of mathematics that relies on non-standard models and the

transfer principle

. A field which

satisfies

the transfer principle for

real numbers

is a hyperreal field, and non-standard real analysis uses these fields as non-standard models of the real numbers.

Non-standard analysis was introduced in the early 1960s by the mathematician Abraham Robinson. Robinson’s original approach was based on these non-standard models of the

field of real

numbers. His classic foundational book on the subject

Non-standard Analysis

was published in 1966 and is still in print.

Several technical issues must be addressed to develop a calculus of infinitesimals. For example, it is not enough to construct an ordered field with infinitesimals. See the article on

hyperreal numbers

for a discussion of some of the relevant ideas.

Given any set

S

, the

superstructure

over a set

S

is the set

V

⁢

(

S

)

defined by the conditions

V

0

⁢

(

𝐒

)

=

𝐒

V

n

+

1

⁢

(

𝐒

)

=

V

n

⁢

(

𝐒

)

∪

2

V

n

⁢

(

𝐒

)

V

⁢

(

𝐒

)

=

⋃

n

∈

ℕ

V

n

⁢

(

𝐒

)

Thus the superstructure over

S

is obtained by starting from

S

and iterating the

operation

of adjoining the

power set

of

S

and taking the union of the resulting

sequence

. The superstructure over the real numbers includes a wealth of mathematical structures: For

instance

, it contains

isomorphic

copies of all

separable

metric spaces and metrizable

topological vector spaces

. Virtually all of mathematics that interests an analyst goes on within

V

⁢

(

R

)

.

The working view of

nonstandard analysis

is a set

*

R

and a

mapping

*

:

V

(

ℝ

)

→

V

(

*

ℝ

)

which satisfies some additional properties.

*

ℝ

is of course embedded in

ℝ

.

To formulate these principles we state first some definitions:
A

formula

has bounded quantification if and only if the only

quantifiers

which

occur in

the formula have range restricted over sets, that is are all of the form:

∀

x

∈

A

,

Φ

⁢

(

x

,

α

1

,

…

,

α

n

)

∃

x

∈

A

,

Φ

⁢

(

x

,

α

1

,

…

,

α

n

)

For example, the formula

∀

x

∈

A

,

∃

y

∈

2

B

,

x

∈

y

has bounded quantification, the universally quantified

variable

x

ranges over

A

, the existentially quantified variable

y

ranges over the powerset of

B

. On the other hand,

∀

x

∈

A

,

∃

y

,

x

∈

y

does not have bounded quantification because the quantification of

y

is unrestricted.

A set

x

is internal if and only if x is an element of

*

A

for some element

A

of

V

⁢

(

R

)

.

*

A

itself is internal if

A

belongs to

V

⁢

(

R

)

.

We now formulate the basic logical framework of nonstandard analysis:

Extension

principle: The mapping

*

is the

identity

on

R

.

Transfer principle: For any formula

P

⁢

(

x

1

,

…

,

x

n

)

with bounded quantification and with

free variables

x

1

,

…

,

x

n

, and for any elements

A

1

,

…

,

A

n

of

V

⁢

(

R

)

, the following equivalence holds:
:

P

(

A

1

,

…

,

A

n

)

⇔

P

(

*

A

1

,

…

,

*

A

n

)

Countable

saturation: If

A

k

k

is a

decreasing sequence

of nonempty internal sets, with

k

ranging over the natural numbers, then
:

⋂

k

A

k

≠

∅

One can show using

ultraproducts

that such a map * exists. Elements of

V

⁢

(

R

)

are called standard. Elements of

*

R

are called hyperreal numbers.

The symbol

*

N

denotes the nonstandard natural numbers. By the extension principle, this is a

superset

of

N

. The set

*

N

-

N

is not empty. To see this, apply countable saturation to the sequence of internal sets

A

k

=

{

k

∈

*

ℕ

:

k

≥

n

}

The sequence

A

k

k

is in

N

has a non-empty

intersection

, proving the result.

We begin with some definitions: Hyperreals

r

,

s

are infinitely close if and only if

r

≅

s

⇔

∀

θ

∈

ℝ

+

,

|

r

-

s

|

≤

θ

A hyperreal

r

is infinitesimal if and only if it is infinitely close to 0.

r

is limited or

bounded

if and only if its absolute value is dominated by a standard

integer

.
The bounded hyperreals form a subring of

*

R

containing the reals. In this ring, the infinitesimal hyperreals are an ideal. For example, if

n

is an element of

*

N

-

N

, then

1

n

is an infinitesimal.

The set of bounded hyperreals or the set of infinitesimal hyperreals are external subsets of

V

(

*

R

)

; what this means in practice is that bounded quantification, where the bound is an internal set, never ranges over these sets.

This entry was adapted from the Wikipedia article

http://en.wikipedia.org/wiki/Nonstandard_analysis

Non-standard analysis as of December 19, 2006.

Title

non-standard analysis

Canonical name

NonstandardAnalysis

Date of creation

2013-03-22 16:28:26

Last modified on

2013-03-22 16:28:26

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

13

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 03H05

Synonym

nonstandard analysis

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