Ultra product

An ultra product is a construct in the field of model theory , a branch of mathematics . The objective of the construction is to obtain, in addition to one model (or many models) for a given system of axioms , another that has unusual properties that cannot be formalized in the language of the axiom system. The idea of the construction is to define relations for sequences through a kind of majority decision.

definition

Some first-order language is given . Be an infinite index set, an ultrafilter on which no main filter is. For everyone is a model of language . On the Cartesian product ${\ displaystyle L}$ ${\ displaystyle I}$ ${\ displaystyle U}$ ${\ displaystyle I}$ ${\ displaystyle i \ in I}$ ${\ displaystyle A_ {i}}$ ${\ displaystyle L}$

{\ displaystyle P: = \ prod _ {i \ in I} A_ {i}}

we define an equivalence relation by

{\ displaystyle (a_ {i}) _ {i \ in I} \ sim (b_ {i}) _ {i \ in I}}

exactly when

{\ displaystyle \ left \ {i \ in I: a_ {i} = b_ {i} \ right \} \ in U}

and define the following interpretation of the symbols of the language on the basis of the set of equivalence classes : Links are made by component; for each relation symbol applies ${\ displaystyle R}$

{\ displaystyle (a_ {i}) _ {i \ in I} R (b_ {i}) _ {i \ in I}}

exactly when .

{\ displaystyle \ left \ {i \ in I: a_ {i} \, R \, b_ {i} \ right \} \ in U}

(In particular, this is consistent with the definition of equality). Then the set of all equivalence classes of modulo ~ forms a model of the given language ; it's called the ultra product of . ${\ displaystyle P}$ ${\ displaystyle L}$ ${\ displaystyle A_ {i}}$

properties

Every formula of language that is fulfilled in every component is also valid for the ultra-product itself. If all of them fulfill a given system of axioms of the first order, so does the ultra-product. For example, the ultra-product of bodies is a body, the ultra-product of ordered quantities is an ordered quantity, and so on. ${\ displaystyle L}$ ${\ displaystyle A_ {i}}$

On the other hand, this need not apply to statements that cannot be formalized in. For example, the induction axiom is a statement about subsets (and not elements) of the set of natural numbers and is not fulfilled in an ultra product of an infinite number of copies of the set of natural numbers. ${\ displaystyle L}$

The construction depends on; this leads z. Partly on very specific set theoretical questions from the theory of ultrafilter. ${\ displaystyle U}$

Ultra potencies

Often one chooses the same model for everyone and then obtains what is known as an ultra - potency of this model. One example is the hyper-real numbers . An analogous construction for the natural numbers gives a non-standard model of Peano arithmetic . ${\ displaystyle i}$

Embedding a structure in its ultra-potency is elementary .

Assuming the continuum hypothesis , it can be shown that certain ultrapotencies are isomorphic.

Ultraproduct and Ultralimes of metric spaces

If each a metric space is, one can on the Ultraproduct a pseudometric by ${\ displaystyle A_ {i}}$ ${\ displaystyle (A_ {i}, d_ {i})}$

{\ displaystyle D (a, b) = \ lim _ {U} d_ {i} (x_ {i}, y_ {i})}

,

d. i.e., an element is off such that for every neighborhood of : ${\ displaystyle D (a, b)}$ ${\ displaystyle \ left [0, \ infty \ right]}$ ${\ displaystyle V}$ ${\ displaystyle D (a, b)}$

{\ displaystyle \ left \ {i \ in I \ colon d_ {i} (x_ {i}, y_ {i}) \ in V \ right \} \ in U}

.

Choose an "observation point"; i.e., a sequence with . Then you can see the set of all equivalence classes of consequences with regard. The pseudometric assumes only finite values on this subset . ${\ displaystyle p = (p_ {i}) _ {i \ in I}}$ ${\ displaystyle p_ {i} \ in A_ {i}}$ ${\ displaystyle a = (a_ {i}) _ {i \ in I}}$ ${\ displaystyle D (a, p) <\ infty}$ ${\ displaystyle D}$

The metric space that is obtained as the quotient of this subset under the equivalence relation with the metric induced by is called the ultralimes of the sequence relative to the observation point . ${\ displaystyle \ left \ {(A_ {i}, d_ {i}) \ right \} _ {i \ in I}}$ ${\ displaystyle p = (p_ {i}) _ {i \ in I}}$ ${\ displaystyle a \ sim b \ Leftrightarrow D (a, b) = 0}$ ${\ displaystyle D}$

literature

Peter G. Hinman: Fundamentals of Mathematical Logic . AK Peters, Wellesley 2005, ISBN 1-56881-262-0 .
Wolfgang Rautenberg : Introduction to Mathematical Logic . 3. Edition. Vieweg + Teubner , Wiesbaden 2008, ISBN 978-3-8348-0578-2 .

Individual evidence

↑ Rautenberg (2008), p. 164.

[1] Rautenberg (2008), p. 164.