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A mathematical structure used to define the real numbers in Alfred Tarski’s axioms for the real numbers.
A Tarski group is a pointed commutative invertible semigroup with a dense linear order such that , implies or , and implies .
As a result, every Tarski group is an abelian group with identity element , and a nontrivial ordered group.
Tarski’s axioms for the real numbers are as follows:
Axiom 5 says that is a commutative semigroup.
Axiom 6 with axiom 5 together say that is a commutative invertible semigroup.
Axiom 8 says that is a pointed set, which with axioms 5 and 6 imply that is an abelian group.
Axiom 1 says that has a connected relation .
Axiom 2 says that is an asymmetric relation and thus an irreflexive relation.
The fragment of Axiom 4 that only refers to singleton subsets says that is a comparison, making into a strict linear order. (the full axiom 4 is the Dedekind completeness condition).
Axiom 3 says that is a dense linear order.
Axiom 7 says that for all , implies that or , which imply that is a linearly ordered group.
Axiom 9 says that , which indicates that is not trivial.
Let us denote the Dedekind-complete Tarski group as . There is an archimedean field structure on .
is a Archimedean ordered abelian group.
Since is Dedekind-complete and a strictly ordered abelian group, is Archimedean, because the Dedekind-completion of any totally ordered abelian group with infinite elements or infinitesimals is not an abelian group, and the Dedekind-completion of any Archimedean ordered abelian group is still Archimedean.
has a complete metric
Since is strictly ordered, it is a totally ordered abelian group. As a result, there exist maximum and minimum binary functions and , with the absolute value function defined as .
Since is Dedekind-complete, Archimedean, and a totally ordered abelian group, is a metric space with respect to the absolute value and thus a Hausdorff space, and every Cauchy net in converges to a unique element of , and thus the absolute value is a complete metric on .
embeds in .
Since is an abelian group, it is a -module, and since is totally ordered, it is a torsion-free module and thus a torsion-free abelian group, which means that the integers embed in , with injective group homomorphism where and . As a result, for every integer and the affine functions are well defined in .
Since is Dedekind-complete, Archimedean, and a totally ordered abelian group, any closed interval on is compact and conencted. Since is also a complete metric space, the intermediate value theorem is satisfied for every function from a closed interval to . Because are monotonic for , and for the function is just the negation of a monotonic function, have a root? for . Thus is a divisible group and a -vector space, with an injective group homomorphism where and , and embeds in .
is a commutative ring.
Since every Cauchy net in converges to a unique element of , for every directed set and Cauchy net in the rational numbers, there exists a Cauchy net of linear functions defined as . The limit of the Cauchy net exists and is a unique function . Since every real number is the limit of a Cauchy net of rational numbers, there is an -action which takes a real number to the linear function , with being the identity function. The uncurrying of leads to a bilinear function called multiplication of the real numbers, defined on the entire domain of the binary function. Since linear functions in the function space with function composition? and the identity function is a commutative monoid, with multiplication and the multiplicative identity element is also commutative monoid, which means that is a commutative ring.
is a field
Since is a commutative ring, power series are well defined, and because all Cauchy nets converge in , all Cauchy sequences and all Cauchy power series converge in . In particular, every geometric series is a Cauchy power series and the limit of the geometric series
and
converges in the open interval . Thus let us define functions and as
Let us define the function
for and
for . These are functions which converge on the open interval for and for , and satisfy the identity for all and , and for all and , by definition of the geometric series.
The real reciprocal function is piecewise defined as
As limits preserve multiplication, for all . Thus, is a field.
Alfred Tarski, Introduction to Logic and to the Methodology of Deductive Sciences (4th edition). Oxford University Press. (1994) doi:10.2307/2180610, ISBN 978-0-19-504472-0
Ucsnay, Stefanie (Jan 2008), A Note on Tarski’s Note. The American Mathematical Monthly, Vol 115 No. 1, pg 66–68. JSTOR 27642393
Last revised on December 24, 2023 at 21:15:04. See the history of this page for a list of all contributions to it.