nLab
Tarski group

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Context

Group Theory

Analysis

Contents

Idea

A mathematical structure used to define the real numbers in Alfred Tarski‘s axioms for the real numbers.

Definition

A Tarski group is a pointed commutative invertible semigroup (G,+,,1)(G, +, -, 1) with a dense linear order <\lt such that 1<1+11 \lt 1 + 1, a<b+c+aa \lt b + c + a implies a<b+aa \lt b + a or a<c+aa \lt c + a, and a<ba \lt b implies c+a<c+bc + a \lt c + b.

As a result, every Tarski group is an abelian group with identity element 0110 \coloneqq 1 - 1, and a nontrivial ordered group.

Tarski’s axioms for the real numbers

Tarski’s axioms for the real numbers are as follows:

The archimedean field structure on the Dedekind-complete Tarski group

Let us denote the Dedekind-complete Tarski group as \mathbb{R}. There is an archimedean field structure on \mathbb{R}.

Proposition

\mathbb{R} is a Archimedean ordered abelian group.

Proof

Since \mathbb{R} is Dedekind-complete and a strictly ordered abelian group, \mathbb{R} 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.

Proposition

\mathbb{R} has a complete metric

Proof

Since \mathbb{R} is strictly ordered, it is a totally ordered abelian group. As a result, there exist maximum and minimum binary functions max:×\max:\mathbb{R} \times \mathbb{R} \to \mathbb{R} and min:×\min:\mathbb{R} \times \mathbb{R} \to \mathbb{R}, with the absolute value function defined as |x|=max(x,x)\vert x \vert = \max(x, -x).

Since \mathbb{R} is Dedekind-complete, Archimedean, and a totally ordered abelian group, \mathbb{R} is a metric space with respect to the absolute value |x|\vert x \vert and thus a Hausdorff space, and every Cauchy net in \mathbb{R} converges to a unique element of \mathbb{R}, and thus the absolute value |x|\vert x \vert is a complete metric on \mathbb{R}.

Proposition

\mathbb{Q} embeds in \mathbb{R}.

Proof

Since \mathbb{R} is an abelian group, it is a \mathbb{Z}-module, and since \mathbb{R} is totally ordered, it is a torsion-free module and thus a torsion-free abelian group, which means that the integers \mathbb{Z} embed in \mathbb{R}, with injective group homomorphism f:f:\mathbb{Z} \to \mathbb{R} where f(0)=0f(0) = 0 and f(1)=1f(1) = 1. As a result, for every integer aa \in \mathbb{Z} and bb \in \mathbb{Z} the affine functions xax+bx \mapsto a x + b are well defined in \mathbb{R}.

Since \mathbb{R} is Dedekind-complete, Archimedean, and a totally ordered abelian group, any closed interval [a,b][a, b] on \mathbb{R} is compact and conencted. Since \mathbb{R} is also a complete metric space, the intermediate value theorem is satisfied for every function from a closed interval [a,b][a, b] to \mathbb{R}. Because xax+bx \mapsto a x + b are monotonic for a>0a \gt 0, and for a<0a \lt 0 the function is just the negation of a monotonic function, xax+bx \mapsto a x + b have a root? for |a|>0\vert a \vert \gt 0. Thus \mathbb{R} is a divisible group and a \mathbb{Q}-vector space, with an injective group homomorphism f:f:\mathbb{Q} \to \mathbb{R} where f(0)=0f(0) = 0 and f(1)=1f(1) = 1, and \mathbb{Q} embeds in \mathbb{R}.

Proposition

\mathbb{R} is a commutative ring.

Proof

Since every Cauchy net in \mathbb{R} converges to a unique element of \mathbb{R}, for every directed set AA and Cauchy net (a i) iA(a_i)_{i \in A} in the rational numbers, there exists a Cauchy net of linear functions (f i) iA(f_i)_{i \in A} defined as f i(x)=a ixf_i(x) = a_i x. The limit of the Cauchy net lim iA(f i) i\lim_{i \in A} (f_i)_i exists and is a unique function g(x)=lim iA(a i) ixg(x) = \lim_{i \in A} (a_i)_i x. Since every real number is the limit of a Cauchy net of rational numbers, there is an \mathbb{R}-action μ:()\mu:\mathbb{R} \to (\mathbb{R} \to \mathbb{R}) which takes a real number rr to the linear function xrxx \mapsto r x, with α(1)=xx\alpha(1) = x \mapsto x being the identity function. The uncurrying of α\alpha leads to a bilinear function ()():×(-)(-):\mathbb{R} \times \mathbb{R} \to \mathbb{R} 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, \mathbb{R} with multiplication and the multiplicative identity element 11 is also commutative monoid, which means that \mathbb{R} is a commutative ring.

Proposition

\mathbb{R} is a field

Proof

Since \mathbb{R} is a commutative ring, power series are well defined, and because all Cauchy nets converge in \mathbb{R}, all Cauchy sequences and all Cauchy power series converge in \mathbb{R}. In particular, every geometric series is a Cauchy power series and the limit of the geometric series

n=0 x n\sum_{n=0}^\infty x^n

and

n=0 (1) nx n\sum_{n=0}^\infty (-1)^n x^n

converges in the open interval (1,1)(-1, 1). Thus let us define functions f:(1,1)f:(-1, 1) \to \mathbb{R} and g:(1,1)g:(-1, 1) \to \mathbb{R} as

f(x):= n=0 x nf(x) := \sum_{n=0}^\infty x^n
g(x):= n=0 (1) nx ng(x) := \sum_{n=0}^\infty (-1)^n x^n

Let us define the function

h(x,a):=(a) n=0 a n(x+f(a+1)) nh(x, a) := (-a) \sum_{n=0}^\infty a^n (x + f(a + 1))^n

for a<0a \lt 0 and

k(x,a):=a n=0 (a) n(x+g(a1)) nk(x, a) := a \sum_{n=0}^\infty (-a)^n (x + g(a - 1))^n

for a>0a \gt 0. These are functions which converge on the open interval (1/a,0)(1/a, 0)for h(x,a)h(x, a) and (0,1/a)(0,1/a) for k(x,a)k(x, a), and satisfy the identity h(x,a)x=1h(x, a) x = 1 for all a<0a \lt 0 and x(1/a,0)x \in (1/a, 0), and k(x,a)x=1k(x, a) x = 1 for all a>0a \gt 0 and x(0,1/a)x \in (0,1/a), by definition of the geometric series.

The reciprocal is piecewise defined as

1x={lim a0 h(x,a) ifx<0 lim a0 +k(x,a) ifx>0 \frac{1}{x} = \begin{cases} \lim_{a \to 0^-} h(x, a) & \mathrm{if}\ x \lt 0 \\ \lim_{a \to 0^+} k(x, a) & \mathrm{if}\ x \gt 0 \\ \end{cases}

As limits preserve multiplication, 1xx=1\frac{1}{x} x = 1 for all xx \in \mathbb{R}. Thus, \mathbb{R} is a field.

See also

 References

  • 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 May 19, 2022 at 10:06:03. See the history of this page for a list of all contributions to it.