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In 1936, Alfred Tarski gave an axiomatization of the real numbers and their arithmetic, consisting of only the eight axioms shown below and a mere four primitive notions:^{[1]} the set of reals denoted **R**, a binary relation over **R**, denoted by infix <, a binary operation of addition over **R**, denoted by infix +, and the constant 1.

Tarski's axiomatization, which is a second-order theory, can be seen as a version of the more usual definition of real numbers as the unique Dedekind-complete ordered field; it is however made much more concise by avoiding multiplication altogether and using unorthodox variants of standard algebraic axioms and other subtle tricks. Tarski did not supply a proof that his axioms are sufficient or a definition for the multiplication of real numbers in his system.

Tarski also studied the first-order theory of the structure (**R**, +, ·, <), leading to a set of axioms for this theory and to the concept of real closed fields.

- Axiom 1
- If
*x*<*y*, then not*y*<*x*.- [That is, "<" is an asymmetric relation. This implies that "<" is irreflexive, i.e., for all
*x*, not*x*<*x*.]

- [That is, "<" is an asymmetric relation. This implies that "<" is irreflexive, i.e., for all

- Axiom 2
- If
*x*<*z*, there exists a*y*such that*x*<*y*and*y*<*z*. - Axiom 3
- For all subsets
*X*,*Y*⊆**R**, if for all*x*∈*X*and*y*∈*Y*,*x*<*y*, then there exists a*z*such that for all*x*∈*X*and*y*∈*Y*, if*x*≠*z*and*y*≠*z*, then*x*<*z*and*z*<*y*.- [In other words, "<" is Dedekind-complete, or informally: "If a set of reals
*X*precedes another set of reals*Y*, then there exists at least one real number*z*separating the two sets." - This is a second-order axiom as it refers to sets and not just elements.]

- [In other words, "<" is Dedekind-complete, or informally: "If a set of reals

- Axiom 4
*x*+ (*y*+*z*) = (*x*+*z*) +*y*.- [Note that this is an unorthodox mixture of associativity and commutativity.]

- Axiom 5
- For all
*x*,*y*, there exists a*z*such that*x*+*z*=*y*.- [This allows subtraction and also gives a 0.]

- Axiom 6
- If
*x*+*y*<*z*+*w*, then*x*<*z*or*y*<*w*.- [This is the contrapositive of a standard axiom for ordered groups.]

- Axiom 7
- 1 ∈
**R**. - Axiom 8
- 1 < 1 + 1.

Tarski stated, without proof, that these axioms turn the relation < into a total ordering. The missing component was supplied in 2008 by Stefanie Ucsnay.^{[2]}

The axioms then imply that **R** is a linearly ordered abelian group under addition with distinguished positive element 1**,** and that this group is Dedekind-complete, divisible, and Archimedean.

Tarski never proved that these axioms and primitives imply the existence of a binary operation called multiplication that has the expected properties, so that **R** becomes a complete ordered field under addition and multiplication. It is possible to define this multiplication operation by considering certain order-preserving homomorphisms of the ordered group (**R**,+,<).^{[3]}

**^**Tarski, Alfred (24 March 1994).*Introduction to Logic and to the Methodology of Deductive Sciences*(4 ed.). Oxford University Press. ISBN 978-0-19-504472-0.**^**Ucsnay, Stefanie (Jan 2008). "A Note on Tarski's Note".*The American Mathematical Monthly*.**115**(1): 66–68. JSTOR 27642393.**^**Arthan, Rob D. (2001). "An Irrational Construction of ℝ from ℤ" (PDF).*Theorem Proving in Higher Order Logics*. Lecture Notes in Computer Science. Berlin, Heidelberg: Springer: 43–58. doi:10.1007/3-540-44755-5_5. Section 4