Tarski's axiomatization of the reals
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.
The axioms
Axioms of order (primitives: R, <)
- 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.]
- 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.]
Axioms of addition (primitives: R, <, +)
- 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.]
Axioms for 1 (primitives: R, <, +, 1)
- Axiom 7
- 1 ∈ R.
- Axiom 8
- 1 < 1 + 1.
Discussion
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]
References
- ↑ 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. https://archive.org/search?query=Tarski+Introduction+to+Logic+and+to+the+Methodology+of+Deductive+Sciences.
- ↑ Ucsnay, Stefanie (Jan 2008). "A Note on Tarski's Note". The American Mathematical Monthly 115 (1): 66–68.
- ↑ Arthan, Rob D. (2001). "An Irrational Construction of ℝ from ℤ" (in en). Theorem Proving in Higher Order Logics. Lecture Notes in Computer Science (Berlin, Heidelberg: Springer): 43–58. doi:10.1007/3-540-44755-5_5. https://www.lemma-one.com/papers/47.pdf. Section 4
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