Tarski's Axiomatization Of The Reals
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In 1936,
Alfred Tarski Alfred Tarski (; ; born Alfred Teitelbaum;School of Mathematics and Statistics, University of St Andrews ''School of Mathematics and Statistics, University of St Andrews''. January 14, 1901 – October 26, 1983) was a Polish-American logician ...
gave an axiomatization of the
real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s and their arithmetic, consisting of only the eight
axiom An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or ...
s shown below and a mere four primitive notions: the set of reals denoted R, a
binary relation In mathematics, a binary relation associates some elements of one Set (mathematics), set called the ''domain'' with some elements of another set called the ''codomain''. Precisely, a binary relation over sets X and Y is a set of ordered pairs ...
over R, denoted by
infix An infix is an affix inserted inside a word stem (an existing word or the core of a family of words). It contrasts with '' adfix,'' a rare term for an affix attached to the outside of a stem, such as a prefix or suffix. When marking text for ...
<, a
binary operation In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two. More specifically, 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 In mathematics, an ordered field is a field together with a total ordering of its elements that is compatible with the field operations. Basic examples of ordered fields are the rational numbers and the real numbers, both with their standard ord ...
; 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 field In mathematics, a real closed field is a field F that has the same first-order properties as the field of real numbers. Some examples are the field of real numbers, the field of real algebraic numbers, and the field of hyperreal numbers. Def ...
s.


The axioms


Axioms of order (primitives: R, <)

;Axiom 1 :If ''x'' < ''y'', then not ''y'' < ''x''. :: hat is, "<" is an asymmetric relation. This implies that "<" is reflexive relation">irreflexive, i.e., for all ''x'', not ''x'' < ''x''.">asymmetric relation">hat is, "<" is an asymmetric relation. This implies that "<" is reflexive relation">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 In mathematics, the associative property is a property of some binary operations that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a Validity (logic), valid rule of replaceme ...
and commutativity.] ;Axiom 5 :For all ''x'', ''y'', there exists a ''z'' such that ''x'' + ''z'' = ''y''. :: his allows subtraction and also gives a 0.;Axiom 6 :If ''x'' + ''y'' < ''z'' + ''w'', then ''x'' < ''z'' or ''y'' < ''w''. :: his is the contrapositive of a standard axiom for ordered groups.">contrapositive.html" ;"title="his is the contrapositive">his 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. The axioms then imply that R is a linearly ordered group, linearly ordered abelian group under addition with distinguished positive element 1, and that this group is Dedekind-complete,
divisible In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a '' multiple'' of m. An integer n is divisible or evenly divisibl ...
, and Archimedean. Tarski never proved that these axioms and primitives imply the existence of a
binary operation In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two. More specifically, a binary operation ...
called multiplication that has the expected properties, so that R becomes a complete
ordered field In mathematics, an ordered field is a field together with a total ordering of its elements that is compatible with the field operations. Basic examples of ordered fields are the rational numbers and the real numbers, both with their standard ord ...
under addition and multiplication. It is possible to define this multiplication operation by considering certain order-preserving homomorphisms of the ordered group (R,+,<). See in particular Section 4.


References

{{Real numbers Real numbers Ordered groups Mathematical axioms Formal theories of arithmetic