View source for Organization:Tarski's high school algebra problem

{{Short description|Mathematical problem}}
In [[Mathematical logic|mathematical logic]], '''Tarski's high school algebra problem''' was a question posed by [[Biography:Alfred Tarski|Alfred Tarski]].  It asks whether there are [[Identity (mathematics)|identities]] involving [[Addition|addition]], [[Multiplication|multiplication]], and [[Exponentiation|exponentiation]] over the positive [[Integer|integer]]s that cannot be [[Mathematical proof|proved]] using eleven [[Axiom|axiom]]s about these operations that are taught in high-school-level [[Mathematics|mathematics]].  The question was solved in 1980 by Alex Wilkie, who showed that such unprovable identities do exist.

==Statement of the problem==

Tarski considered the following eleven axioms about addition ('+'), multiplication ('·'), and exponentiation to be standard axioms taught in high school:
# ''x''&nbsp;+&nbsp;''y''&nbsp;=&nbsp;''y''&nbsp;+&nbsp;''x''
# (''x''&nbsp;+&nbsp;''y'')&nbsp;+&nbsp;''z''&nbsp;=&nbsp;''x''&nbsp;+&nbsp;(''y''&nbsp;+&nbsp;''z'')
# ''x''&nbsp;·&nbsp;1&nbsp;=&nbsp;''x''
# ''x''&nbsp;·&nbsp;''y''&nbsp;=&nbsp;''y''&nbsp;·&nbsp;''x''
# (''x''&nbsp;·&nbsp;''y'')&nbsp;·&nbsp;''z''&nbsp;=&nbsp;''x''&nbsp;·&nbsp;(''y''&nbsp;·&nbsp;''z'')
# ''x''&nbsp;·&nbsp;(''y''&nbsp;+&nbsp;''z'')&nbsp;=&nbsp;''x''&nbsp;·&nbsp;''y''&nbsp;+&nbsp;''x''&nbsp;·''z''
# 1<sup>''x''</sup>&nbsp;=&thinsp;1
# ''x''<sup>1</sup>&thinsp;=&nbsp;''x''
# ''x''<sup>''y''&thinsp;+&thinsp;''z''</sup>&nbsp;=&nbsp;''x''<sup>''y''</sup>&nbsp;·&nbsp;''x''<sup>''z''</sup>
# (''x''&thinsp;·&thinsp;''y'')<sup>''z''</sup>&nbsp;=&nbsp;''x''<sup>''z''</sup>&nbsp;·&nbsp;''y''<sup>''z''</sup>
# (''x''<sup>''y''</sup>)<sup>''z''</sup>&nbsp;=&nbsp;''x''<sup>''y''&thinsp;·&thinsp;''z''</sup> 

These eleven axioms, sometimes called the ''high school identities'',<ref name="BurrisLee">Stanley Burris, Simon Lee, ''Tarski's high school identities'', [[American Mathematical Monthly]], '''100''', (1993), no.3, pp.&nbsp;231&ndash;236.</ref> are related to the axioms of a bicartesian closed category or an [[Exponential field|exponential ring]].<ref>Strictly speaking an exponential ring has an exponential function ''E'' that takes each element ''x'' to something that acts like ''a''<sup>''x''</sup> for a fixed number ''a''.  But a slight generalisation gives the axioms for the binary operation of exponentiation. The lack of axioms about additive inverses means the axioms would have described an exponential [[Semiring|commutative semiring]], except there are no axioms about additive identities in Tarski's axioms either. However, some authors use the term rig to mean a semiring with additive identities, and reserve the term semiring for the general case not necessarily having additive identities. To those authors, the axioms do describe an exponential commutative semiring.</ref> Tarski's problem then becomes: are there identities involving only addition, multiplication, and exponentiation, that are true for all positive integers, but that cannot be [[Mathematical proof|proved]] using only the axioms 1&ndash;11?

==Example of a provable identity==

Since the axioms seem to list all the basic facts about the operations in question, it is not immediately obvious that there should be anything provably true one can state using only the three operations, but cannot prove with the axioms.  However, proving seemingly innocuous statements can require long proofs using only the above eleven axioms.  Consider the following proof that <math>(x + 1)^2 = x^2 + 2 \cdot x + 1:</math>

<math display=block>\begin{align} 
(x+1)^2 &= (x+1)^{1+1}
\\ &= (x+1)^1 \cdot (x+1)^1 &&\text{by 9.}
\\ &= (x+1) \cdot (x+1)  &&\text{by two applications of 8.}
\\ &= (x+1) \cdot x + (x+1) \cdot 1 &&\text{by 6.}
\\ &= x \cdot (x+1) + (x + 1) &&\text{by 4. and 3.}
\\ &= (x \cdot x + x \cdot 1) + (x \cdot 1 + 1) &&\text{by 6. and 3.}
\\ &= x \cdot x + (x \cdot 1 + x \cdot 1) + 1 &&\text{by two applications of 2.}
\\ &= x^1 \cdot x^1 + x \cdot (1+1) + 1 &&\text{by 6. and two applications of 8.}
\\ &= x^{1+1} + x \cdot 2 + 1 &&\text{by 9.}
\\ &= x^2 + 2 \cdot x + 1 &&\text{by 4.}
\end{align}</math>

Strictly we should not write sums of more than two terms without brackets, and therefore a completely [[Formal proof|formal proof]] would prove the identity <math>(x + 1)^2 = \left(x^2 + 2 \cdot x\right) + 1</math> (or <math>(x + 1)^2 = x^2 + (2 \cdot x + 1)</math>) and would have an extra set of brackets in each line from <math>x \cdot x + (x \cdot 1 + x \cdot 1) + 1</math> onwards.

The length of proofs is not an issue; proofs of similar identities to that above for things like <math>(x + y)^{100}</math> would take a lot of lines, but would really involve little more than the above proof.

==History of the problem==

The list of eleven axioms can be found explicitly written down in the works of [[Biography:Richard Dedekind|Richard Dedekind]],<ref>Richard Dedekind, ''Was sind und was sollen die Zahlen?'', 8te unveränderte Aufl. Friedr. Vieweg & Sohn, Braunschweig (1960). English translation: ''What are numbers and what should they be?'' Revised, edited, and translated from the German by H. A. Pogorzelski, W.
Ryan, and W. Snyder, RIM Monographs in Mathematics, Research Institute for Mathematics, (1995).</ref> although they were obviously known and used by mathematicians long before then.  Dedekind was the first, though, who seemed to be asking if these axioms were somehow sufficient to tell us everything we could want to know about the integers.  The question was put on a firm footing as a problem in [[Mathematical logic|logic]] and [[Model theory|model theory]] sometime in the 1960s by Alfred Tarski,<ref name="BurrisLee"/><ref name="Gurevic">R. Gurevič, ''Equational theory of positive numbers with exponentiation'', Proc. Amer. Math. Soc. '''94''' no.1, (1985), pp.135&ndash;141.</ref> and by the 1980s it had become known as Tarski's high school algebra problem.

==Solution==

In 1980 Alex Wilkie proved that not every identity in question can be proved using the axioms above.<ref>A.J. Wilkie, ''On exponentiation &ndash; a solution to Tarski's high school algebra problem'', Connections between model theory and algebraic and analytic geometry, Quad. Mat., '''6''', Dept. Math., Seconda Univ. Napoli, Caserta, (2000), pp.&nbsp;107&ndash;129.</ref> He did this by explicitly finding such an identity.  By introducing new function symbols corresponding to [[Polynomial|polynomial]]s that map positive numbers to positive numbers he proved this identity, and showed that these functions together with the eleven axioms above were both sufficient and necessary to prove it.  The identity in question is
<math display=block>\begin{align}
    & \left((1+x)^y + (1+x+x^2)^y\right)^x \cdot \left((1+x^3)^x + (1+x^2+x^4)^x\right)^y \\
={} & \left((1+x)^x + (1+x+x^2)^x\right)^y \cdot \left((1+x^3)^y + (1+x^2+x^4)^y\right)^x.
\end{align}</math>
This identity is usually denoted <math>W(x, y)</math> and is true for all positive integers <math>x</math> and <math>y,</math> as can be seen by factoring <math>(1-x+x^2)^{xy}</math> out of the second factor on each side; yet it cannot be proved true using the eleven high school axioms.

Intuitively, the identity cannot be proved because the high school axioms can't be used to discuss the polynomial <math>1-x+x^2.</math>  Reasoning about that polynomial and the subterm <math>-x</math> requires a concept of [[Additive inverse|negation]] or [[Subtraction|subtraction]], and these are not present in the high school axioms.  Lacking this, it is then impossible to use the axioms to manipulate the polynomial and prove true properties about it.  Wilkie's results from his paper show, in more formal language, that the "only gap" in the high school axioms is the inability to manipulate polynomials with negative [[Coefficient|coefficient]]s.

R. Gurevič showed in 1988 that there is no finite axiomatization for the valid equations for the positive [[Natural number|natural number]]s with 1, addition, multiplication, and exponentiation.<ref name=Gurevic2>R. Gurevič, ''Equational theory of positive numbers with exponentiation is not finitely axiomatizable'', Annals of Pure and Applied Logic, 49:1–30, 1990.</ref><ref name="Fiore">Fiore, Cosmo, and Balat.  Remarks on Isomorphisms in Typed Lambda Calculi with Empty and Sum Types [http://www.dicosmo.org/Papers/lics02.pdf]</ref>

==Generalisations==

Wilkie proved that there are statements about the positive integers that cannot be proved using the eleven axioms above and showed what extra information is needed before such statements can be proved.  Using [[Nevanlinna theory]] it has also been proved that if one restricts the kinds of exponential one takes then the above eleven axioms are sufficient to prove every true statement.<ref>C. Ward Henson, Lee A. Rubel, ''Some applications of Nevanlinna theory to mathematical logic: Identities of exponential functions'', [[Organization:Transactions of the American Mathematical Society|Transactions of the American Mathematical Society]], vol.282 '''1''', (1984), pp.&nbsp;1&ndash;32.</ref>

Another problem stemming from Wilkie's result, which remains open, is that which asks what the smallest [[Algebra (ring theory)|algebra]] is for which <math>W(x, y)</math> is not true but the eleven axioms above are.  In 1985 an algebra with 59 elements was found that satisfied the axioms but for which <math>W(x, y)</math> was false.<ref name="Gurevic"/>  Smaller such algebras have since been found, and it is now known that the smallest such one must have either 11 or 12 elements.<ref>Jian Zhang, ''Computer search for counterexamples to Wilkie's identity'', Automated Deduction – CADE-20, [[Company:Springer Science+Business Media|Springer]] (2005), pp.&nbsp;441&ndash;451, {{doi|10.1007/11532231_32}}.</ref>

==See also==

* {{annotated link|Elementary function}}
* {{annotated link|Elementary function arithmetic}}
* {{annotated link|Liouville's theorem (differential algebra)}}
* {{annotated link|Nonelementary integral}}
* {{annotated link|Richardson's theorem}}

==Notes==

{{reflist}}

==References==

* Stanley N. Burris, [[Biography:Karen Yeats|Karen A. Yeats]], ''The saga of the high school identities'', [[Algebra Universalis]] '''52''' no.2&ndash;3, (2004), pp.&nbsp;325&ndash;342, {{MathSciNet | id = 2161657}}.

{{Mathematical logic}}
{{DEFAULTSORT:Tarski's High School Algebra Problem}}

[[Category:Theorems in the foundations of mathematics]]
[[Category:Universal algebra]]

{{Sourceattribution|Tarski's high school algebra problem}}