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In mathematical logic, Tarski's high school algebra problem was a question posed by Alfred Tarski. It asks whether there are identities involving addition, multiplication, and exponentiation over the positive integers that cannot be proved using eleven axioms about these operations that are taught in high-school-level 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'' + ''y'' = ''y'' + ''x'' # (''x'' + ''y'') + ''z'' = ''x'' + (''y'' + ''z'') # ''x'' · 1 = ''x'' # ''x'' · ''y'' = ''y'' · ''x'' # (''x'' · ''y'') · ''z'' = ''x'' · (''y'' · ''z'') # ''x'' · (''y'' + ''z'') = ''x'' · ''y'' + ''x'' ·''z'' # 1''x'' = 1 # ''x''1 = ''x'' # ''x''''y'' + ''z'' = ''x''''y'' · ''x''''z'' # (''x'' · ''y'')''z'' = ''x''''z'' · ''y''''z'' # (''x''''y'')''z'' = ''x''''y'' · ''z''. These eleven axioms, sometimes called the high school identities,〔Stanley Burris, Simon Lee, ''Tarski's high school identities'', American Mathematical Monthly, 100, (1993), no.3, pp.231–236.〕 are related to the axioms of an exponential ring.〔Strictly speaking an exponential ring has an exponential function ''E'' that takes each element ''x'' to something that acts like ''a''''x'' for a fixed number ''a''. But a slight generalisation gives the axioms listed here. The lack of axioms about additive inverses means the axioms actually describe an exponential commutative semiring.〕 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 proved using only the axioms 1–11? 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Tarski's high school algebra problem」の詳細全文を読む スポンサード リンク
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