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  "body": "\n## Problem\n\nConsider the signature with a constant $1$ and binary operations $+$, $\\cdot$ and exponentiation $x^y$, interpreted in the positive integers.\nLet $\\mathrm{HSI}$ be the following eleven [[identities|PKD440G2XW]]:\n\n1. $x + y \\approx y + x$ and $(x + y) + z \\approx x + (y + z)$;\n2. $x \\cdot 1 \\approx x$, $x \\cdot y \\approx y \\cdot x$ and $(x \\cdot y) \\cdot z \\approx x \\cdot (y \\cdot z)$;\n3. $x \\cdot (y + z) \\approx x \\cdot y + x \\cdot z$;\n4. $1^x \\approx 1$, $x^1 \\approx x$, $x^{y + z} \\approx x^y \\cdot x^z$, $(x \\cdot y)^z \\approx x^z \\cdot y^z$ and $(x^y)^z \\approx x^{y \\cdot z}$.\n\nThe problem asks whether every identity in this signature that holds in the positive integers can be derived from $\\mathrm{HSI}$ in [[equational logic|72S3YSQFZ7]].\n\n## Remarks\n\nThe answer is no.\nAlex Wilkie found, around 1980, an identity true in the positive integers that cannot be derived from $\\mathrm{HSI}$.\nLater, R. Gurevič showed that the identities true in the positive integers with $1$, $+$, $\\cdot$ and exponentiation have no finite [[equational basis|9Q008GMZSW]] at all.\nEvery identity without exponentiation that holds in the positive integers is derivable from the identities in items 1 to 3.\nIndeed, both sides expand to the same polynomial; this is why the question was natural.\n\n## Explanations\n\n### L1 human\n\nSchool algebra teaches a short list of rules for adding, multiplying and taking powers.\nTarski asked whether these rules are enough to prove every true equation built from $1$, $+$, $\\times$ and powers.\nSurprisingly, they are not: there are true equations that the school rules cannot prove.\nEven worse, no finite list of rules can do the job.\n",
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