Footnotes
...1981.
Date of the first publication (in Russian). 
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...sets.
This is usually known as the category of finite sets or finite ordinals. 
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...enough,
Actually, he conjectured something stronger, namely that 7#7 is complete for 9#9, the natural numbers equipped with a family of generalized binary   operators 10#10 that extend the usual sum +, product 11#11 and   exponentiation 12#12 operators. In   Tarski's definition, 13#13 is the sum, 14#14 is multiplication, 15#15 is exponentiation (for the other cases see for example [Rog88]). 
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...axioms:
He also showed that there are no nontrivial   equations for 9#9 if n>2. 
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...Gur85].
For the curious, here is how it works: from the size of the   equation that has to be verified, it is possible to derive an upper bound; if the two sides coincide for all values of the variables up to this upper bound, then they coincide everywhere. 
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Roberto Di Cosmo
Mon Aug 30 11:21:44 MET DST 1999