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In this paper, we are concerned with the arithmetical definability of certain notions of integers and rationals in terms of other notions. The results derived will be applied to obtain a negative solution of corresponding decision problems.In Section 1, we show that addition of positive integers can be defined arithmetically in terms of multiplication and the unary operation of successor S (where Sa = a + 1). Also, it is shown that both addition and multiplication can be defined arithmetically in terms of successor and the relation of divisibility | (where x|y means x divides y).

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A formal, metal-free, [2 + 2 + 2] cycloaddition strategy is described based on a cascade of two pericyclic processes. The first step involves an intramolecular propargylic ene reaction of a 1,6-diyne to generate a vinylallene, which then reacts in an inter-or intramolecular Diels-Alder reaction with an alkenyl or alkynyl dienophile. Reactions involving unsymmetrical alkenyl and alkynyl dienophiles proceed with good to excellent regioselectivity, and the diastereoselectivity in the Diels-Alder step is also high, with endo cycloadducts produced as the exclusive products of the reaction. In the case of alkynyl dienophiles, [4 + 2] cycloaddition initially generates an isotoluene-type intermediate that isomerizes to the isolated aromatic product upon exposure to a catalytic amount of DBU at room temperature. The mechanism of several earlier fully intramolecular related transformations have been shown to involve an analogous process rather than the diradical-mediated pathways proposed previously.