Open Least Element Principle and Bounded Query Computation
We show that elementaryarithmetic formulated in the language with a free function symbol f and the least element principle for open formulas (where we assume that the symbols for all elementary functions are included in the language) does not prove the least element principle for bounded formulas in the same language. A related result is that composition and anyn umber of unnested applications of bounded minimum operator are, in general, insufficient to generate the elementary closure of a function, even if all elementaryfunctions are available. Thus, unnested bounded minimum operator is weaker than unnested bounded recursion.
KeywordsElementary Function Function Symbol Proof Theory Simple Program Peano Arithmetic
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