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Q− is a weaker variant of Robinson arithmetic Q in which addition and multiplication are partial functions, i.e. ternary relations that are graphs of possibly non-total functions. We show that Q is interpretable in Q−. This gives an alternative answer to a question of A. Grzegorczyk whether Q− is essentially undecidable.
We prove that a variant of Robinson arithmetic Q with non-total operations is interpretable in the theory of concatenation TC introduced by A. Grzegorczyk. Since Q is known to be interpretable in that non-total variant, our result gives a positive answer to the problem whether Q is interpretable in TC. An immediate consequence is essential undecidability
The recursion theoretic limit lemma, saying that each function with a Σn+2 graph is a limit of certain function with a ∆n+1 graph, is provable in BΣn+1.