Hilbert and Husserl presented axiomatic arithmetic theories in different ways and proposed two different notions of u27completenessu27 for arithmetic, at the turning of the 20th Century (1900-1901). The former led to the completion axiom, the latter completion of rewriting. We look into the latter in comparison with the former. The key notion to understand the latter is the notion of definite multiplicity or manifold (Mannigfaltigkeit). We show that his notion of multiplicity is understood by means of term rewrite theory in a very coherent manner, and that his notion of u27definiteu27 multiplicity is understood as the relational web (or tissue) structure, the core part of which is a u27convergentu27 term rewrite proof structure. We examine how Husserl introduced his term rewrite theory in 1901 in the context of a controversy with Hilbert on the notion of completeness, and in the context of solving the justification problem of the use of imaginaries in mathematics, which was an important issue in the foundations of mathematics in the period.
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