Past exam of the model theory and Gödel's incompleteness theorems course of the University of Oxford / 2007 / A / 2 / c

A logical formula $D(\vec{x},y)$ weakly defines a total numerical function $f$ in $S$ if, for every standard input tuple $\vec{n}$ and $m=f(\vec{n})$,Strong definability of a numerical function additionally requires the internal uniqueness statementTogether with the positive instance, this is equivalent to proving $\forall y(D(\overline{\stackrel{⃗}{n}},y)\leftrightarrow y=\bar{m})$ for each standard input. The universal quantifier is the extra strength: it covers nonstandard potential outputs in first-order models of $S$. These are the weak and strong function conventions used here. ^[ref]

Suppose the graph of the total function is Sigma-1. Its complement is also Sigma-1, sinceReplace the graph assertion on the right by its Sigma-1 definition; existential quantifiers can be collected into one block. Bounded completeness of an arithmetic theory supplies numeralwise stage enumerators for each of these two disjoint complementary sets. Apply (b) to them. Its separating logical formula is provable exactly at the required graph tuples in the positive direction and refutable at all the other standard tuples, proving weak definability of a numerical function.

Using the graph logical formula alone would only supply positive instances from true existential witnesses; the separation construction is what supplies the negative-instance formal proofs without assuming Sigma-1 soundness or completeness for false Sigma-1 statements.