 for all \(\varphi(\vec{x}) \in \Sigma^0_n\) and \(\vec{m} \in \mathbb{N}^k\) (and where we have also defined our Gödel numbering to agree with the indexation of \(\Sigma^0_n\)-formulas introduced above). Now consider the \(\Pi^0_n\)-formula \(\lambda(x) = \neg \sigma_{2,n}(x,x) \in \Pi^0_n\) and let \(A\) be the set defined by \(\lambda(x)\). A standard diagonal argument shows that \(A\) cannot be \(\Sigma^0_n\)-definable and also that if \(\ulcorner \sigma_{2,n}(x,x) \urcorner = l\) in the enumeration of \(\Sigma^0_n\)-formulas then \(\neg \sigma_{2,n}(l,l)\) is a \(\Pi^0_n\)-formula which cannot be provably equivalent to a \(\Sigma^0_k\)-formula (see, e.g., Kaye 1991: ch. 9.3). Thus as Kleene (1943, 64) observed, part of the significance of the Hierarchy Theorem is that it illustrates how the Liar Paradox may be formalized to yield a stratified form of Tarski’s Theorem on the undefinability of truth  (see the entry on self-reference).