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In this paper, we study the metamathematics of consistent arithmetical theories $T$ (containing $\textsf {I}\varSigma _{1}$); we investigate numerical properties based on proof predicates that depend on numerations of the axioms. Numeral Completeness. For every true (in $\mathbb {N}$) sentence $\vec {Q}\vec {x}.\varphi (\vec {x})$, with $\varphi (\vec {x})$ a $\varSigma _{1}(\textsf {I}\varSigma _1)$-formula, there is a numeration $\tau $ of the axioms of $T$ such that $\textsf {I}\varSigma _1\vdash \vec {Q}\vec {x}. \texttt {Pr}_{\tau }(\ulcorner \varphi (\overset {\text{.} }{\vec {x}})\urcorner )$, where $\texttt {Pr}_{\tau }$ is the provability predicate for the numeration $\tau $. Numeral Consistency. If $T$ is consistent, there is a $\varSigma _{1}(\textsf {I}\varSigma _1)$-numeration $\tau $ of the axioms of $\textsf {I}\varSigma _{1}$ such that $\textsf {I}\varSigma _1\vdash \forall\, x. \texttt {Pr}_{\tau }(\ulcorner \neg \textit {Prf}(\ulcorner \perp \urcorner , \overset {\text{.}}{x})\urcorner )$, where $\textit {Prf}(x,y)$ denotes a $\varDelta _{1}(\textsf {I}\varSigma _1)$-definition of ‘$y$ is a $T$-proof of $x$’. Finitist consistency is addressed by generalizing a result of Artemov: Partial finitism. If $T$ is consistent, there is a primitive recursive function $f$ such that, for all $n\in \mathbb {N}$, $f(n)$ is the code of an $\textsf {I}\varSigma _{1}$-proof of $\neg\, \textit{Prf}(\ulcorner \perp \urcorner ,\overline {n})$. These results are not in conflict with Gödel’s Incompleteness Theorems. Rather, they allow to extend their usual interpretation and show a deep connection to reflections in Hilbert’s last papers of 1931.

The use of diagonalization reasonings is transversal to the Mathematical practise. Since Cantor, diagonalization reasonings are used in a great variety of areas that vanish from Topology to Logic. The objective of the present thesis was to study the formal aspects of diagonalization in Logic and more generally in the Mathematical practise. The main goal was to find a formal theory that is behind important diagonalization phenomena in Mathematics.We started by the study of diagonalization in theories of Arithmetic: Diagonalization Lemma and self-reference. In particular, we argued that important properties related to self-reference are not decidable, and we argued that the diagonalization of formulas is substantially different from the diagonalization of terms, more precisely, the Diagonal Lemma cannot prove the Strong Diagonal Lemma.We studied in detail Yablo's Paradox. By presenting a minimal theory to express Yablo's Paradox, we argued that Yablo's Paradox is not a paradox about Arithmetic. From that theory and with the help of some notions of Temporal Logic, we claimed that Yablo's Paradox is self-referential.After that, we studied several paradoxes -the Liar, Russell's Paradox, and Curry's Paradox-and Löb's Theorem, and we presented a common origin to those paradoxes and theorem: Curry System. Curry Systems were studied in detail and a consistency result for specific conditions was offered. Finally, we presented a general theory of diagonalization, we exemplified the formal use of the theory, and we studied some results of Mathematics using that general theory.All the work that we present on this thesis is original. The fourth chapter gave rise to a paper by the author ([SK17]) and the third chapter will also give rise in a short period of time to a paper. Regarding the other chapters, the author, together with his Advisors, is also preparing a paper.

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