A carregar...
Publicação
Diagonalization in Formal Mathematics
| Nome: | Descrição: | Tamanho: | Formato: | |
|---|---|---|---|---|
| 974.51 KB | Adobe PDF |
Orientador(es)
Resumo(s)
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.
Descrição
Palavras-chave
Diagonalization General Theory of Diagonalization Self-Reference Diagonalization Lemma Strong Diagonalization Lemma Paradox