Incompleteness theorems (1)

1 Março 2023, 15:30 • Ricardo Santos

Introductory section to Gödel’s Incompleteness theorems. The following topics have been covered: (i) Review of the notion of conservative extension; (ii) Hilbert’s programme; (iii) primitive recursive functions; (iv) Primitive Recursive Arithmetic; (v) Distinction between incompleteness and essential incompleteness.

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Hilbert’s programme

27 Fevereiro 2023, 15:30 • Ricardo Santos

We review Hilbert’s programme, with emphasis on the technical side of the previously introduced concepts. We provide precise definitions of recursive functions and discuss in some detail examples of primitive recursive functions. We discuss why primitive recursive arithmetic may be a theory suitable for developing Hilbert’s programme. We present intuitively the issue of incompleteness, reviewing the concepts of conservative extension and distinguishing between incompleteness and essential incompleteness, and address the question of what incompleteness may mean for Hilbert’s programme.

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Hilbertian Finitism

22 Fevereiro 2023, 15:30 • Ricardo Santos

Session on Hilbert’s ‘On the Infinite’ introducing Hilbertian Finitism and Hilbert’s programme. The following topics have been covered: (i) Term formalism; (ii) type-token distinction; (iii) Hilbert’s term formalism about the natural numbers; (iv) Hilbert’s rejection of the meaningfulness of ‘infinity’; (v) characterization of the notion of conservative extension; (vi) conservative extension as a strategy for legitimizing infinitary mathematics on the basis of finitary mathematics; (vii) sketch of the main idea behind Hilbert’s programme; (viii) quick presentation of primitive recursive arithmetic, and reasons for thinking that it is a finitary theory.

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Truth and knowledge in mathematics

15 Fevereiro 2023, 15:30 • Ricardo Santos

Session on Benacerraf’s “Mathematical truth”. Ontological platonism in tension with mathematical knowledge. Platonism vs. formalism: what is a combinatorial philosophy of mathematics. Truth in mathematics; Tarskian theory of truth; semantic realism. Versions of mathematical formalism. The causal theory of knowledge Benaceraf’s dilemma beyond the causal theory of knowledge.

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NeoFregean Logicism

13 Fevereiro 2023, 15:30 • Ricardo Santos

Session on Hale & Wright’s ‘Logicism in the Twenty-First Century’. The following topics have been covered: (i) what is neoFregeanism?; (ii) in what sense do neoFregeans propose a reduction of arithmetic to logic?; (iii) What was Frege’s conception of meaning (distinction between sense and reference), in particular of the meaning of sentences (reference: truth-value; sense: mode of presentation); (iv) why do neoFregeans think that Hume’s principle is analytic, and how can it be seen as a definition of the cardinal operator? (v) what is the Caesar problem, and how can it be used to object to neoFregean logicism?

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