Sumários
Categoricity and Benacerraf’s ‘What Numbers Could Not Be’
12 Abril 2023, 15:30 • Ricardo Santos
Presentation of the model theory for (full) second-order logic. Definition of categoricity. Presentation of Dedekind’s categoricity theorem for second-order Peano Arithmetic. Significance of Dedekind’s categoricity theorem for the indeterminacy of reference to mathematical structures. Relationship between categoricity and Gödel’s First Incompleteness Theorem and reassessment of the significance of Dedekind’s categoricity theorem for the indeterminacy of reference to mathematical structures. Discussion of Benacerraf’s paper ‘What Numbers Could Not Be’. Among other things, the following topics were covered: (i) Benacerraf’s argument against the identification of numbers with sets; (ii) Benacerraf’s argument against the existence of numbers; (iii) the “structuralist” elements of Benacerraf’s views on arithmetic.
Basic Model Theory II and Referential Indeterminacy
10 Abril 2023, 15:30 • Ricardo Santos
Second session introducing model theory, and discussion of some model-theoretic results vis-à-vis the indeterminacy of reference to mathematical objects. The following topics were covered: (i) Isomorphism and elementary equivalence – definitions and examples; (ii) the equinumerous sets have isomorphic models based on them theorem and its significance vis-à-vis the indeterminacy of reference; (iii) the existence and structure of nonstandard models of arithmetic theorem, and its significance vis-à-vis the indeterminacy of reference to mathematical structure.
The iterative conception of set II & Basic Model Theory
29 Março 2023, 15:30 • Ricardo Santos
Second and last session on the iterative conception of set. Presentation of ZFC’s axioms of Separation and Replacement. The iterative conception’s justification of Separation. No justification for Replacement afforded by the iterative conception. Justification for Replacement by the lights of a different conception of set – the Limitation of Size conception. Two potential roles for set theory in the foundations of mathematics: (i) Fundamental theory role – the view that mathematics is nothing but set theory; (ii) Model-theoretic – Set theory as representing what mathematical theories require of the world for their truth. Brief introduction to model theory: signatures, models, satisfaction for first-order languages, model-theoretic consequence for first-order languages.
The iterative conception of set
27 Março 2023, 15:30 • Ricardo Santos
We begin by
recapitulating Russell’s paradox and the conception of set that underscores it.
We bring up the axiom of (unrestricted) comprehension and briefly discuss it in
connection with philosophical positions like that of Poincare’s. We introduce
the generic idea of the iterative conception of sets: sets are constructed in
stages, indexed by the ordinals. We show how this can avoid the paradoxes. We
present the axioms of ZF set theory.
Dummett on intuitionism
22 Março 2023, 15:30 • Ricardo Santos
We discuss
Dummett’s paper “The philosophical basis of intuitionistic logic”. We review
the main tenets of intuitionism and their impact upon logic. We present
Dummett’s novel, meaning-theoretic (and hence non-mathematical) argument in
favour of intuitionism. We distinguish between semantic anti/realism and other,
in particular, ontological underpinnings for these metaphysical positions. We
discuss Dummett’s manifestability argument.