Workshop

Plakat2

Programme and abstracts can be found here.

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Workshop

PaC

Workshop
Friday, October 6th
Sala riunioni (2° piano)
Department of Computer Science

09:30   Thomas Streicher – An Effective Version of the Spectral Theorem
10:20   Hajime Ishihara – The Hahn-Banach theorem, constructively revisited
11:10   Coffee break
11:50   Ihsen Yengui – Algorithms for computing syzygies over V[X1,…,Xn], V a valuation ring
12:40   Takako Nemoto – Finite sets and infinite sets in weak intuitionistic arithmetic
13:30   Lunch break
15:30   Fabio Pasquali – Choice principles and the tripos-to-topos construction
16:40   Chuangjie Xu – Unifying (Herbrand) functional interpretations of (nonstandard) arithmetic

Abstracts:

Hajime Ishihara
The Hahn-Banach theorem, constructively revisited

We review known constructive versions, such as the separation theorem and continuous extension theorem, of the Hahn-Banach theorem and their proofs, and consider new versions including the dominated extension theorem and their proofs.

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Takako Nemoto
Finite sets and infinite sets in weak intuitionistic arithmetic

We consider, for a set A of natural numbers, the following notions of finiteness

FIN1: There are k and m0,…,mk-1 such that A={m0,…,mk-1} ;
FIN2: There is an upper bound for A;
FIN3: There is m such that for all B ⊆ A (|B|<m);
FIN4: It is not the case that, for all x, there is y such that y∈ A;
FIN5: It is not the case that, for all m, there is B ⊆ A such that |B|=m,

and infiniteness

INF1: There are no k and m0,…,mk-1 such that A={m0,…,mk-1};
INF2: There is no upper bound for A;
INF3: There is no m such that for all B ⊆ A (|B|<m);
INF4: For all y, there is x>y such that x∈ A;
INF5: For all m, there is B ⊆ A such that |B|=m.

We systematically compare them in the method of constructive reverse mathematics. We show that the equivalence among them can be characterized by various combinations of induction axioms and non-constructive principles, including the axiom called bounded comprehension.

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Fabio Pasquali
Choice principles and the tripos-to-topos construction

In this talk we study the connections between choice principles expressed in categorical logic and the Tripos-to-Topos construction [HJP80].
This is a joint work with M. E. Maietti and G. Rosolini based on [MPR, MR16, Pas17].

[HJP80] J.M.E. Hyland, P.T. Johnstone and A.M. Pitts. Tripos Theory. Math. Proc. Camb. Phil. Soc., 88:205-232, 1980.
[MR16] M.E. Maietti and G. Rosolini. Relating quotient completions via categorical logic. In Dieter Probst and Peter Schuster (eds), editors, Concepts of Proof in Mathematics, Philosophy and Computer Science, pages 229 – 250. De Gruyter, 2016.
[Pas17] F. Pasquali. Hilbert’s epsilon-operator in Doctrines. IfCoLog Journal of Logics and their Applications. Vol 4, Num 2: 381-400, 2017
[MPR] M.E. Maietti, F. Pasquali and G. Rosolini. Triposes, exact completions and Hilbert’s epsilon operator. In preparation.

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Chuangjie Xu
Unifying (Herbrand) functional interpretations of (nonstandard) arithmetic

We extend Oliva’s method [3] to obtain a parametrised functional interpretation for nonstandard arithmetic. By instantiating the parametrised relations, we obtain the Herbrand functional interpretations introduced in [2,4] for nonstandard arithmetic as well as the usual, well-known ones for Berger’s uniform Heyting arithmetic [1].

[1] U. Berger, Uniform Heyting arithmetic, Annals of Pure and Applied Logic 133 (2005), no. 1, 125-148.
[2] F. Ferreira and J. Gaspar, Nonstandardness and the bounded functional interpretation, Annals of
Pure and Applied Logic 166 (2015), no. 6, 701-712.
[3] P. Oliva, Unifying functional interpretations, Notre Dame J. Formal Logic 47 (2006), no. 2, 263-290.
[4] B. van den Berg, E. Briseid, and P. Safarik, A functional interpretation for nonstandard arithmetic, Annals of Pure and Applied Logic 163 (2012), no. 12, 1962-1994.

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Organised by Peter Schuster and Daniel Wessel.

 

Please note further that on Thursday, October 5th, 4pm,
there will be a departmental seminar with Olivia Caramello:
The proof-theoretic relevance of Grothendieck topologies

Past events

11. May 2017

Lorenzo Rossi (Paris Lodron University of Salzburg)
A unified approach to truth and implication 

The truth predicate is commonly thought to be symmetric, at least in the sense that for every sentence A, A and “A is true” should be inter-substitutable salva veritate. In this paper, we study an object-linguistic predicate for implication obeying similar symmetry requirements. While several non-classical logics are compatible with symmetric truth, theories of symmetric implication can only be formulated in a small class of substructural logics. We present an axiomatic theory of symmetric implication and truth over Peano Arithmetic, called SyIT, formulated in a non-reflexive logic, and we study its semantics and proof-theory. First, we show that SyIT axiomatizes a class of fixed-point models that generalize Saul Kripke’s (1975) fixed points for symmetric truth. Second, we compare SyIT with the theory PKF (Halbach and Horsten 2006), an axiomatization of Kripke’s fixed points for truth in strong Kleene logic, and we show that SyIT and PKF are proof-theoretically equivalent. The latter result shows that going substructural and adding a symmetric implication predicate to a theory of symmetric truth comes at no proof-theoretical costs.
(Joint work with Carlo NicolaiLMU Munich)

slides

8. May 2017

Categories, Constructions, Sets and Types
Minisymposium