September 19: Dana Scott: Discussing Higher-Order Modal Modeling
16.00--17.00
Abstract:
In my lecture at the recent E.W. Beth Centenary Conference in Amsterdam, I suggested that the algebraic semantics for logic using complete lattices shows that higher-order logic has "natural" models different from the standard, classical, two-valued semantics. In my view, I think not as much attention has been given to the analogous models for modal logic. At the seminar a quick review of algebraic semantics will be given and some details about how this semantics can be used will be discussed. There are many questions needing answers -- especially in connection with the extensive work over many years on topos theory.
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September 22: Xavier Caicedo Ferrer: A characterization of first order Lukasiewicz Logic
16.00--17.00
Abstract:
We consider some model theoretic properties of infinitely valued first order Lukasiewicz logic. Under appropriate reformulation, the analogue of Lindstrom's first characterization of first order logic and some characterizations of compactness properties hold for this logic and its [0,1]-valued extensions. Similar results hold for finitely valued Lukasiewicz logics.
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October 17: Adrian Mathias: Rudimentary recursion, provident sets and forcing
16.00--17.00
Abstract:
The class of rudimentary recursive set functions is strictly larger than the Gandy-Jensen class of rudimentary functions used by Jensen in the study of the fine structure of Gödel's constructible hierarchy, but very much smaller than the Jensen-Karp class of primitive recursive set functions.
Provident sets are those transitive sets closed under rudimentary recursive functions allowing parameters from the set in question.
Friday's talk will establish uniform affine bounds on the behaviour of rudimentary recursive functions and use them to give characterizations of provident sets.
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February 20: Martin Davis: Goedel's Developing Platonism
16.00--17.00
Abstract:
In Gödel's (unsent) reply to the questionaire sent to him by B.D. Grandjean he asserted that since 1925 he had held a position of "mathematical realism" whereby "mathematical concepts [and sets] and theorems are describing objects of some kind". (The words in square brackets were added by Gödel.) A more nuanced story emerges from the hints made available with the publication of the magnificent five volume set of Gödel's Collected Works.
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March 13: Wilfrid Hodges: Where Frege is coming from
14.00--15.00
Abstract:
What came next in logic after Frege is well known, even if his direct influence is often exaggerated. But there is less agreement about how he relates to his predecessors. Comparing him with German-speaking logicians and philosophers of his times, one easily gets the impression that besides giving original answers, he also devised his own questions. But if one takes for comparison the high points of the Aristotelian Logic tradition - and for this purpose I would include Ammonius, Ibn Sina, Leibniz and the Port-Royal Logic - we get a different picture. Frege's starting point seems to have been his own formulation of largely unspoken fundamental assumptions of the Aristotelians. Looking at the same tradition, his best predecessors often gave similar formulations. Seeing that, we can better appreciate the originality of his answers.
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March 13: Jouko Väänänen: Second order logic, set theory and foundations of mathematics
15.30--16.30
Abstract:
I will give the set theorist's approach to foundations of mathematics. I will argue that second order logic does not give an essentially different foundation. I will also argue that the existence of non-standard models of set theory does not undermine its foundational importance.
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March 13: Carsten Held: Frege and Second-Order Logic
15.00--16.45
Abstract:
Frege, in his Begriffsschrift (1879) and Grundgesetze der Arithmetik (1893, 1903) develops a logical calculus containing many elements of modern second-order logic. Famously, this calculus is proven inconsistent by the Russell antinomy. Frege himself, as well as his interpreters, immediately put the blame on Basic Law V.
In this talk I try to develop a different perspective on this issue. I take a fresh look at the considerations about name and predicate in Frege's 1892 article "Über Begriff und Gegenstand". I argue that if Frege followed his own analysis then he would have to give a different interpretation to second-order formulae. In particular, a formula containing the Russell predicate in this interpretation will not count as well-formed and cannot be substituted into another formula.
Basic Law V, though looking slightly differently in this interpretation, does not have to be rejected.
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May 6: Kenneth Manders: Knot Representation
16.00--17.00
Abstract:
By analysing the contributions of some of the basic representations used in Knot theory, we show how representation uses affect what we can understand mathematically. This illustrates general concepts we are developing for judging the contributions of representations. In contrast, the details of mathematical ontology appear rather irrelevant in this case.
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May 27: Juha Kontinen: Regular representations of uniform TC^0
15.00--16.00
Abstract:
The circuit complexity class DLOGTIME-uniform AC0 is known to be a modest subclass of DLOGTIME-uniform TC0. The weakness of AC0 is caused by the fact that AC0 is not closed under restricting AC0-computable queries into AC0-computable substrings of the input. Analogously, in descriptive complexity, the logics corresponding to DLOGTIME-uniform AC0 do not have the relativization property and hence they are not regular. This weakness of DLOGTIME-uniform AC0 has been elaborated in the line of research on the Crane Beach Conjecture. The conjecture (which was refuted by Barrington, Immerman, Lautemann and Schweikardt in 2001) was that if a language L has a neutral letter, then L can be defined in FO_B, first-order logic with the collection of all numeric built-in predicates B, iff L can be already defined in FO_{<}, first-order logic with order.

We consider logics in the range of DLOGTIME-uniform AC0 and TC0. First we show that DLOGTIME-uniform TC0 can be logically characterized in terms of quantifier logics with cardinality quantifiers FO_{<}(C_S), where S is the range of some polynomial with positive integer coefficients of degree at least two. In the second part of the paper we first adapt the key properties of general logics to accomodate built-in relations. Then we define the regular interior R-int(L) and regular closure R-cl(L), of a logic L, and show that the Crane Beach Conjecture can be interpreted as a statement concerning R- int(FO_B). In particular, by the result of Barrington, Immerman, Lautemann and Schweikardt, if B contains only unary relations (besides <) then R-int(FO_B)=FO_{<} on strings. In contrast, if B contains < and the range of a polynomial of degree at least two, then R-cl(FO_B) includes all languages in DLOGTIME-uniform TC0.

Joint work with Lauri Hella and Kerkko Luosto.
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June 12: Bas Spitters: Constructive Theory of Banach algebras
16.00--17.00
Abstract:
We present a way to organize a constructive development of the theory of Banach algebras, inspired by the works of Cohen, de Bruijn and Bishop. We illustrate this by giving elementary proofs of Wiener's result on the inverse of Fourier series and Wiener's Tauberian Theorem, and we show how this can be used in a localic, or point-free, description of the spectrum of a Banach algebra. This may be seen as a contribution to Hilbert's program to uncover the hidden constructions in abstract proofs.

Joint work with Thierry Coquand.
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