MAXIMALITY OF LOGIC WITHOUT IDENTITY

Lindström’s theorem obviously fails as a characterization of first-order logic without identity ( L − ωω ). In this note, we provide a fix: we show that L − ωω is a maximal abstract logic satisfying a weak form of the isomorphism property (suitable for identity-free languages and studied in [11]), the Löwenheim–Skolem property, and compactness. Furthermore, we show that compactness can be replaced by being recursively enumerable for validity under certain conditions. In the proofs, we use a form of strong upwards Löwenheim–Skolem theorem not available in the framework with identity

Compactness and Löwenheim-Skolem theorems in extensions of first-order logic

Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2019, Director: Enrique Casanovas Ruiz-Fornells[en] Lindström’s theorem characterizes first-order logic as the most expressive among those that satisfy the countable Compactness and downward Löwenheim-Skolem theorems. Given the importance of this results in model theory, Lindström’s theorem justifies, to some extent, the privileged position of first-order logic in contemporary mathematics. Even though Lindström’s theorem gives a negative answer to the problem of finding a proper extension of first-order logic satisfying the same model-theoretical properties, the study of these extensions has been of great importance during the second half of the XX. century: logicians were trying to find systems that kept a balance between expressive power and rich model-theoretical properties. The goal of this essay is to prove Lindström’s theorem, along with its prerequisites, and to give weaker versions of the Compactness and Löwenheim-Skolem theorems for the logic L ( Q 1 ) (first-order logic with the quantifier "there exist uncountably many"), which we present as an example of extended logic with good model-theoretical properties

Compacidad en lógicas con cuantificadores cardinales

Hacemos una presentación del problema de la compacidad paraalgunas lógicas con cuantificadores generalizados. Se recurre luego a las ideas de la prueba de compacidad de Fraïssé para dar una demostración topológica de la compacidad enumerable de los fragmentos monádicos de las lógicas con cuantificadores cardinales. Se obtienen otros resultados usando la misma construcción

Persistence of Lower Dimensional Tori of General Types in Hamiltonian Systems

2000 Mathematics Subject Classification. 37J40.The work is a generalization to [40] in which we study the persistence of lower dimensional tori of general type in Hamiltonian systems of general normal forms. By introducing a modified linear KAM iterative scheme to deal with small divisors, we shall prove a persistence result, under a Melnikov type of non-resonance condition, which particularly allows multiple and degenerate normal frequencies of the unperturbed lower dimensional tori.The first author was partially supported by NSFC grant 19971042, National 973 Project of China: Nonlinearity, the outstanding young's project of Ministry of Education of China, and National outstanding young's award of China. The second author was partially supported by NSF grants DMS9803581 and DMS-0204119