Abstract Model–Theory and Set–Theoretic Multiverses Antonio Vincenzi A New Deal for Abstract Model Theory?? Altonaer Stiftung für philosophische Grundlagenforschung
Model–Theoretic Logics Higer order logics II L II L L ∞∞ II ( Q 2 ) L L κκ M M κκ κ κ + IIw L L ( H ) L <∞ω MM ) ( Q 1 L L κ κ + 1 L ( A,R ) κ L L L ( aa ) ( I ) L L + L L ∞ω κ L ( β ,<) L cf ) ( Q 1 L WO ) L ( Q L ω G 1 L ( ω ,<) L ( Q ) L ( Q 1 ) L κλ L λ + + (2 ) λ L ( Q ) L κω 1 ibol ) L COMP ( Q L + INT λ ω L ( Q 0 ) L 1 L IHYP L Infinitary logics generalized quantifiers ROB
Episodic Relationships? Assuming that the set–theoretic background of each logic is a set–theoretic universe M = ( M, ∈ ,… ) • The sensibility of Compactness, Löwenheim–Skolem,… to M , namely the fact that, for some logic L , these properties can ◯ hold if M satisfies � , GCH ,… contains some large cardinal… ◯ fail otherwise • Set–theoretic constraints on relationships among model– theoretic properties, like Robinson = Compactness + Interpolation • The relationships between logics L = vocabulary + grammar + interpretation and inner models C ( L ) of M
Set–Theoretic Logics I NTERPENETRATION OF I NSTRUMENTS AND C ONCEPTS Construction Concept r y c x X Y P R
• a model–theoretic logic L generates class of constructions C ( L ) of M • a set–theoretic logic K is a class of constructions K of M FORM K + K L fixed point new L next rules basic rules formal L prev tree A graft A basic symbols new symbols MEANING If the operations used in K are inductive then L and K are equivalent
Basics on Set–Theoretic Logics Conceptually • The notion of formal language is not conceptually necessary to define the notion of logic. • The formal language of a logic can be naturally modified by its set–theoretic background • It is a technical trick that simplify some logic operation (analogous to the table for the connectives) Technically Väänänen Problem: the second order logic is the only logic that • Satisfies one–sorted interpolation • Does not satisfy many–sorter interpolation Is different from its delta closure •
Multiverses Logics Since each set–theoretic logic K is relative to a set–theoretic universe M that contains K assume that • a model–theoretic property ( MTP ) of ( K , M ) is a property ◯ defined using the satisfaction relation of K , ◯ which possibly depends on some properties of M . • a multiverse logic ( K , M ) is a collection of ( K , M ) such that ( K , M ) satisfies MTP ⇔ ( K , M ) satisfies MTP for each M ⇔ ‘ ( K , M ) ⎪ = MTP’ is M –absolute • in particular M = Mod ( ZFC ), M = Mod ( KP ), …
W ITTGENSTEIN –C HOMSKY L ANGUAGES F RAMEWORKS Surface Framework Deep Framework Form Meaning Syntax Semantic An MTP is surface if satisfies the following equivalent properties • ◯ the failure of MTP can be repaired acting only on K ◯ the failure of MTP is M –absolute ◯ its relative form is only MTP (( K + , M ),( K , M )) with K< K * An MTP is deep if satisfies the following equivalent properties • ◯ the failure of MTP can be repaired acting only on M ◯ the failure of MTP is M –relative ◯ its relative form is only MTP (( K , M ),( K , M *)) with M * < M
Compactification V OPENKA –L IKE C ONPACTIFICATION T HEOREM . M ↾ κ = the strengthening of M in which each M contains κ If [ λ ]– COMP( K , M ) fails , then there is a Vopenka cardinal κ such that [ λ ]– COMP( K , M ↾ κ ) holds. N ON –S TANDARD C OMPACTIFICATION T HEOREM . * M = non standard M in which each M is substituted by * M If λ be a regular cardinal and ( K , M ) be a set–theoretic logic with dependence number ≤ λ . • If [ λ ]– COMP( K , M ) fails and AM( K , M ) holds then there is a * M such that ‘cofinality λ ’ is * M – absolute and [ λ ]– COMP( K , * M ) holds . • If both [ λ ]– COMP( K , M ) and AM( K , M ) fail then there is a * M such that ‘cofinality λ ’ is * M – absolute and [ λ ]– COMP(( K , M ) , ( K , * M )) holds.
Inexpressibility H EISENBERG – LIKE U NCERTAINTY A class of detectors of a fixed 0 0 3 dimension cannot detect precisely 3 objects of their dimension 2 2 4 1 1 4 Consistency Inconsistency
I NEXPRESSIBILITY T HEOREM . For each bounded and relativizable logic ( K , M ) and each infinite cardinal λ the following are equivalent: ( K , M ) is [ λ ] –Compact. • ( λ , < ) is not Σ 1 ( K , M ) –characterizable . • The class of initial segments of ( λ , < ) is not Σ 1 ( K , M ) –characterizable . • ( λ , < ) cannot be Σ 1 ( K , M ) –pinned down . • In particular, if λ is regular, then the above properties are equivalent to • ( λ , < ) is not cofinally Σ 1 ( K , M ) –characterizable. Alternatively, if λ is singular, then the same properties are equivalent to • ( λ , < ) is not cardinallike Σ 1 ( K , M ) –characterizable. EXPRESSIVITY an object is ( K , M )– expressible ⇕ ( K , M ) individuates univocally a class of objects containing this object
λ –LS λ –LST λ inexpressible [ λ ] –COMP P ROJECT . Use the dichotomy λ regular — λ cofinally characterizable λ singular — λ cardinallike characterizable to generalize the above results to each infinite cardinal
Abstract Deep Properties Y Inconsistency X Consistency Compactness X Y [ λ , κ ] H ( λ ) H ( κ ) Barwise A admissible Σ 1 ( A ) Abstract H � H P ROJECT : Find the properties of H and H � that allow prove Abstract Inexpressibility, Compactification… results
Perspectives SURFACE MTP Interpolation, Definability,… Inexpressibility TRASVERSE MTP Robinson… DEEP MTP Compactness, Löwenheim–Skolem,…
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