Robert McNaughton. Some formal relative consistency proofs. The journal of symbolic logic, vol. 18 (1953), pp. 136–144.

1960 ◽  
Vol 25 (4) ◽  
pp. 353-354
Author(s):  
Azriel Lévy
Keyword(s):  
Symbolic Logic ◽  
Relative Consistency ◽  
Consistency Proofs

A lattice of interpretability types of theories

1977 ◽  
Vol 42 (2) ◽  
pp. 297-305 ◽  
Author(s):  
Jan Mycielski
Keyword(s):  
Order Theory ◽  
Partial Ordering ◽  
First Order ◽  
First Order Theory ◽  
Relative Consistency ◽  
Infinite Distributivity ◽  
Closed Fields ◽  
Image Position ◽  
Interpretability Type ◽  
Consistency Proofs

We consider first-order logic only. A theory S will be called locally interpretable in a theory T if every theorem of S is interpretable in T. If S is locally interpretable in T and T is consistent then S is consistent. Most known relative consistency proofs can be viewed as local interpretations. The classic examples are the cartesian interpretation of the elementary theorems of Euclidean n-dimensional geometry into the first-order theory of real closed fields, the interpretation of the arithmetic of integers (rational numbers) into the arithmetic of positive integers, the interpretation of ZF + (V = L) into ZF, the interpretation of analysis into ZFC, relative consistency proofs by forcing, etc. Those interpretations are global. Under fairly general conditions local interpretability implies global interpretability; see Remarks (7), (8), and (9) below.We define the type (interpretability type) of a theory S to be the class of all theories T such that S is locally interpretable in T and vice versa. There happen to be such types and they are partially ordered by the relation of local interpretability. This partial ordering is of lattice type and has the following form:The lattice is distributive and complete and satisfies the infinite distributivity law of Brouwerian lattices:We do not know if the dual lawis true. We will show that the lattice is algebraic and that its compact elements form a sublattice and are precisely the types of finitely axiomatizable theories, and several other facts.

Fred Appenzeller. An independence result in quadratic form theory: infinitary combinatorics applied to ε-Hermitian spaces. The journal of symbolic logic, vol. 54 (1989), pp. 689–699. - Otmar Spinas. Linear topologies on sesquilinear spaces of uncountable dimension. Fundamenta mathematicae, vol. 139 (1991), pp. 119–132. - James E. Baumgartner, Matthew Foreman, and Otmar Spinas. The spectrum of the Γ-invariant of a bilinear space. Journal of algebra, vol. 189 (1997), pp. 406–418. - James E. Baumgartner and Otmar Spinas. Independence and consistency proofs in quadratic form theory. The journal of symbolic logic, vol. 56 (1991), pp. 1195–1211. - Otmar Spinas. Iterated forcing in quadratic form theory. Israel journal of mathematics, vol. 79 (1992), pp. 297–315. - Otmar Spinas. Cardinal invariants and quadratic forms. Set theory of the reals, edited by Haim Judah, Israel mathematical conference proceedings, vol. 6, Gelbart Research Institute for Mathematical Sciences, Bar-Ilan University, Ramat-Gan 1993, distributed by the American Mathematical Society, Providence, pp. 563–581. - Saharon Shelah and Otmar Spinas. Gross spaces. Transactions of the American Mathematical Society, vol. 348 (1996), pp. 4257–4277.

2001 ◽  
Vol 7 (2) ◽  
pp. 285-286
Author(s):  
Paul C. Eklof
Keyword(s):  
Quadratic Form ◽  
Quadratic Forms ◽  
American Mathematical Society ◽  
Mathematical Society ◽  
Symbolic Logic ◽  
Iterated Forcing ◽  
Independence Result ◽  
Form Theory ◽  
Mathematical Sciences ◽  
Consistency Proofs

2004 Spring Meeting of the Association for Symbolic Logic

2004 ◽  
Vol 10 (3) ◽  
pp. 438-446
Author(s):  
Michael Kremer
Keyword(s):  
Symbolic Logic ◽  
Spring Meeting

Association for Symbolic Logic

2008 ◽  
Vol 14 (4) ◽  
pp. 553-558
Keyword(s):  
Symbolic Logic

2008 Annual Meeting of the Association for Symbolic Logic

2008 ◽  
Vol 14 (3) ◽  
pp. 418-437
Author(s):  
Chris Laskowski
Keyword(s):  
Annual Meeting ◽  
Symbolic Logic

2008–2009 Winter Meeting of the Association for Symbolic Logic

2009 ◽  
Vol 15 (2) ◽  
pp. 237-245
Author(s):  
Ali Enayat
Keyword(s):  
Symbolic Logic
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