On the existence of a connected component
... Weihrauch reductions as translations One goal of Reverse Mathematics is to study the underlying combinatorics of mathematical principles. Proof-theoretic equivalence gives only a very coarse way of measuring this combinatorial similarity. Strong Weihrauch reductions are somewhat analogous to biject ...
... Weihrauch reductions as translations One goal of Reverse Mathematics is to study the underlying combinatorics of mathematical principles. Proof-theoretic equivalence gives only a very coarse way of measuring this combinatorial similarity. Strong Weihrauch reductions are somewhat analogous to biject ...
Arithmetic in Metamath, Case Study: Bertrand`s Postulate
... integer representations but rather base-4 representation. It is of course necessary to commit to a base to work in for practical purposes, and the most logical decision for a system such as set.mm which prioritizes abstract reasoning over numerical calculation is base 10. The reason this choice was ...
... integer representations but rather base-4 representation. It is of course necessary to commit to a base to work in for practical purposes, and the most logical decision for a system such as set.mm which prioritizes abstract reasoning over numerical calculation is base 10. The reason this choice was ...
Multiverse Set Theory and Absolutely Undecidable Propositions
... formulate V1 and V2 inside ZFC in any reasonable way, modeling the fact that they are two “parallel” versions of V , it is hard to avoid the conclusion that V1 = V2 , simply because V is “everything”. This is why the working set theorist will not be able to recognize whether he or she has one or sev ...
... formulate V1 and V2 inside ZFC in any reasonable way, modeling the fact that they are two “parallel” versions of V , it is hard to avoid the conclusion that V1 = V2 , simply because V is “everything”. This is why the working set theorist will not be able to recognize whether he or she has one or sev ...
Math 318 Class notes
... Definition. If f : A → B, g : B → C, then their composition is4 g f = {( a, c) ∈ A × C : ∃b ∈ B, ( a, b) ∈ f , (b, c) ∈ g} Definition. Given a function f : A → B and function g : B → A. g is called a left inverse of f if g f = id A ; g is called a right inverse of f if f g = id B . Proposition 3.11. ...
... Definition. If f : A → B, g : B → C, then their composition is4 g f = {( a, c) ∈ A × C : ∃b ∈ B, ( a, b) ∈ f , (b, c) ∈ g} Definition. Given a function f : A → B and function g : B → A. g is called a left inverse of f if g f = id A ; g is called a right inverse of f if f g = id B . Proposition 3.11. ...
POSSIBLE WORLDS SEMANTICS AND THE LIAR Reflections on a
... intensional constructions — is an area that is thriving on paradoxes and puzzles but is also haunted by them. There are, of course, the puzzles of intensionality that started the whole enterprise and constitute its raison d’être: Frege’s puzzle about the information value of true identity statement ...
... intensional constructions — is an area that is thriving on paradoxes and puzzles but is also haunted by them. There are, of course, the puzzles of intensionality that started the whole enterprise and constitute its raison d’être: Frege’s puzzle about the information value of true identity statement ...
LINEAR LOGIC AS A FRAMEWORK FOR SPECIFYING SEQUENT
... where ∆ is the multiset union of ∆1 and ∆2 , and A is the multiset union of A1 and A2 . In other words, those subgoals immediately to the left of an ⇒ are attempted with empty bounded contexts: the bounded contexts, here ∆ and A, are divided up to be used to prove those goals immediately to the left ...
... where ∆ is the multiset union of ∆1 and ∆2 , and A is the multiset union of A1 and A2 . In other words, those subgoals immediately to the left of an ⇒ are attempted with empty bounded contexts: the bounded contexts, here ∆ and A, are divided up to be used to prove those goals immediately to the left ...
Proofs - Stanford University
... Intuitively, the concept of proof should already be familiar. We all like to assert things, and few of us like to say things that turn out to be false. A proof provides a means for guaranteeing such claims. Proofs in mathematics and computer science require a precisely stated proposition to be prove ...
... Intuitively, the concept of proof should already be familiar. We all like to assert things, and few of us like to say things that turn out to be false. A proof provides a means for guaranteeing such claims. Proofs in mathematics and computer science require a precisely stated proposition to be prove ...
- Free Documents
... by computerized calculations. For that purpose di erent semantic characterizations of interpretable theories and exact formulas in terms of Kripkemodels have been developed which are of interest in their own right. It turns out that an important role is played by maximal exact formulas, i.e. exact f ...
... by computerized calculations. For that purpose di erent semantic characterizations of interpretable theories and exact formulas in terms of Kripkemodels have been developed which are of interest in their own right. It turns out that an important role is played by maximal exact formulas, i.e. exact f ...
Local deduction, deductive interpolation and amalgamation in
... yes, when the maximal spectrum is onedimensional THEOREM (L.Cabrer, D.M.) Suppose the maximal spectrum of A is one-dimensional. Then A is ngenerated projective if and only if A is isomorphic to M(P) for some contractible strongly regular rational polyhedron in [0,1]n containing a vertex of [0,1]n. ...
... yes, when the maximal spectrum is onedimensional THEOREM (L.Cabrer, D.M.) Suppose the maximal spectrum of A is one-dimensional. Then A is ngenerated projective if and only if A is isomorphic to M(P) for some contractible strongly regular rational polyhedron in [0,1]n containing a vertex of [0,1]n. ...
Modal Consequence Relations
... of ‘argument’ one also speaks of a ‘rule’ or an ‘inference’ and says that the rule is valid. This approach culminated in the notion of a consequence relation, which is a relation between sets of formulae and a single formula. A consequence relation ` specifies which arguments are valid; the argument ...
... of ‘argument’ one also speaks of a ‘rule’ or an ‘inference’ and says that the rule is valid. This approach culminated in the notion of a consequence relation, which is a relation between sets of formulae and a single formula. A consequence relation ` specifies which arguments are valid; the argument ...
First-Order Proof Theory of Arithmetic
... can prove the arithmetized version of the cut-elimination theorem and those which cannot; in practice, this is equivalent to whether the theory can prove that the superexponential function i 7→ 21i is total. The very weak theories are theories which do not admit any induction axioms. Non-logical sym ...
... can prove the arithmetized version of the cut-elimination theorem and those which cannot; in practice, this is equivalent to whether the theory can prove that the superexponential function i 7→ 21i is total. The very weak theories are theories which do not admit any induction axioms. Non-logical sym ...
Kripke models for subtheories of CZF
... set theory. Kripke models are a useful tool to study constructive theories and because of their simplicity have been applied with great success to non-classical logics in general, and intuitionistic logic and Heyting Arithmetic in particular. Although nowadays there exist various models of CZF, such ...
... set theory. Kripke models are a useful tool to study constructive theories and because of their simplicity have been applied with great success to non-classical logics in general, and intuitionistic logic and Heyting Arithmetic in particular. Although nowadays there exist various models of CZF, such ...
Query Answering for OWL-DL with Rules
... universal quantification, but allow for the interaction of variables in arbitrary ways. Clearly, a combination of OWL-DL and rules is desirable for building Semantic Web ontologies, and several such combinations have already been discussed. However, such a combination might easily lead to the undeci ...
... universal quantification, but allow for the interaction of variables in arbitrary ways. Clearly, a combination of OWL-DL and rules is desirable for building Semantic Web ontologies, and several such combinations have already been discussed. However, such a combination might easily lead to the undeci ...
