Building explicit induction schemas for cyclic induction reasoning
... of induction hypotheses representing ‘not yet proved’ formulas. The induction hypotheses can be defined before their use, by explicit induction schemas that can be directly embedded in inference systems using explicit induction rules. On the other hand, the induction hypotheses can also be defined b ...
... of induction hypotheses representing ‘not yet proved’ formulas. The induction hypotheses can be defined before their use, by explicit induction schemas that can be directly embedded in inference systems using explicit induction rules. On the other hand, the induction hypotheses can also be defined b ...
The Premiss-Based Approach to Logical Aggregation Franz Dietrich & Philippe Mongin
... natural approach consists in singling out those propositions - to be called premisses - which are in some sense basic to the other judgments, aggregating them in some way, and then see what eventually results for the others. We will assume that if a formula p represents a premiss, a decision on p is ...
... natural approach consists in singling out those propositions - to be called premisses - which are in some sense basic to the other judgments, aggregating them in some way, and then see what eventually results for the others. We will assume that if a formula p represents a premiss, a decision on p is ...
1-2 Note page
... Unit 6 Notes: Polynomials Standard form - terms are in alphabetical order - terms decrease in degree from left to right - no terms have the same degree (when more than one variable, with respect to the first variable in the alphabet) Write each polynomial in standard form, then name each by its degr ...
... Unit 6 Notes: Polynomials Standard form - terms are in alphabetical order - terms decrease in degree from left to right - no terms have the same degree (when more than one variable, with respect to the first variable in the alphabet) Write each polynomial in standard form, then name each by its degr ...
Gödel Without (Too Many) Tears
... Gödel’s doctoral dissertation, written when he was 23, established the completeness theorem for the first-order predicate calculus (i.e. a standard proof system for first-order logic indeed captures all the semantically valid inferences). Later he would do immensely important work on set theory, as ...
... Gödel’s doctoral dissertation, written when he was 23, established the completeness theorem for the first-order predicate calculus (i.e. a standard proof system for first-order logic indeed captures all the semantically valid inferences). Later he would do immensely important work on set theory, as ...
1 Introduction to Categories and Categorical Logic
... Note that our first class of examples illustrate the idea of categories as mathematical contexts; settings in which various mathematical theories can be developed. Thus for example, Top is the context for general topology, Grp is the context for group theory, etc. On the other hand, the last two exa ...
... Note that our first class of examples illustrate the idea of categories as mathematical contexts; settings in which various mathematical theories can be developed. Thus for example, Top is the context for general topology, Grp is the context for group theory, etc. On the other hand, the last two exa ...
My notes - Harvard Mathematics Department
... Given an algebraic curve X, we saw that we can get a Jacobian variety J(X). It is a complex torus (so that it has a natural group structure), and it also has the structure of a projective variety. These two structures are in fact compatible with each other: the addition law is a morphism between alg ...
... Given an algebraic curve X, we saw that we can get a Jacobian variety J(X). It is a complex torus (so that it has a natural group structure), and it also has the structure of a projective variety. These two structures are in fact compatible with each other: the addition law is a morphism between alg ...
Formal Languages and Automata
... denoted ε, no matter which alphabet Σ we are talking about. ! We make no notational distinction between a symbol a ∈ Σ and the string of length 1 containing a. Thus we regard Σ as a subset of Σ∗ . ! ∅, { ε} and ε are three different things! ! ∅ is the (unique) set with no elements, ! { ε } is a set w ...
... denoted ε, no matter which alphabet Σ we are talking about. ! We make no notational distinction between a symbol a ∈ Σ and the string of length 1 containing a. Thus we regard Σ as a subset of Σ∗ . ! ∅, { ε} and ε are three different things! ! ∅ is the (unique) set with no elements, ! { ε } is a set w ...
Simple Lie Algebras over Fields of Prime Characteristic
... Albert and Frank [1] and Frank [12, 13] discovered restricted graded simple subalgebras M of W(m : 1) obtained by taking M = L^ where L is defined by L[_i] = W(m : l)[-ij; £[o] — sl(ra) (Frank [12]), sp(ra) for m even (Albert and Frank [1]), or W(r : 1) + Br acting on Br for m = pr (Frank [13]) and ...
... Albert and Frank [1] and Frank [12, 13] discovered restricted graded simple subalgebras M of W(m : 1) obtained by taking M = L^ where L is defined by L[_i] = W(m : l)[-ij; £[o] — sl(ra) (Frank [12]), sp(ra) for m even (Albert and Frank [1]), or W(r : 1) + Br acting on Br for m = pr (Frank [13]) and ...
PPT - UBC Department of CPSC Undergraduates
... Finishing the Proof Assume for contradiction that 2 is rational. Then, 2 = a/b for a Z, b Z+, where a and b have no common factor except 1. So, a2 = 2b2, and a2 is even. Since a2 is even, a is even (prev. proof!!). a = 2k for some integer k. b2 = a2/2 = (2k)2/2 = 2k2. b2 is even and so is b. ...
... Finishing the Proof Assume for contradiction that 2 is rational. Then, 2 = a/b for a Z, b Z+, where a and b have no common factor except 1. So, a2 = 2b2, and a2 is even. Since a2 is even, a is even (prev. proof!!). a = 2k for some integer k. b2 = a2/2 = (2k)2/2 = 2k2. b2 is even and so is b. ...