Formal Languages and Automata
... definable, up to equivalence, from the basic forms given on Slide 27. For example, if the symbols of the alphabet are ordered in some standard way, it is common to provide a form of pattern for naming ranges of symbols—for example [a-z] might denote a pattern matching any lower-case letter. It is no ...
... definable, up to equivalence, from the basic forms given on Slide 27. For example, if the symbols of the alphabet are ordered in some standard way, it is common to provide a form of pattern for naming ranges of symbols—for example [a-z] might denote a pattern matching any lower-case letter. It is no ...
Intuitionistic Logic
... get a proof of B. So the rules tell us that there is a particular construction, converting proofs of A into proofs of B. This is exactly the justification for the derivation of A → B. For, say, the conjunction rules the analogy is even more striking. We will see below that the analogy can be made ev ...
... get a proof of B. So the rules tell us that there is a particular construction, converting proofs of A into proofs of B. This is exactly the justification for the derivation of A → B. For, say, the conjunction rules the analogy is even more striking. We will see below that the analogy can be made ev ...
Weyl`s Predicative Classical Mathematics as a Logic
... as described below. An LTT consists of a type theory augmented with a separate, primitive mechanism for forming and proving propositions. We introduce a new syntactic class of formulas, and new judgement forms for a formula being a well-formed proposition, and for a proposition being provable from g ...
... as described below. An LTT consists of a type theory augmented with a separate, primitive mechanism for forming and proving propositions. We introduce a new syntactic class of formulas, and new judgement forms for a formula being a well-formed proposition, and for a proposition being provable from g ...
Introduction to Mathematical Logic
... Mathematical logic is a mathematical investigation of this subject, similarly as number theory is the mathematical investigation of the natural numbers. Developing such a theory one can use the full machinery mathematics offers, including, for example, infinite sets, general algebraic constructions, ...
... Mathematical logic is a mathematical investigation of this subject, similarly as number theory is the mathematical investigation of the natural numbers. Developing such a theory one can use the full machinery mathematics offers, including, for example, infinite sets, general algebraic constructions, ...
Lectures on Proof Theory - Create and Use Your home.uchicago
... adopted which, although ‘set theoretic’ in one sense, have nothing to do with the source of the paradoxes. A clear statement of some of these concerns is given in a somewhat later paper by Hermann Weyl “On the new foundational crisis of mathematics” [Weyl, 1921]. Two of the main issues of concern, t ...
... adopted which, although ‘set theoretic’ in one sense, have nothing to do with the source of the paradoxes. A clear statement of some of these concerns is given in a somewhat later paper by Hermann Weyl “On the new foundational crisis of mathematics” [Weyl, 1921]. Two of the main issues of concern, t ...
Document
... • To prove that every string x Expr satisfies a condition P(x), use structural induction: show that – P(a) is true – For every x and every y in Expr, if P(x) and P(y) are true, then P(x ◦ y) and P(x • y) are true – For every x Expr, if P(x) is true, then P(◊(x)) is true • In other words, show th ...
... • To prove that every string x Expr satisfies a condition P(x), use structural induction: show that – P(a) is true – For every x and every y in Expr, if P(x) and P(y) are true, then P(x ◦ y) and P(x • y) are true – For every x Expr, if P(x) is true, then P(◊(x)) is true • In other words, show th ...
Proof Pearl: Defining Functions over Finite Sets
... (useful for defining minimum). Both require distinct fold functionals and their own theory. In the development of these theories we demonstrate locales, a lesserknown Isabelle feature. As we go along, we compare our approach with the one in HOL4 [7] and PVS [13], both of which provide their own libra ...
... (useful for defining minimum). Both require distinct fold functionals and their own theory. In the development of these theories we demonstrate locales, a lesserknown Isabelle feature. As we go along, we compare our approach with the one in HOL4 [7] and PVS [13], both of which provide their own libra ...
Frege, Boolos, and Logical Objects
... V with second-order logic. Recently, there has been a renaissance of research on consistent Frege-style systems.2 In an important series of papers, George Boolos also developed systems for reconstructing Frege’s work. We’ll focus on the work in Boolos [1986], [1987], [1989], and [1993]. Although in ...
... V with second-order logic. Recently, there has been a renaissance of research on consistent Frege-style systems.2 In an important series of papers, George Boolos also developed systems for reconstructing Frege’s work. We’ll focus on the work in Boolos [1986], [1987], [1989], and [1993]. Although in ...
REVERSE MATHEMATICS, WELL-QUASI
... If Q is a countable quasi-order, then Pf (Q) is also countable and hence easy to manage in secondorder arithmetic. The spaces A(Pf[ (Q)), U(Pf[ (Q)), and U(Pf] (Q)) fit very nicely into Dorais’s framework of countable second-countable spaces in second-order arithmetic [Dor11], and so we consider the ...
... If Q is a countable quasi-order, then Pf (Q) is also countable and hence easy to manage in secondorder arithmetic. The spaces A(Pf[ (Q)), U(Pf[ (Q)), and U(Pf] (Q)) fit very nicely into Dorais’s framework of countable second-countable spaces in second-order arithmetic [Dor11], and so we consider the ...
Fibonacci Pitch Sets: Beyond Mod 12
... In the mod 24 collection, two new important Fibonacci sequences come to our attention. The sequence {3, 1, 4, 5, 9, 14, 23, 37, 60, 97, 157, 254, …}, which we will call en,2 has the relation fn + ln + 1 = en + 2. And the sequence {5, 2, 7, 9, 16, 25, 41, 66, 107, 173, 280}, we will call mn, has the ...
... In the mod 24 collection, two new important Fibonacci sequences come to our attention. The sequence {3, 1, 4, 5, 9, 14, 23, 37, 60, 97, 157, 254, …}, which we will call en,2 has the relation fn + ln + 1 = en + 2. And the sequence {5, 2, 7, 9, 16, 25, 41, 66, 107, 173, 280}, we will call mn, has the ...
Completeness or Incompleteness of Basic Mathematical Concepts
... truth values for all sentences of first-order arithmetic. That is, it implies each first-order sentence or its negation. In fact I think that the concept of the natural numbers has a stronger property than first-order completeness. I will discuss this property, which I call “full determinateness” in ...
... truth values for all sentences of first-order arithmetic. That is, it implies each first-order sentence or its negation. In fact I think that the concept of the natural numbers has a stronger property than first-order completeness. I will discuss this property, which I call “full determinateness” in ...
Yablo`s paradox
... The rest of the argument is as before. Construing the argument in this way, we do not have to talk of satisfaction. There is therefore no predicate involved, and a fortiori no fixed-point predicate. We therefore have a paradox without circularity.6 Such a suggestion would be disingenuous, though. As ...
... The rest of the argument is as before. Construing the argument in this way, we do not have to talk of satisfaction. There is therefore no predicate involved, and a fortiori no fixed-point predicate. We therefore have a paradox without circularity.6 Such a suggestion would be disingenuous, though. As ...
In terlea v ed
... situation is reminiscent of the concurrent execution of several independent programs on a single processor (see e.g. [2]). In a popular formal model concurrency is represented by interleaving . This means that parallel processes are never executed at precisely the same instant, but take turns in ex ...
... situation is reminiscent of the concurrent execution of several independent programs on a single processor (see e.g. [2]). In a popular formal model concurrency is represented by interleaving . This means that parallel processes are never executed at precisely the same instant, but take turns in ex ...