How many numbers there are?
... to have also the negative rational numbers (for reasons similar as in the case of Z), we will allow as a rational number any fraction m n where m, n are in Z and n 6= 0. The collection of all rational numbers is denoted as Q. Unlike the case of N or Z we can no longer enumerate elements of Q in a si ...
... to have also the negative rational numbers (for reasons similar as in the case of Z), we will allow as a rational number any fraction m n where m, n are in Z and n 6= 0. The collection of all rational numbers is denoted as Q. Unlike the case of N or Z we can no longer enumerate elements of Q in a si ...
Remarks on Second-Order Consequence
... cardinality of the power set of the natural numbers. Thus (since the power set operation is built-in in second-order logic), sets of the cardinality of the real numbers can be defined in second-order logic. Since bijectability can also be handled, the content of CH can be rendered in a second-order ...
... cardinality of the power set of the natural numbers. Thus (since the power set operation is built-in in second-order logic), sets of the cardinality of the real numbers can be defined in second-order logic. Since bijectability can also be handled, the content of CH can be rendered in a second-order ...
MATH 312H–FOUNDATIONS
... Logic is a separate (non-mathematical) discipline. We will use only basic facts from logic which are (more or less) obvious by common sense. We are dealing with statements (understood only as a mathematical statement) which are true or false (there is no other possibility, expressed in Latin as the ...
... Logic is a separate (non-mathematical) discipline. We will use only basic facts from logic which are (more or less) obvious by common sense. We are dealing with statements (understood only as a mathematical statement) which are true or false (there is no other possibility, expressed in Latin as the ...
The Art of Ordinal Analysis
... proof, Gentzen used his sequent calculus and employed the technique of cut elimination. As this is a tool of utmost importance in proof theory and ordinal analysis, a rough outline of the underlying ideas will be discussed next. The most common logical calculi are Hilbert-style systems. They are spe ...
... proof, Gentzen used his sequent calculus and employed the technique of cut elimination. As this is a tool of utmost importance in proof theory and ordinal analysis, a rough outline of the underlying ideas will be discussed next. The most common logical calculi are Hilbert-style systems. They are spe ...
Completeness through Flatness in Two
... Two-dimensional temporal logic In the last twenty years, various disciplines related to logic have seen the idea arising to develop a framework for modal or temporal logic in which the possible worlds are pairs of elements of the model instead of the elements themselves. Often the motivation for dev ...
... Two-dimensional temporal logic In the last twenty years, various disciplines related to logic have seen the idea arising to develop a framework for modal or temporal logic in which the possible worlds are pairs of elements of the model instead of the elements themselves. Often the motivation for dev ...
Notes on Classical Propositional Logic
... What I will do is extract their mathematical essence, because it will be convenient later on. Let us assume we have two truth values, true and false. (Exactly what these are is not important, only that they are different from each other.) We assume there are various operations defined on the set {tr ...
... What I will do is extract their mathematical essence, because it will be convenient later on. Let us assume we have two truth values, true and false. (Exactly what these are is not important, only that they are different from each other.) We assume there are various operations defined on the set {tr ...
Elements of Finite Model Theory
... structures via the use of pseudo-finite structures (a structure is pseudo-finite, if every first-order sentence it satisfies is true in some finite structure), the breadth of such application seems limited. The Chapter then introduces a fundamental technique for establishing the inexpressibility of ...
... structures via the use of pseudo-finite structures (a structure is pseudo-finite, if every first-order sentence it satisfies is true in some finite structure), the breadth of such application seems limited. The Chapter then introduces a fundamental technique for establishing the inexpressibility of ...
Lecture 6: End and cofinal extensions
... The aim of these exercises is to prove the Gaifman Splitting Theorem [6, 7] assuming MRDP holds in I∆0 + exp. The main difference between our Splitting Theorem and Gaifman’s is that his theorem does not start with any elementarity. Gaifman Splitting Theorem. If M, K |= PA such that M ⊆ K, then M = s ...
... The aim of these exercises is to prove the Gaifman Splitting Theorem [6, 7] assuming MRDP holds in I∆0 + exp. The main difference between our Splitting Theorem and Gaifman’s is that his theorem does not start with any elementarity. Gaifman Splitting Theorem. If M, K |= PA such that M ⊆ K, then M = s ...
Hilbert Calculus
... Completeness - Proof sketch (1) S |= F iff S ∪ {¬F } is unsatisfiable. (Trivial) (2) Definition: S is inconsistent if there is a formula F such that S ⊢ F and S ⊢ ¬F . (3) S ⊢ F iff S ∪ {¬F } is inconsistent. (To be proved!) (4) Unsatisfiable sets are inconsistent. (To be proved!) Proof sketch: Ass ...
... Completeness - Proof sketch (1) S |= F iff S ∪ {¬F } is unsatisfiable. (Trivial) (2) Definition: S is inconsistent if there is a formula F such that S ⊢ F and S ⊢ ¬F . (3) S ⊢ F iff S ∪ {¬F } is inconsistent. (To be proved!) (4) Unsatisfiable sets are inconsistent. (To be proved!) Proof sketch: Ass ...
First-Order Logic with Dependent Types
... for the sort S. o is the type of formulas. The remainder of the signature encodes the usual grammar for FOL formulas. Higher-order abstract syntax is used, i.e., λ is used to bind the free variables in a formula, and quantifiers are operators taking a λ expression as an argument.2 Quantifiers and th ...
... for the sort S. o is the type of formulas. The remainder of the signature encodes the usual grammar for FOL formulas. Higher-order abstract syntax is used, i.e., λ is used to bind the free variables in a formula, and quantifiers are operators taking a λ expression as an argument.2 Quantifiers and th ...
Document
... Induction has three parts Base Case(s) o Simplest examples Inductive Step o From simpler to more complex ‘Exclusivity’ o Only things defined from the base case and inductive steps are legal o This part can be omitted in definition as we assume it must be true ...
... Induction has three parts Base Case(s) o Simplest examples Inductive Step o From simpler to more complex ‘Exclusivity’ o Only things defined from the base case and inductive steps are legal o This part can be omitted in definition as we assume it must be true ...
Oh Yeah? Well, Prove It.
... A large part of mathematics consists of building up a theoretical framework that allows us to solve problems. This theoretical framework is built upon a set of axioms. Axioms are unproven assumptions that are the foundation of all mathematics. (Not everyone agrees on what axioms should be used, but ...
... A large part of mathematics consists of building up a theoretical framework that allows us to solve problems. This theoretical framework is built upon a set of axioms. Axioms are unproven assumptions that are the foundation of all mathematics. (Not everyone agrees on what axioms should be used, but ...
A Syntactic Characterization of Minimal Entailment
... (the degree undecidability of) cwaS (Σ) is Π1 relative to Cn(Σ), or Π2 relative to Σ, and seemingly there is no good reason why it should be less. On the other hand, all asymptotically decidable problems (in particular those ones asymptotically decidable by finite failure proof procedures) are ∆2 , ...
... (the degree undecidability of) cwaS (Σ) is Π1 relative to Cn(Σ), or Π2 relative to Σ, and seemingly there is no good reason why it should be less. On the other hand, all asymptotically decidable problems (in particular those ones asymptotically decidable by finite failure proof procedures) are ∆2 , ...
Formal methods: lecture notes no
... previously shown. If, on the other hand, x = 0 then it is simply the identity function and thus x(T) will return T. Our previously chosen definitions will also allow us to introduce conditional expressions into our language. An expression of the form (if x then y else z ) will in our language be ((x ...
... previously shown. If, on the other hand, x = 0 then it is simply the identity function and thus x(T) will return T. Our previously chosen definitions will also allow us to introduce conditional expressions into our language. An expression of the form (if x then y else z ) will in our language be ((x ...
full text (.pdf)
... Fagin (1976) showed that every first-order sentence satisfies the zero-one law. Grandjean (1982) showed that the problem of deciding which of the two limit values is correct for a given first-order sentence is PSPACE complete. (We state these results precisely and review their proofs in Sect. 1.) Ka ...
... Fagin (1976) showed that every first-order sentence satisfies the zero-one law. Grandjean (1982) showed that the problem of deciding which of the two limit values is correct for a given first-order sentence is PSPACE complete. (We state these results precisely and review their proofs in Sect. 1.) Ka ...
Is `structure` a clear notion? - University of Illinois at Chicago
... if one reads ‘look at the list’ as ‘consider the natural numbers as a subset of the linearly ordered field of reals’. As Pierce notes, a fundamental difficulty in Spivak’s treatment is the failure to distinguish between the truth of each of these properties on the appropriate expansion of (N, S) and ...
... if one reads ‘look at the list’ as ‘consider the natural numbers as a subset of the linearly ordered field of reals’. As Pierce notes, a fundamental difficulty in Spivak’s treatment is the failure to distinguish between the truth of each of these properties on the appropriate expansion of (N, S) and ...
PDF
... remains is the case when A has the form D. We do induction on the number n of ’s in A. The case when n = 0 means that A is a wff of PLc , and has already been proved. Now suppose A has n + 1 ’s. Then D has n ’s, and so by induction, ` D[B/p] ↔ D[C/p], and therefore ` D[B/p] ↔ D[C/p] by 2. This ...
... remains is the case when A has the form D. We do induction on the number n of ’s in A. The case when n = 0 means that A is a wff of PLc , and has already been proved. Now suppose A has n + 1 ’s. Then D has n ’s, and so by induction, ` D[B/p] ↔ D[C/p], and therefore ` D[B/p] ↔ D[C/p] by 2. This ...
Gödel`s Incompleteness Theorems
... This theorem is quite remarkable in its own right because it shows that Peano’s well-known postulates, which by and large are considered as an axiomatic basis for elementary arithmetic, cannot prove all true statements about natural numbers. But Gödel went even further. He showed that his first inc ...
... This theorem is quite remarkable in its own right because it shows that Peano’s well-known postulates, which by and large are considered as an axiomatic basis for elementary arithmetic, cannot prove all true statements about natural numbers. But Gödel went even further. He showed that his first inc ...
Principle of Mathematical Induction
... The Principle of Mathematical Induction is a method of proof normally used to prove that a proposition is true for all natural numbers 1,2,3,… , although there are many variations of the basic method. The method is particularly important in discrete mathematics, and one often sees theorems proven by ...
... The Principle of Mathematical Induction is a method of proof normally used to prove that a proposition is true for all natural numbers 1,2,3,… , although there are many variations of the basic method. The method is particularly important in discrete mathematics, and one often sees theorems proven by ...
Constructive Mathematics, in Theory and Programming Practice
... The notion defined by dropping from this definition the last clause, about preservation of equality, is called an operation. In the first part of this paper we shall have little to say about operations, but they will have more significance in the second part, when we discuss Martin-Löf’s theory of ...
... The notion defined by dropping from this definition the last clause, about preservation of equality, is called an operation. In the first part of this paper we shall have little to say about operations, but they will have more significance in the second part, when we discuss Martin-Löf’s theory of ...
T - RTU
... in First-Order Logic The semantics of first-order logic provide a basis for a formal theory of logical inference. The ability to infer new correct expressions from a set of true assertions is very important feature of first-order logic. These new expressions are correct in that they are consistent w ...
... in First-Order Logic The semantics of first-order logic provide a basis for a formal theory of logical inference. The ability to infer new correct expressions from a set of true assertions is very important feature of first-order logic. These new expressions are correct in that they are consistent w ...
Lecture 10
... The properties P1 - P13 are referred to as the “axioms of a complete ordered field”. For now, we will assume R is a system of numbers which satisfies P1 - P13. The idea is that these properties are “simple enough” that it appears natural to study such an object. There are a number of objections to t ...
... The properties P1 - P13 are referred to as the “axioms of a complete ordered field”. For now, we will assume R is a system of numbers which satisfies P1 - P13. The idea is that these properties are “simple enough” that it appears natural to study such an object. There are a number of objections to t ...
.pdf
... a model of A. A is valid if A is true under every interpretation. These notions can be extended to sets of formula sin a canonical fashion. It should be noted that there is a fine distinction between boolean valuations and first-order valuations. Boolean valuations can only analyze the propositional ...
... a model of A. A is valid if A is true under every interpretation. These notions can be extended to sets of formula sin a canonical fashion. It should be noted that there is a fine distinction between boolean valuations and first-order valuations. Boolean valuations can only analyze the propositional ...
From proof theory to theories theory
... to cut free proofs, it does not allow to reduce it enough so that the search for a proof of a contradiction in the theory ∀x (P (x) ⇔ P (f (x))) fails in finite time. This proof search method “does not know” [14] that this theory is consistent and indeed the cut elimination theorem for predicate log ...
... to cut free proofs, it does not allow to reduce it enough so that the search for a proof of a contradiction in the theory ∀x (P (x) ⇔ P (f (x))) fails in finite time. This proof search method “does not know” [14] that this theory is consistent and indeed the cut elimination theorem for predicate log ...
Peano axioms
In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are a set of axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano. These axioms have been used nearly unchanged in a number of metamathematical investigations, including research into fundamental questions of whether number theory is consistent and complete.The need to formalize arithmetic was not well appreciated until the work of Hermann Grassmann, who showed in the 1860s that many facts in arithmetic could be derived from more basic facts about the successor operation and induction. In 1881, Charles Sanders Peirce provided an axiomatization of natural-number arithmetic. In 1888, Richard Dedekind proposed another axiomatization of natural-number arithmetic, and in 1889 Peano published a more precisely formulated version of them as a collection of axioms in his book, The principles of arithmetic presented by a new method (Latin: Arithmetices principia, nova methodo exposita).The Peano axioms contain three types of statements. The first axiom asserts the existence of at least one member of the set ""number"". The next four are general statements about equality; in modern treatments these are often not taken as part of the Peano axioms, but rather as axioms of the ""underlying logic"". The next three axioms are first-order statements about natural numbers expressing the fundamental properties of the successor operation. The ninth, final axiom is a second order statement of the principle of mathematical induction over the natural numbers. A weaker first-order system called Peano arithmetic is obtained by explicitly adding the addition and multiplication operation symbols and replacing the second-order induction axiom with a first-order axiom schema.