Mathematical Logic. An Introduction
... The ordered pair {{x}, {x, y}} is denoted by (x, y). Ordered pairs allow to formalize (binary) relations and functions: hrigid| − i a relation is a set R of ordered pairs; hrigid| − i a function is a relation f such that for all x, y, y ′ holds: if (x, y) ∈ f and (x, y ′) ∈ f then y = y ′. Then f(x) ...
... The ordered pair {{x}, {x, y}} is denoted by (x, y). Ordered pairs allow to formalize (binary) relations and functions: hrigid| − i a relation is a set R of ordered pairs; hrigid| − i a function is a relation f such that for all x, y, y ′ holds: if (x, y) ∈ f and (x, y ′) ∈ f then y = y ′. Then f(x) ...
Scattered Sentences have Few Separable Randomizations
... Fix a countable first order signature L. A sentence ϕ of the infinitary logic Lω1 ω is scattered if there is no countable fragment LA of Lω1 ω such that ϕ has a perfect set of countable models that are not LA -equivalent. Scattered sentences were introduced by Morley [M], motivated by Vaught’s conje ...
... Fix a countable first order signature L. A sentence ϕ of the infinitary logic Lω1 ω is scattered if there is no countable fragment LA of Lω1 ω such that ϕ has a perfect set of countable models that are not LA -equivalent. Scattered sentences were introduced by Morley [M], motivated by Vaught’s conje ...
this PDF file
... form ϕ @ ψ is true iff ϕ is true and false iff ψ is false; ϕ / ψ is related to Blamey’s [7] transplication and can be read as ‘ψ, presupposing ϕ’. This formula has the value of ψ if ϕ is true, but is neither true nor false otherwise. The Π and Σ quantifiers are the duals of ∀ and ∃ and correspond to ...
... form ϕ @ ψ is true iff ϕ is true and false iff ψ is false; ϕ / ψ is related to Blamey’s [7] transplication and can be read as ‘ψ, presupposing ϕ’. This formula has the value of ψ if ϕ is true, but is neither true nor false otherwise. The Π and Σ quantifiers are the duals of ∀ and ∃ and correspond to ...
Introduction to Logic
... other than what is stated follows of necessity from their being so.” And, in fact, this intuition not only lies at its origin, ca. 500 BC, but has been the main force motivating its development since that time until the last century. There was a medieval tradition according to which the Greek philos ...
... other than what is stated follows of necessity from their being so.” And, in fact, this intuition not only lies at its origin, ca. 500 BC, but has been the main force motivating its development since that time until the last century. There was a medieval tradition according to which the Greek philos ...
A Concise Introduction to Mathematical Logic
... depends on one’s objectives. Traditional semantics and model theory as essential parts of mathematical logic use stronger set-theoretic tools than does proof theory. In some model-theoretic investigations these are often the strongest possible ones. But on average, little more is assumed than knowle ...
... depends on one’s objectives. Traditional semantics and model theory as essential parts of mathematical logic use stronger set-theoretic tools than does proof theory. In some model-theoretic investigations these are often the strongest possible ones. But on average, little more is assumed than knowle ...
Lectures on Proof Theory - Create and Use Your home.uchicago
... way of putting it is that R(α) is the result P α (∅) of iterating the PowerSet operation s 7→ P (s) α times, starting with the null set ∅. Then ordinary set theory is a theory of pure well-founded sets and its intended models are structures of the form hR(κ), ∈i, where the numbers κ will depend upo ...
... way of putting it is that R(α) is the result P α (∅) of iterating the PowerSet operation s 7→ P (s) α times, starting with the null set ∅. Then ordinary set theory is a theory of pure well-founded sets and its intended models are structures of the form hR(κ), ∈i, where the numbers κ will depend upo ...
Lecture 09
... • In a proof by mathematical induction, we don’t assume that P(k) is true for all positive integers! We show that if we assume that P(k) is true, then P(k + 1) must also be true. • Proofs by mathematical induction do not always start at the integer 1. In such a case, the basis step begins at a start ...
... • In a proof by mathematical induction, we don’t assume that P(k) is true for all positive integers! We show that if we assume that P(k) is true, then P(k + 1) must also be true. • Proofs by mathematical induction do not always start at the integer 1. In such a case, the basis step begins at a start ...
Equality in the Presence of Apartness: An Application of Structural
... the new intuitionistic concepts. We shall here use a uniform notation in which the intuitionistic notion is written with a slash over the classical one, as in a 6= b. The properties of this notion of apartness are, first, irreflexivity ¬ a 6= a, and, secondly, the “splitting” of an apartness a 6= b in ...
... the new intuitionistic concepts. We shall here use a uniform notation in which the intuitionistic notion is written with a slash over the classical one, as in a 6= b. The properties of this notion of apartness are, first, irreflexivity ¬ a 6= a, and, secondly, the “splitting” of an apartness a 6= b in ...
Godel`s Proof
... gous Gödelian leap with respect to computers would be to see that because computers at base manipulate numbers, and because numbers are a universal medium for the embedding of patterns of any sort, computers can deal with arbitrary patterns, whether they are logical or illogical, consistent or inco ...
... gous Gödelian leap with respect to computers would be to see that because computers at base manipulate numbers, and because numbers are a universal medium for the embedding of patterns of any sort, computers can deal with arbitrary patterns, whether they are logical or illogical, consistent or inco ...
PDF
... We briefly review the five standard systems of reverse mathematics. For completeness, we include systems stronger than arithmetical comprehension, but these will play no part in this paper. Details, general background, and results, as well as many examples of reversals, can be found in Simpson [1999 ...
... We briefly review the five standard systems of reverse mathematics. For completeness, we include systems stronger than arithmetical comprehension, but these will play no part in this paper. Details, general background, and results, as well as many examples of reversals, can be found in Simpson [1999 ...
Introduction to Logic
... The term “logic” may be, very roughly and vaguely, associated with something like “correct thinking”. Aristotle defined a syllogism as “discourse in which, certain things being stated something other than what is stated follows of necessity from their being so.” And, in fact, this intuition not only ...
... The term “logic” may be, very roughly and vaguely, associated with something like “correct thinking”. Aristotle defined a syllogism as “discourse in which, certain things being stated something other than what is stated follows of necessity from their being so.” And, in fact, this intuition not only ...
Cylindric Modal Logic - Homepages of UvA/FNWI staff
... relation with temporal logics in computer science (Gabbay [12], Immerman-Kozen [17], Venema [43]). For α ≥ ω the logic is sometimes called the finitary logic of infinitary relations, cf. Sain [34]. Note that as their order is fixed, the variables in atomic relational formulas do not provide any info ...
... relation with temporal logics in computer science (Gabbay [12], Immerman-Kozen [17], Venema [43]). For α ≥ ω the logic is sometimes called the finitary logic of infinitary relations, cf. Sain [34]. Note that as their order is fixed, the variables in atomic relational formulas do not provide any info ...
Classical BI - UCL Computer Science
... nonextensible since r ◦ −r 6= ∅ for all r by Proposition 2.2, and ∞ is the unique element r satisfying −r = e. Thus, in CBI-models, if r |= I and r |= J then r = ∞ and ∞ |= P , so I ∧ J → P is a theorem of CBI. However, it is not a theorem of BBI, since one can easily construct a partial or relation ...
... nonextensible since r ◦ −r 6= ∅ for all r by Proposition 2.2, and ∞ is the unique element r satisfying −r = e. Thus, in CBI-models, if r |= I and r |= J then r = ∞ and ∞ |= P , so I ∧ J → P is a theorem of CBI. However, it is not a theorem of BBI, since one can easily construct a partial or relation ...
(pdf)
... encode sentences as numbers, so that Peano Arithmetic, being a theory of numbers, may indirectly talk about its own sentences. These issues are sidestepped in a theory of sets such as ZFC, because virtually any mathematical object, including systems of sentences, can already be thought of as sets, s ...
... encode sentences as numbers, so that Peano Arithmetic, being a theory of numbers, may indirectly talk about its own sentences. These issues are sidestepped in a theory of sets such as ZFC, because virtually any mathematical object, including systems of sentences, can already be thought of as sets, s ...
Numbers! Steven Charlton - Fachbereich | Mathematik
... v00 ) If, for all y1 , . . . , yk ∈ N, the formula φ(0, y, . . . , yn ) holds, and moreover for all n, φ(n, y1 , . . . , yn ) implies φ(S(n), y1 , . . . , yn ). Then φ(n, y1 , . . . , yk ) is true for all natural numbers. It might seem, at first glance, that the ‘first-order’ axiom scheme is equival ...
... v00 ) If, for all y1 , . . . , yk ∈ N, the formula φ(0, y, . . . , yn ) holds, and moreover for all n, φ(n, y1 , . . . , yn ) implies φ(S(n), y1 , . . . , yn ). Then φ(n, y1 , . . . , yk ) is true for all natural numbers. It might seem, at first glance, that the ‘first-order’ axiom scheme is equival ...
Notes on First Order Logic
... Induction Step Suppose that ϕ is (∀y)ψ. Since τ is substitutable for x in ϕ we have two cases: 1. x does not occur free in ψ. Then ((∀y)ψ)[x/τ ] is the same as (∀y)ψ. Furthermore s and s[x/τ ] agree on all free variables in (∀y)ψ. By Theorem ??, we have A, s |= (∀y)ψ[x/τ ] iff A, s |= (∀y)ψ iff A, ...
... Induction Step Suppose that ϕ is (∀y)ψ. Since τ is substitutable for x in ϕ we have two cases: 1. x does not occur free in ψ. Then ((∀y)ψ)[x/τ ] is the same as (∀y)ψ. Furthermore s and s[x/τ ] agree on all free variables in (∀y)ψ. By Theorem ??, we have A, s |= (∀y)ψ[x/τ ] iff A, s |= (∀y)ψ iff A, ...
Introduction to Logic
... other than what is stated follows of necessity from their being so.” And, in fact, this intuition not only lies at its origin, ca. 500 BC, but has been the main force motivating its development since that time until the last century. There was a medieval tradition according to which the Greek philos ...
... other than what is stated follows of necessity from their being so.” And, in fact, this intuition not only lies at its origin, ca. 500 BC, but has been the main force motivating its development since that time until the last century. There was a medieval tradition according to which the Greek philos ...
The Dedekind Reals in Abstract Stone Duality
... one closed point.) We can form products of spaces, X × Y , and exponentials of the form ΣX , but not arbitrary Y X . The theory also provides certain “Σ-split” subspaces, which we explain with reference to the real line in Sections 3, 5 and 7. All maps between these spaces are continuous, not as a t ...
... one closed point.) We can form products of spaces, X × Y , and exponentials of the form ΣX , but not arbitrary Y X . The theory also provides certain “Σ-split” subspaces, which we explain with reference to the real line in Sections 3, 5 and 7. All maps between these spaces are continuous, not as a t ...
The Logic of Provability
... Classical first-order arithmetic with induction; also called arithmetic or PA. More formally, we take the signature of PA to have ‘0’ as a constant and ‘+’, ‘·’, and ‘<’ as binary function symbols; PA is then the theory axiomatized by the following: • ∀x(sx 6= 0) • ∀x, y(sx = sy → x = y) • For every ...
... Classical first-order arithmetic with induction; also called arithmetic or PA. More formally, we take the signature of PA to have ‘0’ as a constant and ‘+’, ‘·’, and ‘<’ as binary function symbols; PA is then the theory axiomatized by the following: • ∀x(sx 6= 0) • ∀x, y(sx = sy → x = y) • For every ...
Classical Logic and the Curry–Howard Correspondence
... In the past, formal proofs were usually given in the traditional linear style due (or attributed) to Euclid, where a derivation would begin with axioms and proceed line by line, with each line derived from those preceding it by means of some inference rule. Nowadays such logics are known as ‘Hilbert ...
... In the past, formal proofs were usually given in the traditional linear style due (or attributed) to Euclid, where a derivation would begin with axioms and proceed line by line, with each line derived from those preceding it by means of some inference rule. Nowadays such logics are known as ‘Hilbert ...
The logic of negationless mathematics
... are introduced as basic relations of our logical system by means of the axioms A9.020133. x = y and x # y are atomic formulas (cf. D9.020131). ...
... are introduced as basic relations of our logical system by means of the axioms A9.020133. x = y and x # y are atomic formulas (cf. D9.020131). ...
i(k-1)
... By the end of this unit, you should be able to: – Formally prove properties of the non-negative integers (or a subset like integers larger than 3) that have appropriate self-referential structure— including both equalities and inequalities—using either weak or strong induction as needed. – Critique ...
... By the end of this unit, you should be able to: – Formally prove properties of the non-negative integers (or a subset like integers larger than 3) that have appropriate self-referential structure— including both equalities and inequalities—using either weak or strong induction as needed. – Critique ...
Proof analysis beyond geometric theories: from rule systems to
... 2011), systems for collective intentionality (Hakli and Negri 2011), and dynamic logics such as the logic of public announcement (Negri and Maffezioli 2010) and the epistemic logic of programs (Maffezioli and Naibo 2013). In all these applications, the geometric rule scheme suffices for the extra ma ...
... 2011), systems for collective intentionality (Hakli and Negri 2011), and dynamic logics such as the logic of public announcement (Negri and Maffezioli 2010) and the epistemic logic of programs (Maffezioli and Naibo 2013). In all these applications, the geometric rule scheme suffices for the extra ma ...
Gentzen`s original consistency proof and the Bar Theorem
... candidate for such an interpretation was this: A is true precisely if we can state a ‘reduction rule’ for ` A, i.e. a rule for constructing a reduction tree.4 Like Gödel, Gentzen had discovered the double negation interpretation of classical first-order number theory in the corresponding intuition ...
... candidate for such an interpretation was this: A is true precisely if we can state a ‘reduction rule’ for ` A, i.e. a rule for constructing a reduction tree.4 Like Gödel, Gentzen had discovered the double negation interpretation of classical first-order number theory in the corresponding intuition ...
One Step At A Time - Carnegie Mellon School of Computer Science
... have shaken. Statement: The number of people of odd parity must be even. Zero hands have been shaken at the start of a party, so zero people have odd parity. If 2 people of different parities shake, then they both swap parities and the odd parity count is unchanged. If 2 people of the same parity sh ...
... have shaken. Statement: The number of people of odd parity must be even. Zero hands have been shaken at the start of a party, so zero people have odd parity. If 2 people of different parities shake, then they both swap parities and the odd parity count is unchanged. If 2 people of the same parity sh ...
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.