Well-foundedness of Countable Ordinals and the Hydra Game
... • ACA0 , for Arithmetical Comprehension, specifies that SM contains all arithmetical sets, that is the comprehension {x ∈ |M | | φ(x)} exists whenever φ is an arithmetical formula (that is, quantifying only over first-order free variables). Assuming RCA0 , one can prove that ACA0 is equivalent to K ...
... • ACA0 , for Arithmetical Comprehension, specifies that SM contains all arithmetical sets, that is the comprehension {x ∈ |M | | φ(x)} exists whenever φ is an arithmetical formula (that is, quantifying only over first-order free variables). Assuming RCA0 , one can prove that ACA0 is equivalent to K ...
Action Logic and Pure Induction
... adjoint to a ⊗ x, our ax. And it is at the heart of the Curry-Howard “isomorphism”.2 By contrast Tarski’s induction principle, in either form (a→a)∗ →(a→a) or a(a→a)∗ →a, is virtually unknown. In a 15-line announcement Ng and Tarski [NT77] restrict Tarski’s variety RA of relation algebras to the cla ...
... adjoint to a ⊗ x, our ax. And it is at the heart of the Curry-Howard “isomorphism”.2 By contrast Tarski’s induction principle, in either form (a→a)∗ →(a→a) or a(a→a)∗ →a, is virtually unknown. In a 15-line announcement Ng and Tarski [NT77] restrict Tarski’s variety RA of relation algebras to the cla ...
PLATONISM IN MODERN MATHEMATICS A University Thesis
... Thus, the meaning of mathematical symbols are removed, and a math proof is analyzed by deriving such a proof from a series of formal axioms and rules of inference performed in a series of logical steps. The intuitonist school of mathematics simply assumes that mathematics is a man-made activity stem ...
... Thus, the meaning of mathematical symbols are removed, and a math proof is analyzed by deriving such a proof from a series of formal axioms and rules of inference performed in a series of logical steps. The intuitonist school of mathematics simply assumes that mathematics is a man-made activity stem ...
Syllogistic Logic with Complements
... complete proof system for the associated entailment relation. In its details, the work is rather different from previous work in the area (for example, [1, 3, 5, 6, 7] and references therein). Our particular system seems new. In addition, the work here builds models using a representation theorem c ...
... complete proof system for the associated entailment relation. In its details, the work is rather different from previous work in the area (for example, [1, 3, 5, 6, 7] and references therein). Our particular system seems new. In addition, the work here builds models using a representation theorem c ...
Scharp on Replacing Truth
... one often attempts to identify some feature of the liar sentence that is shared by other problematic instances of T (instances involving the Curry sentence, liar pairs, Yablo’s paradox, and so on), but not shared with the unproblematic instances of T (instances such as ‘snow is white’, ‘1+1 = 5’ and ...
... one often attempts to identify some feature of the liar sentence that is shared by other problematic instances of T (instances involving the Curry sentence, liar pairs, Yablo’s paradox, and so on), but not shared with the unproblematic instances of T (instances such as ‘snow is white’, ‘1+1 = 5’ and ...
The semantics of propositional logic
... We cannot prove the principle of mathematical induction; it is part of the definition of natural numbers. (We will make this idea more concrete later on in the course.) One proof by induction you may have seen previously is the proof that the sum of the first n natural numbers is n(n + 1)/2. This i ...
... We cannot prove the principle of mathematical induction; it is part of the definition of natural numbers. (We will make this idea more concrete later on in the course.) One proof by induction you may have seen previously is the proof that the sum of the first n natural numbers is n(n + 1)/2. This i ...
Beyond Quantifier-Free Interpolation in Extensions of Presburger
... uninterpreted functions (UF), this allows us to encode the theory of extensional arrays (AR), using uninterpreted function symbols for read and write operations. Our interpolation procedure extracts an interpolant directly from a proof of A ⇒ C. Starting from a sound and complete proof system based ...
... uninterpreted functions (UF), this allows us to encode the theory of extensional arrays (AR), using uninterpreted function symbols for read and write operations. Our interpolation procedure extracts an interpolant directly from a proof of A ⇒ C. Starting from a sound and complete proof system based ...
Basic Logic and Fregean Set Theory - MSCS
... the interpretations of implication and universal quantification. Usually constructivists expect that a more detailed study of the logical operations will result in an improved interpretation that will confirm, or at least support, intuitionistic first-order logic. Bishop, for example, questioned the ...
... the interpretations of implication and universal quantification. Usually constructivists expect that a more detailed study of the logical operations will result in an improved interpretation that will confirm, or at least support, intuitionistic first-order logic. Bishop, for example, questioned the ...
slides
... ... without having to consider explicitly every store and interpretation! Idea: develop a proof system in which every theorem is a valid partial correctness statement Judgements of the form ⊢ {P} c {Q} De ned inductively using compositional and (mostly) syntax-directed axioms and inference rules ...
... ... without having to consider explicitly every store and interpretation! Idea: develop a proof system in which every theorem is a valid partial correctness statement Judgements of the form ⊢ {P} c {Q} De ned inductively using compositional and (mostly) syntax-directed axioms and inference rules ...
Proofs by induction - Australian Mathematical Sciences Institute
... The Tower of Hanoi is a famous puzzle, sometimes known as the End of the World Puzzle. Legend tells that, at the beginning of time, there was a Hindu temple containing three poles. On one of the poles was a stack of 64 gold discs, each one a little smaller than the one beneath it. The monks of the t ...
... The Tower of Hanoi is a famous puzzle, sometimes known as the End of the World Puzzle. Legend tells that, at the beginning of time, there was a Hindu temple containing three poles. On one of the poles was a stack of 64 gold discs, each one a little smaller than the one beneath it. The monks of the t ...
CHAPTER 10 Gentzen Style Proof Systems for Classical Logic 1
... Hilbert style systems are easy to define and admit a simple proof of the Completeness Theorem but they are difficult to use. By humans, not mentioning computers. Their emphasis is on logical axioms, keeping the rules of inference at a minimum. Gentzen systems reverse this situation by emphasizing th ...
... Hilbert style systems are easy to define and admit a simple proof of the Completeness Theorem but they are difficult to use. By humans, not mentioning computers. Their emphasis is on logical axioms, keeping the rules of inference at a minimum. Gentzen systems reverse this situation by emphasizing th ...
Note 3
... infinitely many natural numbers — induction provides a way to reason about them by finite means. Let us demonstrate the intuition behind induction with an example. Suppose we wish to prove the statement: For all natural numbers n, 0 + 1 + 2 + 3 + · · · + n = n(n+1) 2 . More formally, using the unive ...
... infinitely many natural numbers — induction provides a way to reason about them by finite means. Let us demonstrate the intuition behind induction with an example. Suppose we wish to prove the statement: For all natural numbers n, 0 + 1 + 2 + 3 + · · · + n = n(n+1) 2 . More formally, using the unive ...
WHAT IS THE RIGHT NOTION OF SEQUENTIALITY? 1. Introduction
... means that we want to have sequences involving all objects of the original theory. It is easy to give examples that fail this condition. Do we need more? The good news is that these meager means are sufficient also for coding formulas and the like. We can start with a bare minimum and build everythi ...
... means that we want to have sequences involving all objects of the original theory. It is easy to give examples that fail this condition. Do we need more? The good news is that these meager means are sufficient also for coding formulas and the like. We can start with a bare minimum and build everythi ...
Classical First-Order Logic Introduction
... propositional logic, but also the symbols ∃ and ∀ for “there exists” and “for all”, along with various symbols to represent variables, constants, functions, and relations. ...
... propositional logic, but also the symbols ∃ and ∀ for “there exists” and “for all”, along with various symbols to represent variables, constants, functions, and relations. ...
First-Order Predicate Logic (2) - Department of Computer Science
... over S. Then G follows from X (is a semantic consequence of X) if the following implication holds for every S-structure F : If F |= E for all E ∈ X, then F |= G. This is denoted by X |= G Observations • For any first-order sentence G: ∅ |= G if, and only if, G is a tautology. Since ‘being a tautolog ...
... over S. Then G follows from X (is a semantic consequence of X) if the following implication holds for every S-structure F : If F |= E for all E ∈ X, then F |= G. This is denoted by X |= G Observations • For any first-order sentence G: ∅ |= G if, and only if, G is a tautology. Since ‘being a tautolog ...
CONSTRUCTION OF NUMBER SYSTEMS 1. Peano`s Axioms and
... Caution: Here T is a fixed set defined as above. But, S is a variable subset of N. I will leave it as an exercise to check that 1 ∈ T . Next assume that n ∈ T and we want to show that σ(n) ∈ T . So let S ⊂ N with σ(n) ∈ S. If n ∈ S, then by hypothesis, we know that S has a least element. So assume t ...
... Caution: Here T is a fixed set defined as above. But, S is a variable subset of N. I will leave it as an exercise to check that 1 ∈ T . Next assume that n ∈ T and we want to show that σ(n) ∈ T . So let S ⊂ N with σ(n) ∈ S. If n ∈ S, then by hypothesis, we know that S has a least element. So assume t ...
Programming with Classical Proofs
... In order to formalize first-order logic, we start by defining a natural deduction proof system for the so-called minimal first-order logic (mFOL). Minimal logic, introduced in 1936 by Ingebrigt Johansson [23], is a simplified version of intuitionistic logic where ex falso quodlibet does not hold. In ...
... In order to formalize first-order logic, we start by defining a natural deduction proof system for the so-called minimal first-order logic (mFOL). Minimal logic, introduced in 1936 by Ingebrigt Johansson [23], is a simplified version of intuitionistic logic where ex falso quodlibet does not hold. In ...
The First Incompleteness Theorem
... If proofs aren’t linear in T ’s native form, we can do one of two things. Either (i) we just replace T ’s deductive system using trees, or whatever, with a linear version, and think about the equivalent theory T 0 instead. Or it may be less hassle if (ii) we instead extend our scheme for Gödel-numb ...
... If proofs aren’t linear in T ’s native form, we can do one of two things. Either (i) we just replace T ’s deductive system using trees, or whatever, with a linear version, and think about the equivalent theory T 0 instead. Or it may be less hassle if (ii) we instead extend our scheme for Gödel-numb ...
Cardinality
... Most mathematicians are comfortable with the axiom of choice, however, by using it one can prove: Banach-Tarski Theorem: Any solid sphere can be decomposed into five pieces that can be reassembled into two solid spheres with the same radius as the original. Repeated use of this construction can lead ...
... Most mathematicians are comfortable with the axiom of choice, however, by using it one can prove: Banach-Tarski Theorem: Any solid sphere can be decomposed into five pieces that can be reassembled into two solid spheres with the same radius as the original. Repeated use of this construction can lead ...
STANDARD COMPLETENESS THEOREM FOR ΠMTL 1
... (1) if x, y ∈ F , then x ∗ y ∈ F , (2) if x ∈ F , x ≤ y, then y ∈ F . LEMMA 2.6. For any filter F in a ΠMTL-algebra L, let us define the following equivalence relation in L: x ∼F y iff x → y ∈ F and y → x ∈ F . Then ∼F is a congruence and the quotient L/F is a ΠMTL-algebra. We will denote the equiva ...
... (1) if x, y ∈ F , then x ∗ y ∈ F , (2) if x ∈ F , x ≤ y, then y ∈ F . LEMMA 2.6. For any filter F in a ΠMTL-algebra L, let us define the following equivalence relation in L: x ∼F y iff x → y ∈ F and y → x ∈ F . Then ∼F is a congruence and the quotient L/F is a ΠMTL-algebra. We will denote the equiva ...
CHAPTER 10 Mathematical Induction
... 10.1 Proof by Strong Induction This section describes a useful variation on induction. Occasionally it happens in induction proofs that it is difficult to show that S k forces S k+1 to be true. Instead you may find that you need to use the fact that some “lower” statements S m (with m < k) force S k ...
... 10.1 Proof by Strong Induction This section describes a useful variation on induction. Occasionally it happens in induction proofs that it is difficult to show that S k forces S k+1 to be true. Instead you may find that you need to use the fact that some “lower” statements S m (with m < k) force S k ...
Algebraic Proof Systems
... Integer Linear Programing problem is given by a rational matrix {aij } and ...
... Integer Linear Programing problem is given by a rational matrix {aij } and ...
An Example of Induction: Fibonacci Numbers
... fifth Fibonacci number is a multiple of 5. As usual in mathematics, we have to start by carefully defining the objects we are studying. Definition. The sequence of Fibonacci numbers, F0 , F1 , F2 , . . ., are defined by the following equations: F0 = 0 F1 = 1 Fn + Fn+1 = Fn+2 Theorem 1. The Fibonacci ...
... fifth Fibonacci number is a multiple of 5. As usual in mathematics, we have to start by carefully defining the objects we are studying. Definition. The sequence of Fibonacci numbers, F0 , F1 , F2 , . . ., are defined by the following equations: F0 = 0 F1 = 1 Fn + Fn+1 = Fn+2 Theorem 1. The Fibonacci ...
Number
... is the successor of 3, etc. The reason for using the notation n0 instead of n + 1 is because the addition is yet to be defined. The first two axioms is intended to “build up” all the natural numbers by starting with the initial number 1 and creating the others by repeatedly applying the “successor o ...
... is the successor of 3, etc. The reason for using the notation n0 instead of n + 1 is because the addition is yet to be defined. The first two axioms is intended to “build up” all the natural numbers by starting with the initial number 1 and creating the others by repeatedly applying the “successor o ...
Kripke Models of Transfinite Provability Logic
... with ‘depth’ Θ (i.e., the order-type of <0 ) and ‘length’ Λ (the set of modalities it interprets). IΘ Λ validates all frame conditions except for condition (ii). We shall only approximate it in that we require, for ζ < ξ, v <ξ w ⇒ ∃ v 0 <ζ w such that v 0 -p v. Here p will be a set of parameters and ...
... with ‘depth’ Θ (i.e., the order-type of <0 ) and ‘length’ Λ (the set of modalities it interprets). IΘ Λ validates all frame conditions except for condition (ii). We shall only approximate it in that we require, for ζ < ξ, v <ξ w ⇒ ∃ v 0 <ζ w such that v 0 -p v. Here p will be a set of parameters and ...
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.