A preprint version is available here in pdf.
... Rep(M, N ) and SSy(M ), and that these two are contained in SSy(M, N ). Examples can be given (using the tools outlined in this section and in Section 4) for which these are all different. On the other hand if the pair (M, N ) is recursively saturated then SSy(M, N ) = SSy(M ). Note that SSy(M ) can ...
... Rep(M, N ) and SSy(M ), and that these two are contained in SSy(M, N ). Examples can be given (using the tools outlined in this section and in Section 4) for which these are all different. On the other hand if the pair (M, N ) is recursively saturated then SSy(M, N ) = SSy(M ). Note that SSy(M ) can ...
Lecture 2 - Thursday June 30th
... Sketch Proof. That (1) is true follows from the fact that (1) holds for n = 0, since {k ∈ N|k ≥ 1} is an inductive set, so it equals N which shows k ≥ 1 for all k ∈ N. Therefore since every integer x > 0 is a natural number and we just showed x ≥ 1, (0, 1) contains no integer. Now we consider (n, n ...
... Sketch Proof. That (1) is true follows from the fact that (1) holds for n = 0, since {k ∈ N|k ≥ 1} is an inductive set, so it equals N which shows k ≥ 1 for all k ∈ N. Therefore since every integer x > 0 is a natural number and we just showed x ≥ 1, (0, 1) contains no integer. Now we consider (n, n ...
Semi-constr. theories - Stanford Mathematics
... (i) The systems Res-HAω + (AC) + (MP) + (μ) and Res-PAω + (QF-AC) + (μ) are proof-theoretically equivalent to and conservative extensions of PA; furthermore they are conservative extensions of the 2nd order system ACA0 for ∏12 sentences. (ii) The systems HAω + (AC) + (MP) + (μ) and PAω + (QF-AC) + ...
... (i) The systems Res-HAω + (AC) + (MP) + (μ) and Res-PAω + (QF-AC) + (μ) are proof-theoretically equivalent to and conservative extensions of PA; furthermore they are conservative extensions of the 2nd order system ACA0 for ∏12 sentences. (ii) The systems HAω + (AC) + (MP) + (μ) and PAω + (QF-AC) + ...
Weak Theories and Essential Incompleteness
... essential incompleteness (Tarski, Mostowski, & Robinson, 1953): a theory is essentially incomplete if all its recursively axiomatizable extensions are incomplete. Then Gödel (Rosser) theorem in fact says that a certain weak base theory (which is recursively axiomatizable and of which Peano arithmet ...
... essential incompleteness (Tarski, Mostowski, & Robinson, 1953): a theory is essentially incomplete if all its recursively axiomatizable extensions are incomplete. Then Gödel (Rosser) theorem in fact says that a certain weak base theory (which is recursively axiomatizable and of which Peano arithmet ...
A Note on Naive Set Theory in LP
... The choice of LP as the logic in which to embed a naive set theory is not without justification. As we have noticed, it is easy to work in since models are quite easy to construct. Secondly, it is perhaps the most natural paraconsistent expansion of classical predicate logic. It leaves all things in ...
... The choice of LP as the logic in which to embed a naive set theory is not without justification. As we have noticed, it is easy to work in since models are quite easy to construct. Secondly, it is perhaps the most natural paraconsistent expansion of classical predicate logic. It leaves all things in ...
Herbrand Theorem, Equality, and Compactness
... method (such as truth tables) for that will do. Notice that by propositional compactness, it’s sufficient to consider finite sets Φ0 of ground instances. The Herbrand theorem states that this method is sound and complete. Herbrand Theorem: Let L be a first-order language without = and with at least ...
... method (such as truth tables) for that will do. Notice that by propositional compactness, it’s sufficient to consider finite sets Φ0 of ground instances. The Herbrand theorem states that this method is sound and complete. Herbrand Theorem: Let L be a first-order language without = and with at least ...
Syntax of first order logic.
... a model. By compactness, Φ ∪ Ψ has a model. But this must be infinite. q.e.d. ...
... a model. By compactness, Φ ∪ Ψ has a model. But this must be infinite. q.e.d. ...
Set theory, by Thomas Jech, Academic Press, New York, 1978, xii +
... trivial to do the following. First, write down a list of axioms about sets and membership, enunciating some "obviously true" set-theoretic principles; the most popular Hst today is called ZFC (the Zermelo-Fraenkel axioms with the axiom of Choice). Next, explain how, from ZFC, one may derive all of c ...
... trivial to do the following. First, write down a list of axioms about sets and membership, enunciating some "obviously true" set-theoretic principles; the most popular Hst today is called ZFC (the Zermelo-Fraenkel axioms with the axiom of Choice). Next, explain how, from ZFC, one may derive all of c ...
study guide.
... quantifier. A statement “exists” (“some”, “a”) ∃xP (x) is true whenever P (x) is true for at least one element x in the universe; ∃ is an existential quantifier. The word “any” means sometimes ∃ and sometimes ∀. A domain (universe) of a quantifier, sometimes written as ∃x ∈ D and ∀x ∈ D is the set o ...
... quantifier. A statement “exists” (“some”, “a”) ∃xP (x) is true whenever P (x) is true for at least one element x in the universe; ∃ is an existential quantifier. The word “any” means sometimes ∃ and sometimes ∀. A domain (universe) of a quantifier, sometimes written as ∃x ∈ D and ∀x ∈ D is the set o ...
Graded assignment three
... b) Give some examples of integers that are equivalent to 1 under this relation. ...
... b) Give some examples of integers that are equivalent to 1 under this relation. ...
On interpretations of arithmetic and set theory
... exists and is a transitive superset of x. So Lemma 11 implies x has a transitive closure, completing the induction. The proof of the converse requires the axiom of foundation but is the same as the standard proof of ∈-induction in ZF. See, for example, Chapter 2 of Drake [7]. The statement TC is not ...
... exists and is a transitive superset of x. So Lemma 11 implies x has a transitive closure, completing the induction. The proof of the converse requires the axiom of foundation but is the same as the standard proof of ∈-induction in ZF. See, for example, Chapter 2 of Drake [7]. The statement TC is not ...
Stephen Cook and Phuong Nguyen. Logical foundations of proof
... proofs in P. And a sort of converse to this last statement holds too since the theory T proves the soundness of P. Thus, in the language of the previous paragraph, the proof system P is not only complete, but efficiently so, with respect to the propositional translations of bounded theorems in T . A ...
... proofs in P. And a sort of converse to this last statement holds too since the theory T proves the soundness of P. Thus, in the language of the previous paragraph, the proof system P is not only complete, but efficiently so, with respect to the propositional translations of bounded theorems in T . A ...
Solution 1 - WUSTL Math
... We will assume that our present knowledge is just the following. We have the natural numbers or counting numbers, usually denoted by the letter N. These are just the collection {1, 2, 3, . . .}. These have the following basic properties. Lower case English letters will denote natural numbers in the ...
... We will assume that our present knowledge is just the following. We have the natural numbers or counting numbers, usually denoted by the letter N. These are just the collection {1, 2, 3, . . .}. These have the following basic properties. Lower case English letters will denote natural numbers in the ...
Math 315 Review Homework 1 1. Define Field Axioms
... 1. Define Field Axioms, Positivity Axioms and Completeness Axiom. 2. Prove, directly from the axioms above, the following properties of real numbers: (i) if a > 0, c < 0, then ac < 0; (ii) if a > 0, b > 0 and a < b, then 1/a > 1/b; (iii) there exists a positive real number a such that a2 = 2. 3. Let ...
... 1. Define Field Axioms, Positivity Axioms and Completeness Axiom. 2. Prove, directly from the axioms above, the following properties of real numbers: (i) if a > 0, c < 0, then ac < 0; (ii) if a > 0, b > 0 and a < b, then 1/a > 1/b; (iii) there exists a positive real number a such that a2 = 2. 3. Let ...
Solutions to Exercises Chapter 2: On numbers and counting
... 1 Criticise the following proof that 1 is the largest natural number. Let n be the largest natural number, and suppose than n 6= 1. Then n > 1, and so n2 > n; thus n is not the largest natural number. Of course, the flaw is the assumption that there is a largest natural number. The argument shows th ...
... 1 Criticise the following proof that 1 is the largest natural number. Let n be the largest natural number, and suppose than n 6= 1. Then n > 1, and so n2 > n; thus n is not the largest natural number. Of course, the flaw is the assumption that there is a largest natural number. The argument shows th ...
Lecture 39 Notes
... P (x) is ∀[x : D].P (x), and we know that the evidence is uniform in x as in x:D ∀[x : D].(P (x) ⇒ P (x)) or ∀[x : D].(P (x) & Q(x) ⇒ P (x)). In Lecture 38 there is a discussion of the close connection between programs with assertions (asserted programs) justified by varieties of programming logics ...
... P (x) is ∀[x : D].P (x), and we know that the evidence is uniform in x as in x:D ∀[x : D].(P (x) ⇒ P (x)) or ∀[x : D].(P (x) & Q(x) ⇒ P (x)). In Lecture 38 there is a discussion of the close connection between programs with assertions (asserted programs) justified by varieties of programming logics ...
PROOFS BY INDUCTION AND CONTRADICTION, AND WELL
... Proofs which use this property are called ‘proofs by induction,’ and usually have a common form. The goal is to prove that some property or statement P(k), holds for all k ∈ N, where the property itself depends on k. First one proves the base case, that P(0) holds (or sometimes P(1) instead of or in ...
... Proofs which use this property are called ‘proofs by induction,’ and usually have a common form. The goal is to prove that some property or statement P(k), holds for all k ∈ N, where the property itself depends on k. First one proves the base case, that P(0) holds (or sometimes P(1) instead of or in ...
1 The Natural Numbers
... The natur al number s ar e a set N tog ether with a special element called 0, and a function S : N → N satisfying the following axioms: (N1) S is injective.2 (N2) 0 6∈ S(N).3 (N3) If a subset M of N contains 0 and satisfies S(M ) ⊆ M , then M = N. The function S is called the successor function. The ...
... The natur al number s ar e a set N tog ether with a special element called 0, and a function S : N → N satisfying the following axioms: (N1) S is injective.2 (N2) 0 6∈ S(N).3 (N3) If a subset M of N contains 0 and satisfies S(M ) ⊆ M , then M = N. The function S is called the successor function. The ...
Tautologies Arguments Logical Implication
... A formula A logically implies B if A ⇒ B is a tautology. Theorem: An argument is valid iff the conjunction of its premises logically implies the conclusion. Proof: Suppose the argument is valid. We want to show (A1 ∧ . . . ∧ An) ⇒ B is a tautology. • Do we have to try all 2k truth assignments (where ...
... A formula A logically implies B if A ⇒ B is a tautology. Theorem: An argument is valid iff the conjunction of its premises logically implies the conclusion. Proof: Suppose the argument is valid. We want to show (A1 ∧ . . . ∧ An) ⇒ B is a tautology. • Do we have to try all 2k truth assignments (where ...
A Note on Bootstrapping Intuitionistic Bounded Arithmetic
... BBASIC axioms. Proof (B-1) follows from formula (b) of Proposition 1 and (CU-4). (B-4) is an immediate consequence of (CU-3) and (b) and (c) of Proposition 1. To show CUBASIC+ |= (B-5), first note that x 6= 0 ⊃ 1 ≤ x by (CU-2) and (CU-3); hence x 6= 0 ⊃ 0 6= |2x| by (CU-8) and (e) of Proposition 1 ...
... BBASIC axioms. Proof (B-1) follows from formula (b) of Proposition 1 and (CU-4). (B-4) is an immediate consequence of (CU-3) and (b) and (c) of Proposition 1. To show CUBASIC+ |= (B-5), first note that x 6= 0 ⊃ 1 ≤ x by (CU-2) and (CU-3); hence x 6= 0 ⊃ 0 6= |2x| by (CU-8) and (e) of Proposition 1 ...
Godel incompleteness
... them interesting, and therefore popular... but also misinterpreted! First of all, the theorem does not say that every axiomatic system is necessarily incomplete: for example, absolute geometry with the following rules is incomplete: 1. Any two points can be joined by a straight line. 2. Any straight ...
... them interesting, and therefore popular... but also misinterpreted! First of all, the theorem does not say that every axiomatic system is necessarily incomplete: for example, absolute geometry with the following rules is incomplete: 1. Any two points can be joined by a straight line. 2. Any straight ...
Peano and Heyting Arithmetic
... numbers. Let us name a function π which is an injective map from finite sequences to natural numbers. The range of π should be definable; that is, there should be a formula φπ such that HA ` φπ (n) when n = π(σ) for some σ and HA ` ¬φπ (n) when n is not in the range of π. Then we need the natural op ...
... numbers. Let us name a function π which is an injective map from finite sequences to natural numbers. The range of π should be definable; that is, there should be a formula φπ such that HA ` φπ (n) when n = π(σ) for some σ and HA ` ¬φπ (n) when n is not in the range of π. Then we need the natural op ...
Frege`s Other Program
... not equinumerous. Such a possibility is precluded if numbers are conceived of as objects, and in fact our counterexample crucially depends on assumptions as to what concepts have value ranges. The importance of this sort of counterexample is that it goes straight to the heart of the matter as regard ...
... not equinumerous. Such a possibility is precluded if numbers are conceived of as objects, and in fact our counterexample crucially depends on assumptions as to what concepts have value ranges. The importance of this sort of counterexample is that it goes straight to the heart of the matter as regard ...
POSSIBLE WORLDS AND MANY TRUTH VALUES
... constructed, via ¬, ∨, , from formulas τai pj , and again α00 ⇔ α000 is valid on every frame. Finally, let β be obtained from α000 by replacing each τai pj by a new variable qij , let γ be a formula which “says” that, necessarily, for each j exactly one qij holds, and let α∗ be γ ⇒ β. Then α∗ is a ...
... constructed, via ¬, ∨, , from formulas τai pj , and again α00 ⇔ α000 is valid on every frame. Finally, let β be obtained from α000 by replacing each τai pj by a new variable qij , let γ be a formula which “says” that, necessarily, for each j exactly one qij holds, and let α∗ be γ ⇒ β. Then α∗ is a ...
Wk #2 - MrsJackieBroomall
... Arithmetic Means between two numbers: Numbers which form an arithmetic sequence with the two given numbers. Geometric Means between two numbers: Numbers which form a geometric sequence with the two given numbers. ...
... Arithmetic Means between two numbers: Numbers which form an arithmetic sequence with the two given numbers. Geometric Means between two numbers: Numbers which form a geometric sequence with the two given numbers. ...
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.