Document
... real number 1 < r < 2 with r2 = 2? We need an axiom about real numbers that isn’t taught in high school. Notation: R = the set of real numbers, Q = the set of rational numbers (“Q” for “quotient”), Z = the set of integers (“Z” for “Zahlen” which is German for “number”), and N = {1,2,3,…} the set of ...
... real number 1 < r < 2 with r2 = 2? We need an axiom about real numbers that isn’t taught in high school. Notation: R = the set of real numbers, Q = the set of rational numbers (“Q” for “quotient”), Z = the set of integers (“Z” for “Zahlen” which is German for “number”), and N = {1,2,3,…} the set of ...
[Write on board:
... real number 1 < r < 2 with r2 = 2? We need an axiom about real numbers that isn’t taught in high school. Notation: R = the set of real numbers, Q = the set of rational numbers (“Q” for “quotient”), Z = the set of integers (“Z” for “Zahlen” which is German for “number”), and N = {1,2,3,…} the set of ...
... real number 1 < r < 2 with r2 = 2? We need an axiom about real numbers that isn’t taught in high school. Notation: R = the set of real numbers, Q = the set of rational numbers (“Q” for “quotient”), Z = the set of integers (“Z” for “Zahlen” which is German for “number”), and N = {1,2,3,…} the set of ...
Default Reasoning in a Terminological Logic
... The field of TLs has lately been an active area of research, with the attention of researchers especially focusing on the investigation of their logical and computational properties. Nevertheless, few researchers have addressed the problem of extending these logics with the ability to perform defaul ...
... The field of TLs has lately been an active area of research, with the attention of researchers especially focusing on the investigation of their logical and computational properties. Nevertheless, few researchers have addressed the problem of extending these logics with the ability to perform defaul ...
Linearly Ordered Topological Spaces and Weak
... which will lead to the addition of extra points to a LOTS to obtain a weak domain representable space rather than the LOTS itself being weak domain representable. As is customary we will use ↑ a to represent the set {b ∈ X : a ≤ b} and ↓ a to represent the set {b ∈ X : b ≤ a}. Definition 3. For all ...
... which will lead to the addition of extra points to a LOTS to obtain a weak domain representable space rather than the LOTS itself being weak domain representable. As is customary we will use ↑ a to represent the set {b ∈ X : a ≤ b} and ↓ a to represent the set {b ∈ X : b ≤ a}. Definition 3. For all ...
(pdf)
... Definition 1.9. A model or structure M for some language L is an ordered triple M = (A, I, β), where A is a nonempty set, β is a variable assignment function and I is an interpretation function with domain the set of all constants, relations and function symbols in L such that: (1) For every consta ...
... Definition 1.9. A model or structure M for some language L is an ordered triple M = (A, I, β), where A is a nonempty set, β is a variable assignment function and I is an interpretation function with domain the set of all constants, relations and function symbols in L such that: (1) For every consta ...
countability diagonalization
... Example: if x=a then f(x) is b; if x≠a then f(x) is a. S’ cannot be anywhere in the matrix, since it will differ from every string by at least one symbol. But we have listed all elements in the matrix. Contradiction! The set must be uncountably infinite! CS340-Discrete Structures ...
... Example: if x=a then f(x) is b; if x≠a then f(x) is a. S’ cannot be anywhere in the matrix, since it will differ from every string by at least one symbol. But we have listed all elements in the matrix. Contradiction! The set must be uncountably infinite! CS340-Discrete Structures ...
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 ...
Document
... Example: Prove that there are infinitely many prime numbers Proof: Assume there are not infinitely many prime numbers, therefore they are listable, i.e. p1,p2,…,pn Consider the number q = p1p2…pn+1. q is not divisible by any of the listed primes Therefore, q is a prime. However, it was not ...
... Example: Prove that there are infinitely many prime numbers Proof: Assume there are not infinitely many prime numbers, therefore they are listable, i.e. p1,p2,…,pn Consider the number q = p1p2…pn+1. q is not divisible by any of the listed primes Therefore, q is a prime. However, it was not ...
The Perfect Set Theorem and Definable Wellorderings of the
... THEOREM. Let r be a reasonablepointclass and let M be a perfect set basis for r. If < is a wellorderingof a set of reals and < e r, then the field of < (i.e. the set {a: a < a}) is containedin M. PROOF. Without loss of generality we can assume that the field of < is contained in 20 so that we can wo ...
... THEOREM. Let r be a reasonablepointclass and let M be a perfect set basis for r. If < is a wellorderingof a set of reals and < e r, then the field of < (i.e. the set {a: a < a}) is containedin M. PROOF. Without loss of generality we can assume that the field of < is contained in 20 so that we can wo ...