Herbrands Theorem
... Part 2: Suppose S’ is a finite unsatisfiable set of gr. instances of clauses in S - Every interpretation I of S contains an interpretation I’ of S’ - So, if I’ falsifies S’, then I must also falsify S’ - Since S’ is falsified by every interpretation I’, it must also be falsified by every interpretat ...
... Part 2: Suppose S’ is a finite unsatisfiable set of gr. instances of clauses in S - Every interpretation I of S contains an interpretation I’ of S’ - So, if I’ falsifies S’, then I must also falsify S’ - Since S’ is falsified by every interpretation I’, it must also be falsified by every interpretat ...
The sequences part
... The most important application of sequences is the definition of convergence of an infinite series. From the elementary school you have been dealing with addition of numbers. As you know the addition gets harder as you add more and more numbers. For example it would take some time to add S100 = 1 + ...
... The most important application of sequences is the definition of convergence of an infinite series. From the elementary school you have been dealing with addition of numbers. As you know the addition gets harder as you add more and more numbers. For example it would take some time to add S100 = 1 + ...
1. Number Sense, Properties, and Operations
... c. Perform arithmetic operations with complex numbers. (CCSS: N-CN) i. Define the complex number i such that i2 = –1, and show that every complex number has the form a + bi where a and b are real numbers. (CCSS: N-CN.1) ii. Use the relation i2 = –1 and the commutative, associative, and distributive ...
... c. Perform arithmetic operations with complex numbers. (CCSS: N-CN) i. Define the complex number i such that i2 = –1, and show that every complex number has the form a + bi where a and b are real numbers. (CCSS: N-CN.1) ii. Use the relation i2 = –1 and the commutative, associative, and distributive ...
DOC - John Woods
... Metatheory of CPL A big question is, “Why do we bother with proof theory?” After all, its principal concepts – axiom, theorem, deduction, proof – have no intuitive meaning there. What’s the point? Suppose we could show that for each of these uninterpreted properties of CPL’s proof theory theory is a ...
... Metatheory of CPL A big question is, “Why do we bother with proof theory?” After all, its principal concepts – axiom, theorem, deduction, proof – have no intuitive meaning there. What’s the point? Suppose we could show that for each of these uninterpreted properties of CPL’s proof theory theory is a ...
Document
... Two sets M and N are equivalent … if it is possible to put them, by some law, in such a relation to one another that to every element of each one of them corresponds one and only one element of the other. If M and N are equivalent we often say that they have they have the same cardinality or the sam ...
... Two sets M and N are equivalent … if it is possible to put them, by some law, in such a relation to one another that to every element of each one of them corresponds one and only one element of the other. If M and N are equivalent we often say that they have they have the same cardinality or the sam ...
handout - Colorado Math Circle
... 6. Can you think of a way to use the Zeckendorf representation to convert between miles and kilometers? Negative Fibonacci Indices. We can extend the Fibonacci sequence to negative indices by applying the Fibonacci recursive formula “backwards”. Every number can be uniquely represented as a sum of d ...
... 6. Can you think of a way to use the Zeckendorf representation to convert between miles and kilometers? Negative Fibonacci Indices. We can extend the Fibonacci sequence to negative indices by applying the Fibonacci recursive formula “backwards”. Every number can be uniquely represented as a sum of d ...
A Basis Theorem for Perfect Sets
... integer; we carry the values assigned to Gx , Gy , and t during one step to the beginning of the next step, but these variables do not achieve a limit over the course of the recursion. When we speak of the least Gk , we are referring to the Gk with minimal index. Initially X and Y are not defined an ...
... integer; we carry the values assigned to Gx , Gy , and t during one step to the beginning of the next step, but these variables do not achieve a limit over the course of the recursion. When we speak of the least Gk , we are referring to the Gk with minimal index. Initially X and Y are not defined an ...
1 Basic Combinatorics
... multiplication principle is using decision trees or Cartesian products of sets, but we will not require this formulation here. We will return to this topic later. It is instructive to do another example involving the multiplication principle. Example. Call a sequence (x1 , x2 , . . . , xn−1 , xn ) o ...
... multiplication principle is using decision trees or Cartesian products of sets, but we will not require this formulation here. We will return to this topic later. It is instructive to do another example involving the multiplication principle. Example. Call a sequence (x1 , x2 , . . . , xn−1 , xn ) o ...
The Compactness Theorem 1 The Compactness Theorem
... assignments A and A0 , we say that A0 extends A if dom(A) ⊆ dom(A0 ) and if A[[pi ]] = A0 [[pi ]] for all pi ∈ dom(A). Sometimes we refer to partial assignments simply as assignments. Recall that a set of formulas S is satisfiable if there is an assignment that satisfies every formula in S. For exam ...
... assignments A and A0 , we say that A0 extends A if dom(A) ⊆ dom(A0 ) and if A[[pi ]] = A0 [[pi ]] for all pi ∈ dom(A). Sometimes we refer to partial assignments simply as assignments. Recall that a set of formulas S is satisfiable if there is an assignment that satisfies every formula in S. For exam ...
Non-standard analysis
The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers. The standard way to resolve these debates is to define the operations of calculus using epsilon–delta procedures rather than infinitesimals. Non-standard analysis instead reformulates the calculus using a logically rigorous notion of infinitesimal numbers.Non-standard analysis was originated in the early 1960s by the mathematician Abraham Robinson. He wrote:[...] the idea of infinitely small or infinitesimal quantities seems to appeal naturally to our intuition. At any rate, the use of infinitesimals was widespread during the formative stages of the Differential and Integral Calculus. As for the objection [...] that the distance between two distinct real numbers cannot be infinitely small, Gottfried Wilhelm Leibniz argued that the theory of infinitesimals implies the introduction of ideal numbers which might be infinitely small or infinitely large compared with the real numbers but which were to possess the same properties as the latterRobinson argued that this law of continuity of Leibniz's is a precursor of the transfer principle. Robinson continued:However, neither he nor his disciples and successors were able to give a rational development leading up to a system of this sort. As a result, the theory of infinitesimals gradually fell into disrepute and was replaced eventually by the classical theory of limits.Robinson continues:It is shown in this book that Leibniz's ideas can be fully vindicated and that they lead to a novel and fruitful approach to classical Analysis and to many other branches of mathematics. The key to our method is provided by the detailed analysis of the relation between mathematical languages and mathematical structures which lies at the bottom of contemporary model theory.In 1973, intuitionist Arend Heyting praised non-standard analysis as ""a standard model of important mathematical research"".