Per Lindström FIRST
... definitions of “term” (noun phrase), “formula”, “sentence” (formula without free variables), etc. of l should be explicitly stated (cf. Chapter 1, §1). In formulas we need, in addition to the mathematical symbols, among other things, certain logical symbols such as ¬ (not), ∧ (and), ∃ (there exists) ...
... definitions of “term” (noun phrase), “formula”, “sentence” (formula without free variables), etc. of l should be explicitly stated (cf. Chapter 1, §1). In formulas we need, in addition to the mathematical symbols, among other things, certain logical symbols such as ¬ (not), ∧ (and), ∃ (there exists) ...
Foundations of Mathematics I Set Theory (only a draft)
... We will refrain from giving non mathematical examples such as “the set of pupils in your class is a subset of the set of pupils of your school”. Another (mathematical) example: The set {0, 1} is a subset of {0, 1, 2}. This is clear. Is it an element? No if the set {0, 1} is not equal to one of 0, 1 ...
... We will refrain from giving non mathematical examples such as “the set of pupils in your class is a subset of the set of pupils of your school”. Another (mathematical) example: The set {0, 1} is a subset of {0, 1, 2}. This is clear. Is it an element? No if the set {0, 1} is not equal to one of 0, 1 ...
Bridge to Higher Mathematics
... some people, who hold that or means that exactly one (but not both) of the individual statements is true. There is a mathematical term having this meaning, called ‘exclusive or’ (abbreviated to xor), but it arises relatively infrequently. Just remember that whenever the word or appears in a mathemat ...
... some people, who hold that or means that exactly one (but not both) of the individual statements is true. There is a mathematical term having this meaning, called ‘exclusive or’ (abbreviated to xor), but it arises relatively infrequently. Just remember that whenever the word or appears in a mathemat ...
the existence of fibonacci numbers in the algorithmic generator for
... nature in the forms and designs of many plants and animals and have also been reproduced in various manners in art, architecture, and music. The mathematician Leonardo of Pisa, better known as Fibonacci, had a significant impact on mathematics. His contributions to mathematics have intrigued and ins ...
... nature in the forms and designs of many plants and animals and have also been reproduced in various manners in art, architecture, and music. The mathematician Leonardo of Pisa, better known as Fibonacci, had a significant impact on mathematics. His contributions to mathematics have intrigued and ins ...
31(1)
... first and last bits considered to be adjacent (i.e., the first bit follows the last bit). This condition is visible when the string is displayed in a circle with one bit "capped": the capped bit is the first bit and reading clockwise we see the second bit, the third bit, and so on to the nth bit (th ...
... first and last bits considered to be adjacent (i.e., the first bit follows the last bit). This condition is visible when the string is displayed in a circle with one bit "capped": the capped bit is the first bit and reading clockwise we see the second bit, the third bit, and so on to the nth bit (th ...
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"".