Gödel incompleteness theorems and the limits of their applicability. I
... first theorem, had not yet been created at that time. In fact, it was Gödel’s theorems that to a great extent stimulated the creation of this theory: the notion of general recursive function was introduced in Gödel’s 1934 lectures [5], and the equivalence of this notion to the notion of an (everyw ...
... first theorem, had not yet been created at that time. In fact, it was Gödel’s theorems that to a great extent stimulated the creation of this theory: the notion of general recursive function was introduced in Gödel’s 1934 lectures [5], and the equivalence of this notion to the notion of an (everyw ...
Sets, Logic, Computation
... between and constructions using these. It will be good to have shorthand symbols for these, and think through the general properties of sets, relations, and functions in order to do that. If you are not used to thinking mathematically and to formulating mathematical proofs, then think of the first p ...
... between and constructions using these. It will be good to have shorthand symbols for these, and think through the general properties of sets, relations, and functions in order to do that. If you are not used to thinking mathematically and to formulating mathematical proofs, then think of the first p ...
Rational Functions With Nonnegative Integer Coefficients
... M n. Then n=0 anxn is rational. If the entries of M are nonnegative integers, it is reasonable to say that the an have a combinatorial interpretation. More generally, if M is a matrix whose entries are polynomials with nonnegative coefficients and with no constant term then the entries of (I − M )−1 a ...
... M n. Then n=0 anxn is rational. If the entries of M are nonnegative integers, it is reasonable to say that the an have a combinatorial interpretation. More generally, if M is a matrix whose entries are polynomials with nonnegative coefficients and with no constant term then the entries of (I − M )−1 a ...
Functions defined on intervals: In this section we consider functions
... functions is analogous to the relationship between the integers and rational numbers. For example, just as every integer k can be regarded as a rational number, k = k / 1 , every polynomial p can be regarded as a rational function p( x) = p( x) / 1 . Up to this point our functions have been defined ...
... functions is analogous to the relationship between the integers and rational numbers. For example, just as every integer k can be regarded as a rational number, k = k / 1 , every polynomial p can be regarded as a rational function p( x) = p( x) / 1 . Up to this point our functions have been defined ...
MATH10040: Chapter 0 Mathematics, Logic and Reasoning
... A statement may be true or false (but not both). The statement ‘A and B’ is true if and only A is true and B is true. The statement ‘A or B’ is true if at least one of A and B is true. In particular, if ‘A and B’ is true, then ‘A or B’ is true. In other words, in logic and mathematics, or is ‘inclus ...
... A statement may be true or false (but not both). The statement ‘A and B’ is true if and only A is true and B is true. The statement ‘A or B’ is true if at least one of A and B is true. In particular, if ‘A and B’ is true, then ‘A or B’ is true. In other words, in logic and mathematics, or is ‘inclus ...
Logic for Computer Science. Lecture Notes
... Finally we have to state clearly what kind of opinions (sentences) can be formulated in the language we deal with and, moreover, which of those opinions are true (valid), and which are false (invalid). Now we can investigate the subject of reasoning via the validity of expressed opinions. Such an ab ...
... Finally we have to state clearly what kind of opinions (sentences) can be formulated in the language we deal with and, moreover, which of those opinions are true (valid), and which are false (invalid). Now we can investigate the subject of reasoning via the validity of expressed opinions. Such an ab ...
The Critical Analysis of the Pythagorean Theorem and of the
... problem of truth of a geometrical theory should be solved within the framework of correct methodological basis – the unity of formal logic and of rational dialectics. The unity of formal logic and of rational dialectics represents theoretical generalization of practice. 2. Indissoluble connection, r ...
... problem of truth of a geometrical theory should be solved within the framework of correct methodological basis – the unity of formal logic and of rational dialectics. The unity of formal logic and of rational dialectics represents theoretical generalization of practice. 2. Indissoluble connection, r ...
4. Techniques of Proof: II
... Letting f (k) = sk for 1 k n, we obtain the more familiar notation S = {s1, s2, …, sn}. The same kind of counting process is possible for a denumerable set, and this is why both kinds of sets are called countable. For example, if T is denumerable, then there exists a bijection g : T, and we ma ...
... Letting f (k) = sk for 1 k n, we obtain the more familiar notation S = {s1, s2, …, sn}. The same kind of counting process is possible for a denumerable set, and this is why both kinds of sets are called countable. For example, if T is denumerable, then there exists a bijection g : T, and we ma ...
Principia Mathematica
The Principia Mathematica is a three-volume work on the foundations of mathematics, written by Alfred North Whitehead and Bertrand Russell and published in 1910, 1912, and 1913. In 1927, it appeared in a second edition with an important Introduction To the Second Edition, an Appendix A that replaced ✸9 and an all-new Appendix C.PM, as it is often abbreviated, was an attempt to describe a set of axioms and inference rules in symbolic logic from which all mathematical truths could in principle be proven. As such, this ambitious project is of great importance in the history of mathematics and philosophy, being one of the foremost products of the belief that such an undertaking may be achievable. However, in 1931, Gödel's incompleteness theorem proved definitively that PM, and in fact any other attempt, could never achieve this lofty goal; that is, for any set of axioms and inference rules proposed to encapsulate mathematics, either the system must be inconsistent, or there must in fact be some truths of mathematics which could not be deduced from them.One of the main inspirations and motivations for PM was the earlier work of Gottlob Frege on logic, which Russell discovered allowed for the construction of paradoxical sets. PM sought to avoid this problem by ruling out the unrestricted creation of arbitrary sets. This was achieved by replacing the notion of a general set with the notion of a hierarchy of sets of different 'types', a set of a certain type only allowed to contain sets of strictly lower types. Contemporary mathematics, however, avoids paradoxes such as Russell's in less unwieldy ways, such as the system of Zermelo–Fraenkel set theory.PM is not to be confused with Russell's 1903 Principles of Mathematics. PM states: ""The present work was originally intended by us to be comprised in a second volume of Principles of Mathematics... But as we advanced, it became increasingly evident that the subject is a very much larger one than we had supposed; moreover on many fundamental questions which had been left obscure and doubtful in the former work, we have now arrived at what we believe to be satisfactory solutions.""The Modern Library placed it 23rd in a list of the top 100 English-language nonfiction books of the twentieth century.