Basic Logic and Fregean Set Theory - MSCS
... hand this removes the need for additional evidence as proposed by Kreisel. On the other hand we need a new constructive logic. The following is the interpretation that naturally follows. • ⊤ is true by itself. that is, the empty proof suffices. There is no proof for ⊥. • A proof p of A ∧ B consists ...
... hand this removes the need for additional evidence as proposed by Kreisel. On the other hand we need a new constructive logic. The following is the interpretation that naturally follows. • ⊤ is true by itself. that is, the empty proof suffices. There is no proof for ⊥. • A proof p of A ∧ B consists ...
SOME AXIOMS FOR CONSTRUCTIVE ANALYSIS Introduction
... JOAN RAND MOSCHOVAKIS AND GARYFALLIA VAFEIADOU Abstract. This note explores the common core of constructive, intuitionistic, recursive and classical analysis from an axiomatic standpoint. In addition to clarifying the relation between Kleene’s and Troelstra’s minimal formal theories of numbers and n ...
... JOAN RAND MOSCHOVAKIS AND GARYFALLIA VAFEIADOU Abstract. This note explores the common core of constructive, intuitionistic, recursive and classical analysis from an axiomatic standpoint. In addition to clarifying the relation between Kleene’s and Troelstra’s minimal formal theories of numbers and n ...
Coordinate-free logic - Utrecht University Repository
... different than saying that there are ‘out there’ a less-than relation and a greaterthan relation. In my view, people who think there are really two such relations are misled by language. It seems hard to deny that 4’s being less than 6 is the very same fact as 6’s being greater than 4. In English an ...
... different than saying that there are ‘out there’ a less-than relation and a greaterthan relation. In my view, people who think there are really two such relations are misled by language. It seems hard to deny that 4’s being less than 6 is the very same fact as 6’s being greater than 4. In English an ...
The unintended interpretations of intuitionistic logic
... negation as set theoretic union, intersection, and relative complement. For intuitionistic propositional logic Stone and Tarski obtained a completeness theorem for topological spaces by assigning open sets to atoms and by interpreting disjunction, conjunction, and implication as in the classical cas ...
... negation as set theoretic union, intersection, and relative complement. For intuitionistic propositional logic Stone and Tarski obtained a completeness theorem for topological spaces by assigning open sets to atoms and by interpreting disjunction, conjunction, and implication as in the classical cas ...
Jean Van Heijenoort`s View of Modern Logic
... Thank you very much for your friendly card and the offprints of your papers. I am concurrently sending you reprints of my two essays regarding the fundamentals; several passages therein relate to the results that you obtained. For example, my paper entitled “Über formal unentscheidbare Sätze etc.” ...
... Thank you very much for your friendly card and the offprints of your papers. I am concurrently sending you reprints of my two essays regarding the fundamentals; several passages therein relate to the results that you obtained. For example, my paper entitled “Über formal unentscheidbare Sätze etc.” ...
Translating the Hypergame Paradox - UvA-DARE
... is essentially the same, but it is only concerned with naive set theory. Here the notion of a grounded set is considered: a set z is grounded if there is no sequence (z~)~~N with ~0 = z and Z~+I E LI+~for each natural number n. Consider the set G of all grounded sets. It is easy to see that G is gro ...
... is essentially the same, but it is only concerned with naive set theory. Here the notion of a grounded set is considered: a set z is grounded if there is no sequence (z~)~~N with ~0 = z and Z~+I E LI+~for each natural number n. Consider the set G of all grounded sets. It is easy to see that G is gro ...
Cardinality
... This is a Cherowitzo special definition – you will not find this anywhere in the literature. A denumerable set is one whose elements can written in a list (an infinite list) where all the elements appear somewhere and no element appears twice. If you can create such a list of elements of the set, th ...
... This is a Cherowitzo special definition – you will not find this anywhere in the literature. A denumerable set is one whose elements can written in a list (an infinite list) where all the elements appear somewhere and no element appears twice. If you can create such a list of elements of the set, th ...
Gresham Ideas - Gresham College
... This is the first of three lectures I am giving this term which present different angles on paradoxes. This evening I will be looking at paradoxes in logic. Next month I will be talking about a paradox which has fascinated me since I was first introduced to it almost 40 years ago, and which in recen ...
... This is the first of three lectures I am giving this term which present different angles on paradoxes. This evening I will be looking at paradoxes in logic. Next month I will be talking about a paradox which has fascinated me since I was first introduced to it almost 40 years ago, and which in recen ...
A short article for the Encyclopedia of Artificial Intelligence: Second
... then it is η-convertible to λx(M x), provided x is not free in M ). Many standard proof-theoretic results – such as cut-elimination (Girard, 1986), unification (Huet, 1975), resolution (Andrews, 1971), and Skolemization and Herbrand’s Theorem (Miller, 1987) – have been formulated for this fragment. ...
... then it is η-convertible to λx(M x), provided x is not free in M ). Many standard proof-theoretic results – such as cut-elimination (Girard, 1986), unification (Huet, 1975), resolution (Andrews, 1971), and Skolemization and Herbrand’s Theorem (Miller, 1987) – have been formulated for this fragment. ...
ppt
... Cardinality (size) Power set (and an induction proof) Cartesian products Examples of Set proofs. ...
... Cardinality (size) Power set (and an induction proof) Cartesian products Examples of Set proofs. ...
Here - Dorodnicyn Computing Centre of the Russian Academy of
... Thus, from the main Cantor's theorem it follows that the Cantor's axiomatic statement "all sets are actual" (above) - which is the only basis for all his transfinite ordinal and cardinal constructions is wrong, i.e., according to Poincare, all Cantor's set theory as well as all modern "non-naive" ax ...
... Thus, from the main Cantor's theorem it follows that the Cantor's axiomatic statement "all sets are actual" (above) - which is the only basis for all his transfinite ordinal and cardinal constructions is wrong, i.e., according to Poincare, all Cantor's set theory as well as all modern "non-naive" ax ...
Set theory
Set theory is the branch of mathematical logic that studies sets, which informally are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics. The language of set theory can be used in the definitions of nearly all mathematical objects.The modern study of set theory was initiated by Georg Cantor and Richard Dedekind in the 1870s. After the discovery of paradoxes in naive set theory, numerous axiom systems were proposed in the early twentieth century, of which the Zermelo–Fraenkel axioms, with the axiom of choice, are the best-known.Set theory is commonly employed as a foundational system for mathematics, particularly in the form of Zermelo–Fraenkel set theory with the axiom of choice. Beyond its foundational role, set theory is a branch of mathematics in its own right, with an active research community. Contemporary research into set theory includes a diverse collection of topics, ranging from the structure of the real number line to the study of the consistency of large cardinals.