Variations on a Montagovian Theme
... terminology. A numerical relation R is called recursive if there is a mechanical algorithm to check, for any given numbers, whether they stand in this relation to one another or not. R is recursively enumerable if there is an algorithm for listing all and only the numbers that stand in the relation ...
... terminology. A numerical relation R is called recursive if there is a mechanical algorithm to check, for any given numbers, whether they stand in this relation to one another or not. R is recursively enumerable if there is an algorithm for listing all and only the numbers that stand in the relation ...
connections to higher type Recursion Theory, Proof-Theory
... using the given bijective pairing of numbers. However, since we are interested in higher type computations, as given by the HPEF, we need also other kinds of higher type objects, such as exponentiations, in the category. Unfortunately, there is no general way to enumerate the set of morphisms of two ...
... using the given bijective pairing of numbers. However, since we are interested in higher type computations, as given by the HPEF, we need also other kinds of higher type objects, such as exponentiations, in the category. Unfortunately, there is no general way to enumerate the set of morphisms of two ...
DMT irm 6
... (d) Since each sequence of 2 H’s and 4 T’s corresponds to a 2-element subset of {1, 2, 3, 4, 5, 6}, and vice versa, the number of such sequences is equal to the number of such subsets, which is the combination number C(6, 2). Alternatively, to find a sequence of 6 H’s and T’s, with exactly 2 positio ...
... (d) Since each sequence of 2 H’s and 4 T’s corresponds to a 2-element subset of {1, 2, 3, 4, 5, 6}, and vice versa, the number of such sequences is equal to the number of such subsets, which is the combination number C(6, 2). Alternatively, to find a sequence of 6 H’s and T’s, with exactly 2 positio ...
Basic Logic and Fregean Set Theory - MSCS
... their mathematics, they all agree, be it sometimes grudgingly, on the same first-order constructive logic: Intuitionistic logic. This logic, and some further principles on the existence of natural numbers and sets, is recognized by all leading schools, and may be considered the basic constructive lo ...
... their mathematics, they all agree, be it sometimes grudgingly, on the same first-order constructive logic: Intuitionistic logic. This logic, and some further principles on the existence of natural numbers and sets, is recognized by all leading schools, and may be considered the basic constructive lo ...
The Art of Ordinal Analysis
... In the above, many notions were left unexplained. We will now consider them one by one. The elementary computable functions are exactly the Kalmar elementary functions, i.e. the class of functions which contains the successor, projection, zero, addition, multiplication, and modified subtraction func ...
... In the above, many notions were left unexplained. We will now consider them one by one. The elementary computable functions are exactly the Kalmar elementary functions, i.e. the class of functions which contains the successor, projection, zero, addition, multiplication, and modified subtraction func ...
Interpreting Lattice-Valued Set Theory in Fuzzy Set Theory
... This paper presents a comparison of two axiomatic set theories over two non-classical logics. In particular, it suggests an interpretation of lattice-valued set theory as defined in [16] by S. Titani in fuzzy set theory as defined in [11] by authors of this paper. There are many different conception ...
... This paper presents a comparison of two axiomatic set theories over two non-classical logics. In particular, it suggests an interpretation of lattice-valued set theory as defined in [16] by S. Titani in fuzzy set theory as defined in [11] by authors of this paper. There are many different conception ...
Sets, Infinity, and Mappings - University of Southern California
... The term elements emphasizes that the objects of a set can be viewed as its smallest “atomic” pieces. The union of all pieces forms the whole set. The set A defined in (4) is a 3-element set (it consists of 3 objects). The set B defined in (5) is a 2-element set. The term members is useful when sets ...
... The term elements emphasizes that the objects of a set can be viewed as its smallest “atomic” pieces. The union of all pieces forms the whole set. The set A defined in (4) is a 3-element set (it consists of 3 objects). The set B defined in (5) is a 2-element set. The term members is useful when sets ...
The Surprise Examination Paradox and the Second Incompleteness
... The second incompleteness theorem follows directly from Gödel’s original proof for the first incompleteness theorem. As described above, Gödel expressed the statement “this statement has no proof” and showed that, if the theory is consistent, this is a true statement (over N) that has no proof. In ...
... The second incompleteness theorem follows directly from Gödel’s original proof for the first incompleteness theorem. As described above, Gödel expressed the statement “this statement has no proof” and showed that, if the theory is consistent, this is a true statement (over N) that has no proof. In ...
Assignment MCS-013 Discrete Mathematics Q1: a) Make truth table
... What we do to prove p => q in this method is to assume that q is false and then show that p is false. Example: Prove that ‘If x, y belong Z such that xy is odd then both x and y are odd’. Solution: p: xy is odd. q: both x and y is odd, then ~p: xy is even. ~q:both x and y is even. We want to prove p ...
... What we do to prove p => q in this method is to assume that q is false and then show that p is false. Example: Prove that ‘If x, y belong Z such that xy is odd then both x and y are odd’. Solution: p: xy is odd. q: both x and y is odd, then ~p: xy is even. ~q:both x and y is even. We want to prove p ...