Primitive Recursion Chapter 2
... case a contradiction results. The usual sort of response is to try to excise the contradiction from set theory, leaving everything else more or less intact. But if obvious assumptions led to contradiction before, why can’t the remaining existence assumptions do so again, particularly when unimaginab ...
... case a contradiction results. The usual sort of response is to try to excise the contradiction from set theory, leaving everything else more or less intact. But if obvious assumptions led to contradiction before, why can’t the remaining existence assumptions do so again, particularly when unimaginab ...
Lecture Notes - Department of Mathematics
... alphabet is often used to denote “large” sets.) Is Ω tame? If it is, then it belongs to Ω and hence, Ω is wild. On the other hand, if Ω is wild then Ω is its element. Since by definition, all elements of Ω are tame, Ω is tame as well. Therefore, we proved that wildness of Ω implies its tameness and ...
... alphabet is often used to denote “large” sets.) Is Ω tame? If it is, then it belongs to Ω and hence, Ω is wild. On the other hand, if Ω is wild then Ω is its element. Since by definition, all elements of Ω are tame, Ω is tame as well. Therefore, we proved that wildness of Ω implies its tameness and ...
A Geometric Perspective on the Riemann Zeta Function`s Partial Sums
... 1. The Importance of the Riemann Zeta Function Very complex mathematical ideas often spring from the investigation of questions that are simple to understand. The subject of this article - the behavior of the Riemann Zeta Function - is one such complex mathematical object. However, the study of the ...
... 1. The Importance of the Riemann Zeta Function Very complex mathematical ideas often spring from the investigation of questions that are simple to understand. The subject of this article - the behavior of the Riemann Zeta Function - is one such complex mathematical object. However, the study of the ...
Hidden Periodicity and Chaos in the Sequence of Prime Numbers
... It is believed that properties of the gaps between consecutive primes can provide a lot of information about the primes distribution in the natural sequence. The so-called prime number theorem states that the “average length” of the gap between a prime p and the next prime number is proportional as ...
... It is believed that properties of the gaps between consecutive primes can provide a lot of information about the primes distribution in the natural sequence. The so-called prime number theorem states that the “average length” of the gap between a prime p and the next prime number is proportional as ...
Axiomatic Set Teory P.D.Welch.
... very real substance to Cantor’s efforts to build up a hiearchy of increasing complexity from simple sets.) However it was clear that although the study of such sets was rewarded with a regular picture of their properties, this was far from proving anything about all sets. We now know that Cantor was ...
... very real substance to Cantor’s efforts to build up a hiearchy of increasing complexity from simple sets.) However it was clear that although the study of such sets was rewarded with a regular picture of their properties, this was far from proving anything about all sets. We now know that Cantor was ...
Full text
... necessarily converges to 1 by a finite number of iterations of the process such that, if an odd number is given, multiply by 3 and add 1; if an even number if given, divide by 2. The first step is to show an infinite sequence generated by that iterative process is recursive. For the sake of that obj ...
... necessarily converges to 1 by a finite number of iterations of the process such that, if an odd number is given, multiply by 3 and add 1; if an even number if given, divide by 2. The first step is to show an infinite sequence generated by that iterative process is recursive. For the sake of that obj ...
The Nature of Mathematics
... Axiom A true mathematical statement whose truth is accepted without proof. Theorem A true mathematical statement whose truth can be verified is often referred to as a theorem. Corollary A mathematical result that can be deduced from some earlier result. Lemma A mathematical result that is useful in ...
... Axiom A true mathematical statement whose truth is accepted without proof. Theorem A true mathematical statement whose truth can be verified is often referred to as a theorem. Corollary A mathematical result that can be deduced from some earlier result. Lemma A mathematical result that is useful in ...
The Relative Efficiency of Propositional Proof
... see that the existence of d follows from P = JVP, observe that the problem solved by d is in JVP. In fact, a nondeterministic Turing machine can write any string of length n on its tape and then verify that the string is a proof of the given proposition. For any reasonable logical theory, this verif ...
... see that the existence of d follows from P = JVP, observe that the problem solved by d is in JVP. In fact, a nondeterministic Turing machine can write any string of length n on its tape and then verify that the string is a proof of the given proposition. For any reasonable logical theory, this verif ...
Sharp estimate on the supremum of a class of sums of small i.i.d.
... what can be told if this condition is dropped. This is the subject of my paper [5]. I also discuss some examples in that paper which show that its estimates are sharp, and I compare them with the results of some earlier works. The proofs in [5] are based on Theorem 1 of this work. But since the argu ...
... what can be told if this condition is dropped. This is the subject of my paper [5]. I also discuss some examples in that paper which show that its estimates are sharp, and I compare them with the results of some earlier works. The proofs in [5] are based on Theorem 1 of this work. But since the argu ...
Skewes Numbers
... The average spacing between primes around x is log x. Before a crossover interval, the primes are further apart than this. As we enter a crossover interval, the primes become more crowded: the spacing between primes is a little less than 727 in the first crossover interval. As we leave the interval, ...
... The average spacing between primes around x is log x. Before a crossover interval, the primes are further apart than this. As we enter a crossover interval, the primes become more crowded: the spacing between primes is a little less than 727 in the first crossover interval. As we leave the interval, ...