Set Theory and Logic

2108 Submissions

[4] viXra:2108.0091 [pdf] submitted on 2021-08-18 20:43:46

Asymmetry in the Real Number Line and: A Proof that \pi + e is an Irrational Number

Authors: Alexander C Sarich
Comments: 7 Pages.

The set of all Real numbers, R, consists of all Rational numbers, Q, being any ratio of two Integer numbers that does not divide by 0. All other Real numbers that are not a Rational number are contained in the set of Irrational numbers, R/Q. These two subsets comprising all of the Real numbers are known to have distinct cardinalities of differing magnitudes of infinity[2]. When a consecutive ordering of all Rational numbers is established, whereby any unique Rational number can be shown to be disconnected from all other Rational numbers[3], a theorem regarding asymmetry on the Real number line is established. This theorem simplifies the necessary requirements to prove that the summation of two known Irrational numbers is Rational or Irrational.
Category: Set Theory and Logic

[3] viXra:2108.0063 [pdf] replaced on 2021-09-22 12:35:44

Set Theory Inc^# _∞^# Based on Infinitary Intuitionistic Logic with Restricted Modus Ponens Rule (Part.iii): Hyper Inductive Definitions.application in Transcendental Number Theory.generalized Lindemann-Weierstrass Theorem

Authors: Jaykov Foukzon
Comments: 57 Pages.

In this paper intuitionistic set theory INC# in infinitary set theoretical language is considered. External induction principle in nonstandard intuitionistic arithmetic were derived. Non trivial application in number theory is considered.The Goldbach-Euler theorem is obtained without any references to Catalan conjecture. Main results are: (i) number e^{e}is transcendental; (ii) the both numbers e+pi and e-pi are irrational.
Category: Set Theory and Logic

[2] viXra:2108.0030 [pdf] replaced on 2021-10-07 15:30:59

Logic and Intuition

Authors: Bertrand Wong
Comments: 4 Pages.

This paper aims to inspire thinking on the capabilities and potential of the human brain. The brain apparently has great potential for development and great untapped capabilities. Practically everyone is keen on improving his mental capacity, especially the capability of logical reasoning, that is, the ability in utilising logic to achieve the desired outcomes. It appears that logic is equated with intelligence and is regarded as the most important aspect of thinking by many (though emotional intelligence is now the new kid in the block which seems to be gaining traction). The author here looks at reasoning or logic, as well as intuition, from a different and perhaps unique perspective.
Category: Set Theory and Logic

[1] viXra:2108.0026 [pdf] submitted on 2021-08-08 20:57:58

Some Inconsistence in Logic

Authors: Bertrand Wong
Comments: 5 Pages.

This paper brings up some important points about logic, e.g., mathematical logic, and also an inconsistence in logic as per Godel’s incompleteness theorems which state that there are mathematical truths that are not decidable or provable. These incompleteness theorems have shaken the solid foundation of mathematics where innumerable proofs and theorems have pride of place. The great mathematician David Hilbert had been much disturbed by them. There are much long unsolved famous conjectures in mathematics, e.g., the twin primes conjecture, the Goldbach conjecture, the Riemann hypothesis, et al. Perhaps, by Godel’s incompleteness theorems the proofs for these famous conjectures will not be possible and the numerous mathematicians attempting to find solutions for these conjectures are simply banging their heads against the metaphorical wall. Besides mathematics, Godel’s incompleteness theorems will have ramifications in other areas involving logic. The paper looks at the ramifications of the incompleteness theorems, which pose the serious problem of inconsistency, and offers a solution to this dilemma. The paper also looks into the apparent inconsistence of the axiomatic method in mathematics.
Category: Set Theory and Logic