The Case Against Cantor’s Diagonal Argument

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The way to list all infinite real numbers and to construct a bijection between natural numbers and real numbers

Georg Cantor defined countable and uncountable sets for infinite sets. The set of natural numbers is defined as a countable set, and the set of real numbers is proved to be uncountable by Cantor’s diagonal argument. Most scholars accept that it is impossible to construct a bijection between the set of natural numbers and the set of real numbers. However, the way to construct a bijection between the set of natural numbers and the set of real numbers is proposed in this paper. The set of real numbers can be proved to be countable by Cantor’s definition. Cantor’s diagonal argument is challenged because it also can prove the set of natural numbers to be uncountable. The process of argumentation provides us new perspectives to consider about the size of infinite sets.

A proof for Cantor-Schröder-Bernstein Theorem using the diagonal argument

We prove Cantor-Schröder-Bernstein theorem using the diagonal argument.

Divisibility Networks of the Rational Numbers in the Unit Interval

Divisibility networks of natural numbers present a scale-free distribution as many other process in real life due to human interventions. This was quite unexpected since it is hard to find patterns concerning anything related with prime numbers. However, it is by now unclear if this behavior can also be found in other networks of mathematical nature. Even more, it was yet unknown if such patterns are present in other divisibility networks. We study networks of rational numbers in the unit interval where the edges are defined via the divisibility relation. Since we are dealing with infinite sets, we need to define an increasing covering of subnetworks. This requires an order of the numbers different from the canonical one. Therefore, we propose the construction of four different orders of the rational numbers in the unit interval inspired in Cantor’s diagonal argument. We motivate why these orders are chosen and we compare the topologies of the corresponding divisibility networks showing that all of them have a free-scale distribution. We also discuss which of the four networks should be more suitable for these analyses.

The Case Against Cantor’s Diagonal Argument

We present the case against Cantor’s Diagonal Argument (CDA), exposing a number of fatal inconsistencies.

The Diagonalization Paradox

In 1891 Georg Cantor published his Diagonal Argument which, he asserted, proved that the real numbers cannot be put into a one-to-one correspondence with the natural numbers. In this paper we will see how by varying the initial conditions of the demonstration we can use Cantor’s method to produce a one-to-one correspondence between the set of natural numbers and the set of infinite binary decimals in the open interval (0, 1).

The Diagonal Argument Strikes Again: Why Physical Systems Can Never Be Sure

Chalmers has described the meta-problem of consciousness as the problem of understanding how and why we come to believe that we are conscious. Here we show that the meta-problem of consciousness is intimately related to another problem; the meta-problem of existence, or the problem of understanding how and why we come to believe that we exist. This problem is shown to lead to a version of Russell's paradox which makes it impossible for any physical system ever to be sure that it exists. The problem is illustrated by a thought experiment, the "sleepwalker paradox", which shows that no physical system can ever be sure that it is not in a dreamless sleep.

Generalizations of Rice's Theorem, Applicable to Executable and Non-Executable Formalisms

We formulate and prove two Rice-like theoremsthat characterize limitations on nameability of propertieswithin a given naming scheme for partial functions.Such a naming scheme can, but need not be, an executable formalism.A programming language is an example of an executable naming scheme,where the program text names the partial function it implements.Halting is an example of a propertythat is not nameable in that naming scheme.The proofs reveal requirements on the naming schemeto make the characterization work.Universal programming languages satisfy these requirements,but also other formalisms can satisfy them.We present some non-universal programming languagesand a non-executable specification languagesatisfying these requirements.Our theorems haveTuring's well-known Halting Theorem and Rice's Theorem as special cases,by applying them to a universal programming language or Turing Machines as naming scheme.Thus, our proofs separate the nature of the naming scheme(which can, but need not, coincide with computability) from the diagonal argument.This sheds further light on how far reaching and simple the `diagonal' argument is in itself.