Related papers: Life, The Mind, and Everything
After highlighting the cases in which the semantics of a language cannot be mechanically reproduced (in which case it is called inherent), the main epistemological consequences of the first incompleteness Theorem for the two fundamental…
The Turing machine, as it was presented by Turing himself, models the calculations done by a person. This means that we can compute whatever any Turing machine can compute, and therefore we are Turing complete. The question addressed here…
A century ago, discoveries of a serious kind of logical error made separately by several leading mathematicians led to acceptance of a sharply enhanced standard for rigor within what ultimately became the foundation for Computer Science. By…
The fact that the famous Godel incompleteness theorem and the archetype of all logical paradoxes, that of the Liar, are related closely is, of course, not only well known, but is a part of the common knowledge of logician community.…
An ultimate universal theory -- a complete theory that accounts, via few and simple first principles, for all the phenomena already observed and that will ever be observed -- has been, and still is, the aspiration of most physicists and…
By nature, transmissible human knowledge is enumerable: every sentence, movie, audio record can be encoded in a sufficiently long string of 0's and 1's. The works of G\"odel, Turing and others showed that there are inherent limits and…
One aspect of the algorithmic lens in theoretical computer science is a view on other scientific disciplines that focuses on satisfactory solutions that adhere to real-world constraints, as opposed to solutions that would be optimal…
The world of mathematics is often considered abstract, with its symbols, concepts, and topics appearing unrelated to physical objects. However, it is important to recognize that the development of mathematics is fundamentally influenced by…
It is a widespread belief that results like G\"odel's incompleteness theorems or the intrinsic randomness of quantum mechanics represent fundamental limitations to humanity's strive for scientific knowledge. As the argument goes, there are…
In this paper we prove Chaitin's ``heuristic principle'', {\it the theorems of a finitely-specified theory cannot be significantly more complex than the theory itself}, for an appropriate measure of complexity. We show that the measure is…
From the perspective of the physics of complex systems (1) we deal with the current state of mod-ern physics including the crisis in physics demonstrated through its epistemological, psychological, economical as well as the social context;…
In 1686 in his Discours de Metaphysique, Leibniz points out that if an arbitrarily complex theory is permitted then the notion of "theory" becomes vacuous because there is always a theory. This idea is developed in the modern theory of…
From the perspective of the physics of complex systems (1) we deal with the current state of modern physics including the crisis in physics demonstrated through its epistemological, psychological, economical as well as the social context;…
In 1927 Heisenberg discovered that the ``more precisely the position is determined, the less precisely the momentum is known in this instant, and vice versa''. Four years later G\"odel showed that a finitely specified, consistent formal…
The famous G\"odel incompleteness theorem says that for every sufficiently rich formal theory (containing formal arithmetic in some natural sense) there exist true unprovable statements. Such statements would be natural candidates for being…
There is a cognitive limit in Human Mind. This cognitive limit has played a decisive role in almost all fields including computer sciences. The cognitive limit replicated in computer sciences is responsible for inherent Computational…
A knowledge system S describing a part of real world does in general not contain complete information. Reasoning with incomplete information is prone to errors since any belief derived from S may be false in the present state of the world.…
Hilbert and Ackermann asked for a method to consistently extend incomplete theories to complete theories. G\"odel essentially proved that any theory capable of encoding its own statements and their proofs contains statements that are true…
We introduce some early considerations of physical and mathematical impossibility as preludes to the Goedel incompleteness theorems. We consider some informal aspects of these theorems and their underlying assumptions and discuss some the…
We first partly develop a mathematical notion of stable consistency intended to reflect the actual consistency property of human beings. Then we give a generalization of the first and second G\"odel incompleteness theorem to stably…