Related papers: From Incompleteness Towards Completeness
The pursue of what are properties that can be identified to permit an automated reasoning program to generate and find new and interesting theorems is an interesting research goal (pun intended). The automatic discovery of new theorems is a…
Internal categories feature notions of limit and completeness, as originally proposed in the context of the effective topos. This paper sets out the theory of internal completeness in a general context, spelling out the details of the…
This is a study of S. Kripke's notion of fulfilment. Motivated by Paris-Harrington statement, Kripke was looking for a proof of G\"odel's Incompleteness Theorem which was model-theoretic, natural (without self-reference), and easy.…
It is proved that any polynomial vector field in two complex variables which is complete on a non-algebraic trajectory is complete.
We find all polyhedral graphs such that their complements are still polyhedral. These turn out to be all self-complementary.
This paper provides an algebraic reconstruction of Einstein's own argument for the incompleteness of quantum mechanics -- the one that he thought did not make it into the EPR paper -- in order to clarify the assumptions that underlie an…
A variety V is said to be coherent if any finitely generated subalgebra of a finitely presented member of V is finitely presented. It is shown here that V is coherent if and only if it satisfies a restricted form of uniform deductive…
An alternative proof of the completeness of relational algebra with respect to allowed formulas of first-order logic is presented. The proof relies on the well-known embedding of relational algebra into cylindric algebra, which makes it…
In which a review of the concept of countability is done in mathematics, subjecting review some of the theorems so far accepted, showing their inconsistency and also taking concrete elements on the countability of all the powers of the set…
While the general form of even perfect numbers is well-known, the existence or non-existence of odd perfect numbers is still an open problem. We address this problem and prove that if a natural number is odd, then it's not perfect.
A concept of "guessability" is defined for sets of sequences of naturals. Eventually, these sets are thoroughly characterized. To do this, a nonstandard logic is developed, a logic containing symbols for the ellipsis as well as for…
We give a direct and elementary proof of the theorem on formal functions by studying the behaviour of the Godement resolution of a sheaf of modules under completion.
We present several philosophical ideas emerging from the studies of complex systems. We make a brief introduction to the basic concepts of complex systems, for then defining "abstraction levels". These are useful for representing…
A constructive proof of the Goedel-Rosser incompleteness theorem has been completed using the Coq proof assistant. Some theory of classical first-order logic over an arbitrary language is formalized. A development of primitive recursive…
Model selection and assessment with incomplete data pose challenges in addition to the ones encountered with complete data. There are two main reasons for this. First, many models describe characteristics of the complete data, in spite of…
Despite provable unknowables in recursion theory, indeterminism and randomness in physics is confined to conventions, subjective beliefs and preliminary evidence. The history of the issue is very briefly reviewed, and answers to five…
Recently there have emerged an assortment of theorems relating to the 'absoluteness of emerged events,' and these results have sometimes been used to argue that quantum mechanics may involve some kind of metaphysically radical…
Complementarity is one of the main features of quantum physics that radically departs from classical notions. Here we consider the limitations that this principle imposes due to the unpredictability of measurement outcomes of incompatible…
This article discusses completeness of Boolean Algebra as First Order Theory in Goedel's meaning. If Theory is complete then any possible transformation is equivalent to some transformation using axioms, predicates etc. defined for this…
According to Chaitin, G\"odel once told him "it doesn't matter which paradox you use [to prove the First Incompleteness Theorem]". In this paper I will present a few infinitary paradoxes and show how to "translate" them to some undecidable…