NotaBene .


Has the inference its own system? A new look at Kant's transcendental dialectic

Abstract: This article is dedicated to Kant's exceptional achievement in transcendental logic of Critique of pure reason - the inference as a pure and common logical form, not just the syllogism, is founded and deduced in its own logical formations. This result is accomplished on the basis of two main transformations. The first one is the incorporating the construction of unconditioned and its concepts in the structure of inference which allows Kant to solve the problem for the general logical function of the inference. The second connects the logical inference functions with the system of judgment and in such direction establishes these functions as extension and completion of the basic logical possibilities of the judgment. With that Kant also offers the stunning idea for the whole theoretical and structural integrity of logical thought-forms.

Kew words: Kant, transcendental dialectic, concepts of the unconditioned in the inference structure, the system of inference


Cantor's Continuum Hypothesis and its Transfinite Grounding (Part 1)

Abstract: The present paper will expose Cantor's Theory of transfinite numbers, and more specific his Continuum Hypothesis, on transfinite grounding - upon the constructing of the linear continuum c as "the pure continuous number-domain", with the words of Dedekind. For this purpose, we will unfold Cantor's grounding that cardinality of the continuum c is equal to the cardinality of the sets of natural numbers - N and the rational numbers - Q as basic dimensions in n-dimensionality of the continuum. This is a construction, which poses the known number sets as basic and initial dimensions of the continuum c. These sets are also taken in their inclusion to one another as proper subsets to each other and to the set of real numbers - R. From here, we can deduce the Continuum Hypothesis formula with producing the next Aleph-numbers and to propose a new version of generalized Continuum Hypothesis. Also this will provide the solution and interpretation of Aleph-0 as a number - with its own number value upon the continuum c and as a number of the complex processes and entities in reality.

Going even more further, we can postulate the intensity of an infinite set: as its own ability to be raised to power, or covered, with transfinite cardinal number, in particular to the least transfinite number Aleph-0. So, namely on the intensity of the infinite sets, continuity can be defined and connected with the dimensionality of the continuum. And from here we can formulate the Aleph-enumerability and omega-enumerability of infinite sets as the suggestion of grounding the enumerability of the infinite sets.

All this leads also to the construction of transfinite numbers orders, in this construction we can develop them as higher and higher number classes with increasing cardinalities. These transfinite numbers orders can provide the density of "the whole absolutely infinite aggregate of numbers", with Cantor's formulation. With all this, the new transfinite numbers, introduced by Cantor, are demonstrated as powerful instrument for describing and investigating the reality to the very boundaries of its physical and conceptual comprehension.

Key words: Continuum Hypothesis, Cantor, transfinite numbers, Aleph-0, Theory of numbers, logic of infinity.