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The Objective Scope of Thinking in Kants Transcendental Dialectic

In the Transcendental Dialectic of Critique of Pure Reason, Kant poses a task hitherto unexplored in the territory of logic: to find and derive universal forms in the realization of the supreme unity of thought. In concrete terms, these must be the higher a priori forms of reason or its supreme concepts: transcendental ideas. They base the whole thinking in its higher synthetic functions which are cast into the extension to universality of the inference functions of thinking. On this scale, the projection of the ultimate and general scope of thinking opens, which Kant defines as the totality of the conditions in thinking in general, and as a pure and supreme functional element of the unconditioned of thinking. There is no way that this innovatively set and solved task will not lead to new formations in view of the general nature and action of the higher concepts of reason. These transcendental logical formations and procedures form a new construction of whole thinking and illuminate its general objective possibilities.
Keywords: Kant, transcendental philosophy, transcendental logic, the unconditioned, the principle of determination.

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250 YEARS SINCE THE BIRTH OF GEORG WILHELM FRIEDRICH HEGEL (1770-1831) Hegel and the Summits of the Logical Science (first part)

In logic, Hegel realizes a large-scale program that aims to develop the system of pure thinking. This is a sequel and to some extent a completion of Kants transcendental logic, which Kant designs to embrace and fulfill the whole territory of thinking. Kant puts the concept as the essential logical form of thinking, Hegel responds to this with a logical project in which the concept is the basic working element. Hegel defines the subject matter of logic as the system of pure thought determinations and the development of the logical content of thinking. The thought determinations give the concepts which represent the basic types of object determining in thinking. Hegels logical project results in a complete system of concepts which, until Hegel, have never been developed in logic. The change also reflects on the new vision of the logical form as an infinite form. Hegel undertakes the transformation of the Aristotelian proposition which is criticized by him as finite and fixed and transforms it into a new logical formation with the progress of the inner determinations of the logical subject through the predications and the context.

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250 YEARS SINCE THE BIRTH OF GEORG WILHELM FRIEDRICH HEGEL (1770-1831) Hegel and the Summits of the Logical Science (second part)

In logic, Hegel realizes a large-scale program that aims to develop the system of pure thinking. This is a sequel and to some extent a completion of Kants transcendental logic, which Kant designs to embrace and fulfill the whole territory of thinking. Kant puts the concept as the essential logical form of thinking, Hegel responds to this with a logical project in which the concept is the basic working element. Hegel defines the subject matter of logic as the system of pure thought determinations and the development of the logical content of thinking. The thought determinations give the concepts which represent the basic types of object determining in thinking. Hegels logical project results in a complete system of concepts which, until Hegel, have never been developed in logic. The change also reflects on the new vision of the logical form as an infinite form. Hegel undertakes the transformation of the Aristotelian proposition which is criticized by him as finite and fixed and transforms it into a new logical formation with the progress of the inner determinations of the logical subject through the predications and the context.
Keywords: Hegel, Logic of German Idealism, Science of Logic, truth, logical content, speculative proposition.

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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

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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.

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