Intensional modus in mathematical discourse: A “jump of belief” and mathematical practice

The intensionality of mathematical discourse is vividly represented by the features of the proof for Godel's Second Incompleteness Theorem. An undecidable Godel sentence can have various forms of expression with different meanings. The necessary consideration of this kind of intensionality in usually extensional metamathematical statements is connected with the structural features of the construction of a Godel sentence. Especially important are the general characteristics of the Godelian construction, which combines both the Godel sentence G1 of the First Incompleteness Theorem and G2. In particular, establishing the truth of G1 is related to the intensional nature of mathematical discourse to the same extent as in the case of G2. The analysis of the epistemological aspects of the inten-sionality of the First Theorem is possible through the study of the truth conditions for the Godel sentence. These epistemological consequences include the consideration of the problem of belief in some basic epistemological attitudes in mathematical discourse. It is known that the proof of the existence of an undecidable sentence depends on the assumptions about the validity or consistency of the formal system. The assumption of these concepts gives different strengths to formal constructions, but, in any case, speaking about the truth of the formal system, we must have a guarantee of either validity or consistency, which is intuition based on the belief of mathematicians rooted in mathematical practice. There are two components to establishing the truth of the Godel sentence. First, for all relatively weak formal systems of arithmetic, grasping elementary arithmetic truths, their incompleteness is proved with the corresponding existence of a Godel undecidable sentence. Secondly, the truth of the Godel sentence is determined precisely by the capture of elementary arithmetic truths. In fact, G is both a metamathematical construction and an arithmetic predicate. The incompleteness of relatively weak formal systems of arithmetic is proved in a metamathematical way, and the corresponding Godel's construction G is understood metamathematically. As for the arithmetic predicate G, it is monstrously complex and hardly subject to cognitive understanding. In addition, the transition itself in understanding the nature of G from its metamathematical meaning to the arithmetic one represents a significant “jump”, which is a source of difficulties in understanding this problem. Thus, the intensional aspects of mathematical discourse are manifested particularly in the attribution of truth to G1 by virtue of belief in consistency. In fact, this means a “jump of belief” from extensional mathematical discourse to inten-sional one.

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