Abstract Kurt Gödel did not invent mathematical logic; his famous work in the thirties settled questions which had been clearly formulated in the preceding quarter of this century. Despite sensational presentations by crackpots, philosophers and journalists (or even in poems, for example, by H. M. Enzensberger, set to music by H. W. Henze), Gödel’s results have not revolutionized the silent majority’s conception of mathematics, let alone its practice; much less so than the internal development of the subject since then. Certainly, those results refuted most elegantly each of the grand foundational ‘theories’ current at the time, of which Hilbert’s, on the place of formal rules in mathematical reasoning, and those associated with Frege and Russell, on its reduction to universal systems like set theory, were most popular. (Gödel’s own and related results also deflate the particular ‘anti-formalist’ foundations of the time, Poincaré’s and Brouwer’s constructivist and Zermelo’s infinitistic schemes being extreme examples; they are taken up in the last sections of parts II-IV.) For obvious reasons, in his original publications Gödel made a point of formulating his work in terms acceptable to the theories mentioned, and to stress its bearing on them. But it is fair to say that they were suspect anyway, and—less trivially—that they can be refuted more convincingly by simple constatations rather than by (his) mathematical theorems as explained in more detail in part II. Further, as so often with very grand schemes, the refutations put nothing comparable in the place of the discredited foundational views which are, quite properly, simply ignored in current practice.

This paper examines the limitations of analytical and computational methods in understanding reality, highlighting the secondary role of language and mathematics, which often leads to paradoxes. Gödel’s incompleteness theorems underscore the inherent incompleteness and undecidability in logical and computational systems, such as the Turing machine. We propose that consciousness, operating as a non-material and chaotic finite-state machine (FSM) devoid of self-referencing, can achieve a complete and decidable understanding of reality. This contrasts with the self-referencing nature of logical systems that leads to paradoxes and limitations. Through a conceptual model of the mind inspired by Theravāda Buddhist philosophy, we suggest that awareness of causation is free from self-referencing and coherent with the unpredictable yet causal and deterministic nature of reality. This alignment offers a pathway to a deeper and more comprehensive understanding of causation. The model illustrates the tight integrity between consciousness and causation, proposing that awareness of the present moment of causation can transcend the limitations of Gödel’s incompleteness theorems. This awareness, free from analytical and computational constraints, preserves the integrity of conscious experience and provides a complete and decidable understanding of reality. Future research will focus on developing techniques to sustain this awareness, potentially leading to wisdom and deep insight into the fundamental nature of existence.

Conscious and unconscious brain mechanisms, including cognition, emotions and language are considered in this review. The fundamental mechanisms of cognition include interactions between bottom-up and top-down signals. The modeling of these interactions since the 1960s is briefly reviewed, analyzing the ubiquitous difficulty: incomputable combinatorial complexity (CC). Fundamental reasons for CC are related to the Gödel's difficulties of logic, a most fundamental mathematical result of the 20th century. Many scientists still "believed" in logic because, as the review discusses, logic is related to consciousness; non-logical processes in the brain are unconscious. CC difficulty is overcome in the brain by processes "from vague-unconscious to crisp-conscious" (representations, plans, models, concepts). These processes are modeled by dynamic logic, evolving from vague and unconscious representations toward crisp and conscious thoughts. We discuss experimental proofs and relate dynamic logic to simulators of the perceptual symbol system. "From vague to crisp" explains interactions between cognition and language. Language is mostly conscious, whereas cognition is only rarely so; this clarifies much about the mind that might seem mysterious. All of the above involve emotions of a special kind, aesthetic emotions related to knowledge and to cognitive dissonances. Cognition-language-emotional mechanisms operate throughout the hierarchy of the mind and create all higher mental abilities. The review discusses cognitive functions of the beautiful, sublime, music.

Self-Referential Systems This chapter is largely a review of the essential ideas behind the proofs of Gödel, Rosser and Löb—only presented in a more abstract setting. We believe that it will tie up these ideas in a helpful and instructive manner. We shall first present these ideas in the form of logic puzzles (much in the manner of Smtdlyan [1987]). Then we shall state the results more generally in terms of abstract systems that we call provability systems. These are closely related to certain axiom systems of modal logic, which we briefly discuss at the end of the chapter. In the puzzles to which we now turn, belief will play the rôle of provability. Instead of considering a mathematical system and the sentences provable in it, we consider a logician (sometimes call a reasoner) and the propositions believed by the reasoner. Apart from the heuristic value, these “epistemic” incompleteness theorems appear to be of some interest to those working in artificial intelligence. We shall pay a visit to the Island of Knights and Knaves, in which knights make only true statements and knaves make only false ones. Each inhabitant is either a knight or a knave. No inhabitant can claim that he is not a knight (since a knight would never make such a false claim and a knave would never make such a true claim). A logician visits this island one day and meets a native. All we are told about the logician is that he is completely accurate in his beliefs—he never believes anything false. The native then makes a certain statement X. It then follows that the logician can never believe that the native is a knight nor can he ever believe that the native is a knave.

Kurt Gödel Kurt Gödel (28 April 1906 Brno, then Austria-Hungary, now Czech Republic – 14 January 1978 Princeton, New Jersey) was a logician, mathematician, and philosopher. Impact Some people believe Gödel was one of the most significant logicians of all time. Gödel's work has had a big impact on scientific and philosophical thinking in the 20th century. Many people, such as Bertrand Russell, A. N. Whitehead, and David Hilbert, tried to use logic and set theory at that time. They wanted to understand the foundations of mathematics. Fame Gödel is best known for his two incompleteness theorems. The theorems were published in 1931. He was 25 years of age, and had just finished his doctorate at the University of Vienna one year earlier. The more famous of the two theorems says that if there are consistent axiomatic systems that are powerful enough to describe themselves, there will be things that are true in those systems that can not be proved within the system itself. Proof To prove this theorem, Gödel developed a technique now known as Gödel numbering, which codes formal expressions as natural numbers.

An argument based on Gödel's incompleteness theorem, suggesting that the human mind cannot be computed on a Turing machine that works on Peano arithmetic because the latter cannot see the truth value of its Gödel sentence, while a human mind can.: #: