Metacognition, defined either epistemologically as knowledge about knowledge or operationally as behavior about behavior (Koriat, 2007), presumably enables intelligent agents to self-referentially and, in social contexts (Bahrami et al., 2010, 2012), group-referentially monitor and control emotions, moods, perceptions, memories, reasoning, decisions, and actions. At an epistemological level of description, the nested referent nature of metacognition succumbs to problems originating from recursively enumerable propositional logic; that, as Kurt Godel (1931) first proved for Bertrand Russell and Alfred North Whitehead's axiomatic Principia Mathematica, the meaning of statements created about conditions of a system or set of systems by the respective same system or set of systems can be formally undecidable. Momentarily ignoring peripheral confounds introduced by stochastic and imperfect biological systems, animal and human metacognitive operations and all their possibilities must thus exist in a universe of graded logicomathematical consistency (i.e., All theorems are true syntax-correct propositions of the system.) and completeness (i.e., All true syntax-correct propositions of the system are theorems.) (Kreisel, 1967). Because strong consistency excludes strong completeness, the knotted statements of metacognition may show themselves to be falsehoods, truths, or truths essentially unverifiable in theory as well as in subjective and objective practice (Figure ​(Figure1)1) (Raattkainen, 2005). Figure 1 Diagram outlines relationship of metacognition undecidability and opacity with epistemological and operational definitions, where decidability of logicomathematical formalism is a stronger, more global condition than transparency of empiricism. Interestingly, ... Psychologists well understand the fallibility of formal logic systems and of axiomatic animal and human psychological processes, including, among other phenomena, feature detection, inferential judgments, error diagnosis and correction, concept formation, memory storage and retrieval, and introspection (cf. Nisbett and Ross, 1980; Kahneman et al., 1982; Watanabe and Huber, 2006). We often stipulate—with qualifications—the flawed definition(s) and agent execution of cognition and metacognition (e.g., Shimamura and Metcalfe, 1994; Koriat and Goldsmith, 1996, 1998; Smith, 2009; Terrace and Son, 2009; Frith, 2012; Fleming et al., 2012; Yeung and Summerfield, 2012). For pragmatic reasons, many of us enthusiastic about studying metacognition also avoid the strange, looping causality of self-reference exposed with Godelian numbering to concentrate on solving basic and/or clinical empirical difficulties that arise when trying to identify this stubbornly opaque hypothetical construct of healthy and pathological minds (e.g., Bach and David, 2006; Koren et al., 2006; Vance, 2006; Carruthers, 2009; Gumley, 2011; Brevers et al., 2013). The operational opacity of metacognition becomes arguably most apparent when considering: (1) poor subjective accessibility to the cognition and metacognition of animals and humans of limited or no language proficiency (cf. Hampton, 2009; Fleming and Dolan, 2012; Kepecs and Mainen, 2012; Smith et al., 2012), (2) the synergism and antagonism of unreliable explicit and implicit psychological components that mask real metacognitive abilities and capacities from agent and external observer (cf. Hampton, 2009; Fleming et al., 2012), and (3) the occasional independence between metacognition and cognitive skills which may exacerbate the preceding two problems (cf. Koriat and Goldsmith, 1998; Schneider, 1999). A case illustrating these three classes of problems is found for Paulus et al. (2013), who report evidence of implicit metacognition in normal preschool children performing a paired-associates learning and memory task. The authors confront the dilemma of metacognition opacity with a paradigm common to belief, memory, Theory of Mind, an

Kurt Friedrich Gödel ( GUR-dəl; German: [ˈkʊʁt ˈɡøːdl̩] ; April 28, 1906 – January 14, 1978) was a logician, mathematician, cosmologist, and philosopher. Considered along with Aristotle and Gottlob Frege to be one of the most significant logicians in history, Gödel profoundly influenced scientific and philosophical thinking in the 20th century (at a time when Bertrand Russell, Alfred North Whiteh Kurt Friedrich Gödel ( GUR-dəl; German: [ˈkʊʁt ˈɡøːdl̩] ; April 28, 1906 – January 14, 1978) was a logician, mathematician, cosmologist, and philosopher. Considered along with Aristotle and Gottlob Frege to be one of the most significant logicians in history, Gödel profoundly influenced scientific and philosophical thinking in the 20th century (at a time when Bertrand Russell, Alfred North Whitehead, and David Hilbert were using logic and set theory to investigate the foundations of mathematics), building on earlier work by Frege, Richard Dedekind, and Georg Cantor. Gödel's discoveries in the foundations of mathematics led to the proof of his completeness theorem in 1929 as part of his dissertation to earn a doctorate at the University of Vienna, and the publication of Gödel's incompleteness theorems two years later, in 1931. The incompleteness theorems address limitations of formal axiomatic systems. In particular, they imply that a formal axiomatic system satisfying certain technical conditions cannot decide the truth value of all statements about the natural numbers, and cannot prove that it is itself consistent. To prove this, Gödel developed a technique now known as Gödel numbering, which codes formal expressions as natural numbers. Gödel also showed that neither the axiom of choice nor the continuum hypothesis can be disproved from the accepted Zermelo–Fraenkel set theory, assuming that its axioms are consistent. The former result opened the door for mathematicians to assume the axiom of choice in their proofs. He also made important contributions to proof theory by clarifying the connections between classical logic, intuitionistic logic, and modal logic. Born into a wealthy German-speaking family in Brno, Gödel emigrated to the United States in 1939 to escape the rise of Nazi Germany. Later in life, he suffered from mental illness; believing that his food was being poisoned, he refused to eat and starved to death. A… If a (logical or axiomatic formal) system is omega-consistent, it cannot be syntactically complete. The consistency of axioms cannot be proved within their own system. These theorems ended a half-century of attempts, beginning with the work of…

Or is it that the proof of undecidability is not within ZFC and so ZFC is safe? Exactly. Proofs of undecidability in ZFC always assume a stronger axiom such as Con(ZFC) that is not provable in ZFC. This means that the overall undecidability result is not provable in ZFC, it is only provable in the stronger system. As you saw, if ZFC is inconsistent then it proves every statement in its language. So any proof that demonstrates an undecidable proposition in ZFC must assume ZFC is consistent - this axiom is exactly Con(ZFC). One way of assuming Con(ZFC) is assuming that ZFC has a model (that is, a set that is a model of ZFC). This is equivalent to ZFC being consistent, by the completeness theorem for first-order logic, and so it is not provable in ZFC itself. Thus any undecidability proof that proceeds by constructing a model must assume Con(ZFC). Most set theorists view Con(ZFC) as a relat

The problem is posed of establishing a possible relationship between a new type of Multi-verse representation, Gödel undecidability theorems and the logic of classical, quantum mechanics and quantum gravity. For this purpose example cases of multi-verses are first discussed in the context of non-relativistic classical, quantum mechanics and quantum gravity. As a result, it is confirmed that thanks to Gödel theorems non-relativistic classical and quantum mechanics, as well as quantum gravity theory are incomplete. As a consequence, they necessarily admit undecidable logical propositions and therefore obey a three-way boolean logical, i.e., a propositional logic with the three different logical truth values true, false and undecidable.

Our first proof of the incompleteness of P.A. was based on the assumption that P.A. is correct. Gödel’s proof of the last chapter was based on the metamathematically weaker assumption that P.A. is ω-consistent. Rosser [1936] subsequently showed that P.A. can be proved incomplete under the still weaker metamathematical assumption that P.A. is simply consistent! Now, Rosser did not show that the Gödel sentence G of the last chapter is undecidable on the weaker assumption of simple consistency. He constructed another sentence (a more elaborate one) which he showed undecidable on the basis of simple consistency. Our first proof of the incompleteness of P.A. boils down to finding a formula that expresses the set P̃* (or alternatively one that expresses R*). Gödel’s proof, which we gave in the last chapter, boils down to representing one of the sets P* and R* in P.A. and the only way known in Gödel’s time of doing this involved the assumption of ω-consistency. [This assumption was not needed to show that the sets P* and R* are enumerable in P.A.—it was in passing from the enumerability of these sets to their representability that ω-consistency stepped in.] Now, Rosser did not achieve incompleteness by representing either of the sets P* and R*, but rather by representing some superset of R* disjoint from P*—this can be done under the weaker assumption of simple consistency—and it also serves to establish incompleteness, as we will see. The axiom schemes Ω4 and Ω5 of the system (R) will play a key role in this and the next chapter. We shall say that a system S is an extension of Ω4 and Ω5 if all formulas of Ω4 and Ω5 are provable in S. We will prove the following theorem and its corollaries. Theorem R. Every simply consistent axiomatizable extension of Ω4 and Ω5 in which all Σ1-sets are enumerable must be incomplete. Corollary 1. Every simply consistent axiomatizable extension of Ω4 and Ω5 in which all true Σ0-sentences are provable must be incomplete. Corollary 2. Every simply consistent axiomatizable extension of the system (R) is incomplete.

Based on the MRDP theorem, we introduce the ideas of the proof equation of a formula and universal proof equation of Peano Arithmetic (PA); and then, combining universal proof equation and Gödel's Second Incompleteness Theorem, it is proved that, if PA is consistent, then for every axiom and every theorem of PA, we can construct a corresponding undecidable proposition with Diophantine form. Finally, we present an approach that transforms seeking a proof of a mathematical (set theoretical, number theoretical, algebraic, geometrical, topological, etc) proposition into solving a Diophantine equation.

We shall now turn to a formal axiom system which we call Peano Arithmetic with Exponentiation and which we abbreviate “P.E.”. We take certain correct formulas which we call axioms and provide two inference rules that enable us to prove new correct formulas from correct formulas already proved. The axioms will be infinite in number, but each axiom will be of one of nineteen easily recognizable forms; these forms are called axiom schemes. It will be convenient to classify these nineteen axiom schemes into four groups (cf. discussion that follows the display of the schemes). The axioms of Groups I and II are the so-called logical axioms and constitute a neat formalization of first-order logic with identity due to Kalish and Montague [1965], which is based on an earlier system due to Tarski [1965]. The axioms of Groups III and IV are the so-called arithmetic axioms. In displaying these axiom schemes, F, G and H are any formulas, vi and vj are any variables, and t is any term. For example, the first scheme L1 means that for any formulas F and G, the formula (F ⊃ (G ⊃ F)) is to be taken as an axiom; axiom scheme L4 means that for any variable Vi and any formulas F and G, the formula . . . (∀vi (F ⊃ G) ⊃ (∀vi (F ⊃ ∀vi G) . . . is to be taken as an axiom.