Examples are used to elucidate a theory, but they are not a theory in themselves. A fact on its own, without a theory accompanying it, is not an example of an example, as it elucidates no theory, and is instead only a sign without significance, lacking connotation. This is not the entirely true however, as we are all immersed in a contextual paradigm, and we can bring our own theories to lone facts, and supply our own examples for a theoretical context. A theory is supposed to explain many examples, and in fact, a complete theory will be capable of incorporating every relevant example. Is it possible that a theory can accomplish such a completion, or will there always be an example or two to come along and disprove a complete theory, or prove the incompleteness of a theory? How does a theory prove complete? And would that not be tantamount to the end of time?
Our first problem, how do we decide whether an example provided is relevant to a category when following its formal criteria? Well, you might say, we will need an example of a formal criteria first, as well as a property that has been decided, following this criteria, to be relevant to its operation. Ah, but did you not just ask for an example? How can we decide what is an example of a formal criteria without first having a formal criteria for a formal criteria? Uh oh, we’ve just run into the paradox of self-reference. This could be worse, because we have in the liar’s paradox an example of a formal criteria. Or, more formally, it is an example of a recognized problem, with associative connections between forms. Hey, we’ve stumbled upon a starting place! What we’re interested in is which statements can be formally defined as partaking in the paradox of self-reference. As in, what properties go into a statement that allow us to recognize it as a member of the set of paradoxically recursive utterances. Ok, so what is an example of a statement that can fit the formal criteria for the paradox of self-reference? First, we will need to concern ourselves with what constitutes an example of this type, and what types of examples there might be. How can we list a set of examples for a property we have not defined?
To define our property we will need to construct a format allowing for the disambiguation of a statistical distribution of contextual content. Further, a property will be defined by the function from which its value is derived, following the rules of inference which warrant its containment within a domain that satisfies a sequential limit with convergence upon a factor. The limitations set for a function are dependent on the efficiency of the factor produced as the determining vector approaches the optimal features of an evaluative definition. The determined co-efficient proceeds in line with the evaluative schema of the factor to which belongs the rules of its approximate dominion. Evaluation can be mapped as a function of the sensitivity of a factor in response to the approach of a conditional through which its domain retains, at least, an harmonic constant. This evaluative line is an effective result of the application of a predictive sampling to a distributive model with alternative orders ranked in accordance with their impact on the specific function. The predictive model that appreciates the highest impact will coincide with the distribution sample that can best capture the concentration of the rates under inspection. The model that generates the lowest impact will be in essence of a type of paradox leading to an inflationary coinage by the so-called principle of explosion. Explosive inflation will describe the characteristic of a collection of sets that do not contain the properties of their distribution in a fashion that could render them accountable. The outcome of a collection as an empty set of uncountable (because non-existent) properties is vacuous, as an understatement, and does not pertain to the considered attributes of a singularity vector, as recognized by a delicate (wholesome) re-activity to the inputs that are its necessary condition. In this way, we can describe a miscount as a systematic bias in which every set of a collection of sets is represented by properties they do not contain, or by properties that are said to contain them all, by virtue of an unconditional relation. This type of error is quite homogeneous, and usually entails the assumption of the exhaustion of a subject matter in relation to a set of unrestricted predicates. It is the assertion of an unconditional resemblance between classes, as apparently defined by a structural homogeneity in a proposition string. Contrarily, we have set out to advance our proposition as the codified conditions under which the predicates are judged, through an anonymous, and unidentified, reciprocity of synthetic production. We have compiled a stack of allocated solutions to instances of optimization procedure through the inhabiting of a stratified manifold allowing for the affordance of feature instrumentality. We have not based our proofs on lists of supposed instance-cases, but have chosen a design protocol as a priority of generative induction. This is a system with a much stronger fidelity to the expression of a composite image. The interpretive power can be directly confronted, at each point of departure, as it entails the consequences of its operations upon the model. Succession is a factor of each positive, automatic predictor, with each test case encoding an exclusive settlement for each phase considered, without thereby determining the outcome of the recurrent notion, such that the approval of a phase relies on the compact security of its key detection. The cumulative effect of this appeal, from which derives the source of its expense, supplies each class with the rules of its observance, for which is conserved the choice of an ultimatum, whereby a contract is fulfilled in the relevance of its terms, or otherwise annulled. Each point of the conjecture correlates with a critical feature of the engendered design, through which the map is grafted onto the surface of a serial array, as esteemed by the calculated supposition of its derivative. As such, the effective correlates of our map will exemplify the very features for which they hold distinction, stemming from the notional dependency of their design on the intrinsic qualities of their co-efficient. In sum, the task is to clear the space for the event of their elaboration, in order to quantify the standing of their pairs, to escape the degeneracy of formal paradox, and to codify a sufficient schema for the classification of restrictive ascendancy.