Ripheus23

Metafinity as the foundation of mathematics

Let the finite and the infinite be relative or absolute. By absolute is meant a universal relation: x is absolutely y if in all relevant relations, x is y. For example, if x is to the left of all y, x would be absolutely leftwards (though this is not really possible, let us suppose). Accordingly, we have an order of four metafinite predicates: absolutely finite, relatively finite, relatively infinite, and absolutely infinite.

          Next, allow that there are two fundamental numerical questions: “Which one?” and, “How many?” A number is either an answer to some form of these questions, or eventually derives from such answers. Let us refer to numbers as answers to the first question by the term ordinals, the second cardinals. Numbers as answers, in some way, to both questions will be denoted surdinals (with reference, as will be explained far below, to the concept of surreal numbers). Numbers as answers to neither question, used purely to differentiate between some x and y, will be indexicals. Thence, to use the number 1 as a name for someone, and the number 3 as a name for someone else, is to use these numbers indexically; they are not subject to arithmetic as such, which is to be considered an ordering of numbers in themselves. So while we might add 1 and 3 when these are used as ordinal, cardinal, or surdinal numbers, we would never add them when using them indexically. —The four positive metafinite predicates are then generally correlated with four basic uses of numbers.

          Our axiom system is a set of axioms and their schemas used in mathematical proofs. The most commonly used such system is Zermelo-Fraenkel set theory with the axiom of choice, which has up to nine basic extralogical principles, some of which can perhaps be reduced to others or waived altogether. Now the attempt to provide a foundation for mathematics, using the concept of metafinity, involves the question (roughly enough put), “Assuming that there is some set of axioms and axiom schemas, why is the number of basic principles what it is?” So if ZFC set theory is true, the question is, “Why are there (as many as) nine such principles?” And according to the theory of metafinity, there would not be nine, but only four, generally correlated with the pairings of metafinite predicates and the numerical erotetic dyad. Indeed, this theory is telling us that we know how many axioms and axiom schemas to look for, without directly knowing as such what the system is.

          Let a logically irreducible number be a number occurring in the logical principles of the system. For example, if 0 and 1 are mapped to FALSE and TRUE,[1] then it is unnecessary to derive either number from the other using extralogical principles. Rather than by transconstructing 1 as the simplest successor of 0, for example, we have it immediately, here. Moreover, if this is true, then the classical set-theoretic definition of the ordinal 1 is false: being the simplest successor of 0 characterizes 1, but does not define it. 1 is not actually reducible to the set containing the empty set, or any such thing.

          If there are four axioms or schemas, here, what are they to be expressed as? Axiomatic structures are assertoric: the basic case is subjects and predicates, which in mathematics comes to numbers and operations thereon. At least one axiom will give us some numbers to work with, and at least one axiom will give us operators for them. Now in the metafinite context, we immediately know that we have numbers for all the metafinite predicates, so the classical axiom of infinity is encoded into the context. This gives us our basic case of a relatively infinite number, here  as an ordinal (and surdinal) and  as a cardinal. Arithmetical intuition, and set-theoretic transconstruction, vitiate the idea of a well-ordered sequence of operators, so that our second axiom will be one that gives us the hyperoperation sequence as such.[2] This sequence will be represented under the heading of the axiom of transcension.

          The existence of relevantly basic cases of the other metafinite numbers will generally be referred to under the heading of the axiom of transcardinality. This has it that there are relations among the metafinite predicates such that, starting from the transfinite cardinals, we can go on to situate the absolutely and relatively finite numbers fairly exactly. Until actual examples of what this situation means are provided, of course, this remark is not even quite programmatic but might appear rather mystical (at best).

          Lastly, the axiom of transfinality collates all the relations of finality that appear in the mathematics of the numbers and their sets referred to here, and adds to them, in light of the metafinite context in total: therefore, in quintessential relation to the absolute infinite. Expressing the concept of an absolutely infinite number in a consistent way is the challenge to be met in this case, as the standard vectors of approach give rise to simple contradictions. By way of a very preliminary remark, the axiom of transfinality involves an extreme reimagining of the principle of foundation in set theory. In other words, though a naively infinite and descending sequence of sets is not to be presented, some infinite descending numerical sequence will be. As far as already-established mathematical systems go, a clear example of an analogy with what we will be looking for is the surreal sequence . But a transfinal sequence is meant to be advergent under absolute infinity, which is an absolutely strong relation to enter into, and so we will have cause to analyze phrases that appear in the literature such as “a set that might as well be as large as the universe V of sets” as well as the equation V^V = V (and its cognates, e.g. “”).

          Think of the axioms, then, as introduction rules for types of glyphs. The axiom of infinity introduces the aleph glyphset in general; the axiom of transcension introduces the ascending arrow glyphset; the axiom of transcardinality introduces glyphs for juxtaposing the relatively infinite with the absolutely and relatively finite numbers; and the axiom of transfinality gives us notation for the idea of advergence under the absolute infinite. Logical conjunctions of two or more axioms or schema then yield further analogous or derivative glyphsets. The infinitary context also allows us to apply the axiomatic propositions in an infinitary way. There are therefore 23 such adjunctions of the axioms and their schemes to consider, which gives us much to work with, as will be seen.