Draft:Original research/Metamathematics

Def. a "branch of mathematics dealing with mathematical systems and their nature" is called metamathematics.

Countable
Def. "having a [one-to-one correspondence] (bijection) with a subset of the natural numbers" is called countable.

Def. "capable of being [...] countable" is called enumerable.

Transcendentals
Def. "containing only numbers, letters and arithmetic operators" is called algebraic.

Def. "contains elements that are not algebraic" is called transcendental.

Paradoxes
"Hardly had these theories [e.g., Cantor's general set theory] been consolidated, when the validity of the whole construction was cast into doubt by the discovery of paradoxes or antinomies in the fringes of the theory of sets."

Finite definability
Richard's paradox (1905), "substantially given by Dixon 1906, deals with the notion of finite definability."

"For definiteness, let this refer to a given language, say the English language with a preassigned alphabet, dictionary and grammar. The alphabet we may take as consisting of the blank space (to separate words), the 26 Latin letters, and the comma. By an 'expression' in the language we may understand simply any finite sequence of these 28 symbols not beginning with a blank space. The expressions in the English language can then be enumerated by the device [...] of the matrix [which] constitutes a general one for enumerating [...] algebraic equations [such as] a pairing between members of two collections or 'sets' of objects [called] a one-to-one (1-1) correspondence."

Consider "the function whose value, for any given natural number as argument, is equal to one more than the value, for the given natural number as argument, of the function defined by the expression which corresponds to the given natural number in the last described enumeration".

"In the [above] quoted expression we refer to the above described enumeration of the expressions in the English language defining a number-theoretic function, without defining it. But we could easily have written in the definition of that enumeration in full, as part of the quoted expression. We should then have before us a definition of a function (briefly, the function fn(n) + 1), by an expression in the English language. This function, by its definition, must differ from every function definable by an expression in the English language."

Consider, "the least natural number not nameable in fewer than twenty-two syllables". "This expression names in twenty-one syllables a natural number which by definition cannot be named in fewer than twenty-two syllables! (Berry 1906)."

Impredicative definitions
Def. "definable only in terms of a totality of which it is itself a part" is called impredicative.

"When a set M and a particular object m are so defined that on the one hand m is a member of M, and on the other hand the definition of m depends on M, we say that the procedure (or the definition of m, or the definition of M) is impredicative. Similarly, when a property P is possessed by an object m whose definition depends on P (here M is the set of the objects which possess the property P). An impredicative definition is circular, at least on its face, as what is defined participates in its own definition."

Weyl's constructive continuum
"A fund of operations can be provided for constructing many particular categories of irrationals."

The "three main schools of thought on the foundations of mathematics:"
 * 1) the logicistic school,
 * 2) the intuitionist school, and
 * 3) the formalistic or axiomatic school.

Logicism
"The logistic thesis is that mathematics is a branch of logic. The mathematical notions are to be defined in terms of the logical notions. The theorems of mathematics are to be proved as theorems of logic."

Def. a "generalized kind of number used to denote the size of a set, including infinite sets" is called a cardinal number.

Def. "a cardinal number which possesses every property P such that
 * 1) 0 has the property P and
 * 2) n + 1 has the property P whenever n has the property P" is called a finite cardinal (or natural number).

Def. "a non-negative integer [{0, 1, 2, ...}] " is called a natural number.

Def. "a cardinal number for which mathematical induction holds" is called a natural number.

One way to "adapt the logicistic construction of mathematics to the situation arising from the discovery of the paradoxes" is to exclude "impredicative definitions".

Ramified theory of types
"The primary objects or individuals (i.e. the given things not being subjected to logical analysis) are assigned to one type (say type 0), properties of individuals to type 1, properties of properties of individuals to type 2, etc.; and no properties are admitted which do not fall into one of these logical types (e.g. this puts the properties 'predicable' and 'impredicable' [...] outside the pale of logic)."

Relations "and classes [would be some] admitted types for other objects". To "exclude impredicative definition within a type, the types above type 0 are further separated into orders. Thus for type 1, properties defined without mentioning any totality belong to order 0, and properties defined using the totality of properties of a given order belong to the next higher order. (The logicistic definition of natural number now becomes predicative, when the P in it is specified to range only over properties of a given order, in which case the property of being a natural number is of the next higher order.)"

Axiom of reducibility
To escape that the above "separation into orders makes it impossible to construct the familiar analysis [may require an] axiom of reducibility, which asserts that to any property belonging to an order above the lowest, there is a coextensive property (i.e. one possessed by exactly the same objects) of order 0. If only definable properties are considered to exist, then the axiom means that to every impredicative definition within a given type there is an equivalent predictive one."

The "difficulty is [...]: on what grounds shall we believe in the axiom of reducibility? If properties are to be constructed, the matter should be settled on the basis of constructions, not by an axiom."

"This axiom has a purely pragmatic justification: it leads to the desired results, and to no others [so far as is known]. But clearly it is not the sort of axiom with which we can rest content."

The "desired result and no others can apparently be obtained without the hierarchy of orders (i.e. with a simple theory of types)."

The paradoxes can be classified "into two sorts,
 * 1) 'logical' (e.g. Burali-Forti, Cantor and Russell) and
 * 2) 'epistemological' or 'semantical' (e.g. the Richard and Epimenides); and [...] the logical antinomies are (apparently) stopped by the simple hierarchy of types, and the semantical ones are (apparently) prevented from arising within the symbolic language by the absence therein of the requisite means for referring to expressions of the same language."

Arguments "to justify impredicative definitions within a type entail a conception of the totality of predicates of the type as existing independently of their constructibility or definability."

"Logicism treats the existence of the natural number series as an hypothesis about the actual world ('axiom of infinity')."

"The logicistic thesis can be questioned finally on the ground that logic already presupposes mathematical ideas in its formulation."

Intuitionism
The "belief that the rules of the classical logic, which have come down to us essentially from Aristotle (384-322 B.C.), have an absolute validity, independent of the subject matter to which they are applied" has been challenged.

"According to his view and reading of history, classic logic was abstracted from the mathematics of finite sets and their subsets. ... Forgetful of this limited origin, one afterwards mistook that logic for something above and prior to all mathematics, and finally applied it, without justification to the mathematics of infinite sets."

Examples to show that principles "valid in thinking about finite sets do not necessarily carry over to infinite sets."
 * 1) the whole is greater than any proper part, when applied to 1-1 correspondences between sets,
 * 2) a set of natural numbers contains a greatest.

Law of the excluded middle
For every proposition A, either A or not A.

Let A "be the proposition there exists a member of the set (or domain) D having the Property P."

"Then not A is equivalent to every member of D does not have the Property P, or in other words every member of D has the property not-P."

Either "there exists a member of D having the Property P, or every member of D has the property not-P."

"For definiteness, let us specify P to be a property such that, for any given member of D, we can determine whether that member has the property P or does not."

Suppose "D is a finite set. Then we could examine every member of D in turn, and thus either find a member having the property P, or verify that all members have the property not-P."

"For an infinite set D, the situation is fundamentally different. It is no longer possible in principle to search through the entire set D."

"We may in some cases, i.e. for some sets D and properties P, succeed in finding a member of D having the property P; and in other cases, succeed in showing by mathematical reasoning that every member of D has the property not-P, e.g. by deducing a contradiction from the assumption that an arbitrary (i.e. unspecified) member of D has the property P."

"An example for the second kind of solution is when D is the set of all the ordered pairs (m, n) of positive integers, and P is the property of a pair (m, n) that m2 = 2n2. The result is then Pythagoras' discovery that √2 is irrational."

"But we have no ground for affirming the possibility of obtaining either one or the other of these kinds of solutions in every case."

Metadefinitions
The metadefinition of a particular definition is the set of attributes that address and satisfy a number of purposes or functions for that definition. Once specifics are included in the higher level definition, that definition applies to the lower level situation in which the specifics are characteristic. The specifics are relatively arbitrary within the relationship of the metadefinition, which means the term of the metadefinition may be used in a large number of specific subject areas. The metadefinition is often viewed as a higher level definition that may consist of the definition of several objects within that higher level, or a definitional set of descriptive attributes or characteristics that can be used to produce a variety of related definitions.

Metadefinition theory
The metadefinition layer can be defined as a higher level definition, that may consist of the definition of several objects. When each object is made specific, a specific definition is derived or translated from the metadefinition.

Inducting upward from the definitional layer, the metadefinition of a particular definition is the set of attributes that address and satisfy a number of purposes or functions.

Def. "the set of attributes that address and satisfy a number of purposes or functions" for a particular definition is called a metadefinition for that definition.

A definition serves five functions:


 * 1) a statement of identity,
 * 2) a way to define competitive and cooperative relationships with other terms,
 * 3) a way to end conceptual disputes and thus prepare the way for measurement - its preoperational or premeasurement function,
 * 4) a way to locate a term within a particular context - its orienting or contextual function, and
 * 5) a way to generate new ideas - its generative or revelatory function.

When the concept of the description is relatively arbitrary, the definition containing this concept is in fact a metadefinition. Each arbitrary choice from the metadefinition creates a definition.

Forms
From the theory of a metadefinition, a constituent phrase can be said to have a metadefinition if and only if the set of attributes that address and satisfy a number of purposes or functions for each descriptive definition is related or relatable either to the initial definition or the metadefinition itself through some algorithm such as translation.

"The purpose of the original research is to produce new knowledge, rather than to present the existing knowledge in a new form (e.g., summarized or classified). "

For a word or phrase, once a number of purposes or functions (of the metadefinition) are addressed and satisfied, this set of addressed and satisfied attributes is the definition for the word or phrase. Presenting the existing knowledge of or about the word or phrase in an alternate form (the metadefinition) and summarizing or classifying that existing knowledge into the this form is presenting existing knowledge in an alternate form, perhaps for analysis.

Presenting an alternate form for existing knowledge is not considered a primary source, but producing new knowledge about changes in the attributes over time or as a function of other variables is original research.