Re: Contradicrtion-free mathemattics (The new nonstandard analysis

CONTRADICTION-FREE MATHEMATICS: RECRUIREMENTS
BEYONED PRESENT FOUNDATIONAL FRAMEWORK

1) Tradition says that we cannot help but admit some undefined or ill-defined symbols (concepts) in building a mathematical space. Mathematicians often forget that mathematics is man-made and we can introduce any symbols we want provided we make sure that the axioms that bind the mathematical space together are not only consistent but also well-define every symbol we introduce
(consistent axioms, meaning, no contradiction can be derived from them, cannot include false proposition). Any undefined or ill-defined symbol introduces ambiguity, breeds contradiction and must, therefore, be avoided. A symbol is well-defined if its existence, properties and relationship with other symbols are specified by the
axioms. Existence is required because any vacuous statement, especially, the defining expression of a symbol, is ambiguous. In the case of a decimal it is known or well-defined only if every digit is known or computable (being computable is explained in previous posts).

2) Since every mathematical space and its symbols are well-defined only by its axioms the rules of inference must be specific to and well-defined by the axioms and proof of any theorem rests entirely them. Some implications of this requirement include: a) universal rules of inference such as those provided by formal logic are inadmissible because they are not well-defined by the axioms and b) distinct mathematical spaces are independent; therefore, any proposition involving symbols from two distinct spaces is ambiguous. The undecidable propositions include those involving ambiguous symbols.
It also follows that Godel’s incompleteness theorems are ambiguous and, therefore, flawed, because they involve concepts from two distinct spaces.

3) Among the ambiguous symbols of mathematics is infinite set; I try to mitigate its ambiguity by calling it finite but unbounded, meaning, one can only enumerate finite elements at a time from it but there will always be some element left out. However, there is still inherent
uncertainty because any proposition about its elements involving the universal or existential quantifier is not verifiable and, therefore, ambiguous. Such proposition cannot be used as an axiom because it does not insure the certainty of the conclusion of a theorem. For example, suppose we want to verify that every element x of an
unbounded set has the property P written, P(x). To verify, P(x) we start with some element x_1. If not P(x_1), then the proposition is false. Otherwise, we take another element x_2 and suppose x_2(P_2), etc. Clearly, we cannot verify this proposition.

4) How do we remedy the problem described in 3)? Build a mathematical space on finite symbols and well-define it by consistent axioms subject to the requirements 1), 2), 3). In the case of the system of decimals R*, +, x, the building blocks are the basic digits 0, 1, . . ., 9. The first axiom, that R* contains the basic digits, satisfies the existence requirement of 1). Then the addition and multiplication axioms meet the other requirements and specify the behavior or properties of and relationship among the basic digits and the decimals built on them.
The Cauchy sequences where every term is known or computable generate the nonterminating decimals and the nonstandard decimal d*. At any rate, the mathematical space so built that I call the new real number system is finite but unbounded and, therefore, discrete, has natural ordering, is free from contradiction and enriched by the new integers and the dark and unbounded numbers d* and u* respectively.

E. E. Escultura
.