reality is unrealizable
We want to show that the part of reality that can be understood from set theory can be represented by a specific point in a unit interval, and further, we want to claim that this point is illusive and beyond our grasp.
Since set theory forms the foundations of mathematics, it will not be unreasonable for us to assume that a substantial part of our knowledge of reality can be derived from set-theoretic concepts.
We need the following classification of formulas to proceed further. Originally it was thought that we will be able to show that every formula in set theory is either a theorem or a falsehood, until Goedel showed that there are formulas in set theory which are neither true nor false. Creating a specific formula, he showed that the assumption of the formula itself or its negation will create contradictions in set theory. Self-reference is a crucial concept used by Goedel in the generation of his formula and for that reason we will classify this kind of formulas as introversions. Investigating Cantor's Continuum Hypothesis (CH), Cohen and Goedel have shown that neither CH nor its negation can generate a contradiction in set theory. Accepting that there can be more formulas of this kind, we will classify these formulas as profundities. From all these facts, we conclude that there are four kinds of formulas possible in set theory, namely, theorems, falsehoods, introversions, and profundities. Of course, we assume that there are no contradictions in set theory. If we are able to classify every formula in set theory in one of these categories, we can claim that we have some understanding of reality, at least that part of reality understandable through set theory.
It is a known fact that the formulas of set theory can be enumerated, thereby, assigning a unique positive integer number to every formula. Consider the point in a unit interval, determined by the following rules. The point is specified by a quaternary number, the n-th digit in the number after the quaternary point getting the value 0,1,2, or 3 depending upon whether the n-th formula is a falsehood, theorem, introversion, or profundity, respectively. With all that we know about mathematical logic today, it will not be unreasonable to say that we will never be able to precisely locate this point in the unit interval. If we call the point, Reality, we can reinforce the opinion of many mystics and formulate our predicament as a thesis.
Incomprehensibility Thesis: Reality is Unrealizable.
Mysticism and Logicism: Reality is Unrealizable
Sunday, April 8, 2007
