Science in Christian Perspective

Letter to the Editor


Godels Theorem Misunderstood
Robert A. Herrmann
Mathematics Department
United States Naval Academy
Annapolis, Maryland 21402

From: JASA 33 (December 1981): 255.

Even though the article "A Positive Approach to Creation" in the December 1980 issue of the Journal ASA was exceedingly beneficial and useful to our understanding of certain cosmological concepts, the statements made by the author relative to Godel's incompleteness theorem are incorrect. This misunderstanding of Godel's results is widely held by numerous individuals and has unfortunately permeated much modern literature.

This misunderstanding revolves around the distinct differences between the procedures employed within the discipline of mathematical logic: procedures which mathematically study numerous concepts associated with logical syntactics and the internal logical syntactics itself. Godel's incompleteness theorem,. among others, states in generalization that if we employ a "formally expressible" set of axioms for an (informally) conceived mathematical structure N of a certain complexity and the Godel numbers for this axiom system form a recursive set, among other possible recursive characterizations, then the set of formally obtainable propositions does not include all of the formal propositions which semantically hold true in N. This simply implies that under the "recursive" characteristic state above that the semantical procedures common to most mathematics do not correspond to the syntactical concepts of deduction. Of course, Godel used a weak form of semantics to establish this result, a form that was acceptable to Hilbert.

There is, however, a small contingent of mathematicians who would lead us to believe that the only concepts that should be considered within the mathematical arena are those results which are "recursively" obtainable and thus always meaningful to the workers in "artificial intelligence." A 1980 Pulitzer Prize was awarded to Douglas R. Hofstadter for his highly erudite attempt to philosophically show that the human mind = the biological object known as the human brain = artificial intelligence. Of Course, if mathematicians followed such precepts, then there would be no Godel incompleteness theorem. Fortunately, at least at present, the vast majority of mathematicians do not adhere to these restrictions.

Consequently, for the majority of mathematicians the set S of all formally expressible statements which semantically hold true in N forms a complete system of axioms for N. However, the set S does not have the appropriate "recursive" property required for such incompleteness results. If N were a model of set-theory, then it could indeed contain a mathematical model of the universe, which could answer all mathematically expressible questions that are humanly understandable.