Godel, Goedel, Kurt, theorem, incompleteness theorem, incompleteness, Podnieks, Karlis
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Princeton, 1950 |
Gödel's Theorem Prof. Picture from: MacTutor History of Mathematics archive - Godel Portraits |
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Context - the formal model of science In many texts, Gödel's Incompleteness Theorem is applied without any prior and proper definition of the context. In fact, we are dealing here with a specific mathematical model of science. In this model, scientific theories are replaced by specific models of them - the so-called formal theories. Formal theories are formulated in an extremely precise way - so precise that the reasoning in them can be simulated by a computer. And these formulations never change - if you change the formulation of your theory, then you are trying to propose a new theory! In this formal model of science, Gödel's Theorem is a purely mathematical result, and it sounds as follows: |
The Theorem Gödel's Incompleteness Theorem. If some formal theory is universal enough to allow proving of the simplest properties of natural numbers (1, 2, 3, ...), then, working in this theory, one of the two things will happen inevitably: either we will run into contradictions, or into unsolvable problems. Kurt Gödel^{MacTutor} proved this theorem in Vienna, during the summer of 1930, and he first told about it to his colleagues on Tuesday, August 26 at the Cafe Reichsrat: Picture from an excellent gallery published by BVI. How to find Cafe Reichsrat today? More about chronology of this turning point in the intellectual history of human race - click here. |
What could this theorem mean for us? If we wish to derive from Gödel's Theorem some conclusions "for us", then we must first determine, to what extent our situation corresponds to the formal model of science? Is our theory formulated precisely enough, i.e. can it be represented adequately by a formal theory? If not, then Gödel's Theorem is not applicable to it. But, then, at the same time, we must confess, that, at least in part, we do not understand what our theory is talking about. But, if our theory is formulated precisely enough, then it can be represented adequately by a formal theory. And then, Gödel's Theorem may be applicable to it. Namely: If our theory is universal enough, and we have formulated it precisely enough (and we will not change this formulation in the future), then, working in this theory, one of the two things will happen inevitably: either we will run into contradictions, or into unsolvable problems. An example: set theory. Trying to formulate the principles of set theory precisely for the first time, people ran into contradictions (the most impressive of them is the so-called Russell's Paradox). After that, working with the corrected axioms (the so-called Zermelo-Fraenkel set theory, ZFC), they ran into unsolvable problems (such as the famous Continuum Problem, see below). |
So what? If you think that the above-mentioned perspective is unacceptable, then exactly three ways of stepping back can be noticed in the picture: 1) Let us lower our ambitions, i.e. let us give up building a universal theory. 2) Let us refuse formulating our theory precisely. 3) Let us formulate our theory (universal, or not) precisely, but, if necessary, let us change it in the future. The first way is an individual choice of everyone. Wouldn't the second way expose us to ridicule? The third way is the one of the modern science: keep modeling, but don't make a fetish of your model. All models are wrong, but some are useful. (George E. P. Box) |
From a practical point of view From a practical point of view, Gödel's Theorem makes only a general (though very strong) prediction. But, in the mathematical practice, this general prediction was confirmed many times starting from 1963, when Paul Cohen^{MacTutor} proved that the axioms of set theory (if they do not contain contradictions) cannot solve the famous continuum problem^{Wikipedia}. The inventor of set theory Georg Cantor^{MacTutor} arrived at the continuum problem in 1878: on a straight line, can there exist a set of points containing more points than there are natural numbers, but less points than the entire line? Cantor conjectured that such sets can't exist. However, neither he, nor his successors were able to prove this conjecture. Since 1963, we know the cause: by using the axioms of set theory, one cannot, indeed, neither prove, nor disprove this conjecture. More about these practical confirmations of Gödel's Theorem: List of statements undecidable in ZFC^{Wikipedia} As you may see now, the real significance of Gödel's Theorem differs completely from the funny things written about it in many texts and even textbooks (about mathematical "truths" that cannot be derived from axioms, about superiority of human mind over machines etc.). |
Godel, Goedel, Kurt, theorem, incompleteness theorem, incompleteness, Podnieks, Karlis