The problematic nature of Gödel's theorem
Open Letter
Our paper "The problematic nature of Gödel’s theorem" was rejected by MLQ (Mathematical Logic Quarterly). The managing editor (admittedly) has not read it, and no reviews were provided. His argument in favor of rejection was essentially that (a) "Gödel’s results and techniques of proof
are well-acknowledged by the scientific community...since a lot of versed logicians have given detailed and well-understandable presentations and modifications of these results and proofs in a lot of frequently read textbooks." Additionally (b) the editor claims that in general texts like ours, which try to disprove well-accepted results, "...finally demonstrate only a lack of understanding by the authors. It cannot be the task of editors and referees to disprove all these 'disprovers.' In some cases, their errors are obvious, sometimes it takes a considerable amount of time to point out them. Who would spend this?"
In answer to (a) we would like to note that to our knowledge all secondary authors have directly assumed and sometimes even increased the problems of Gödel’s original work we raise in our paper. As to (b) we would like to state that if indeed all or most such critics hitherto demonstrate only a lack of understanding by the authors, it may be improbable, but not impossible, that our criticism is nonetheless valid. As we are convinced that it actually is we would further like to point out that mathematical truth is not a question of probability. We recognize that many editors and referees, already over-burdened, must necessarily perform a certain degree of "triage" as papers are submitted—but does this not constitute a significant hole in the peer-review system? Is it possible to publish a mathematical paper which challenges the accepted orthodoxy?
We think that our paper could and should create a useful and necessary discussion about Gödel’s theorem although, and in fact precisely because, it is a well established and unquestioned part in mathematical literature.
Abstract
In this paper we will show that some fundamental deductions Gödel’s famous theorem is based upon, are highly problematic: Gödel begins the main part of his theorem by correlating to each formal expression (e.g. the variables) of a formal system a natural number. He calls the totality of these numbers x and he uses this x as a variable in recursive formulas. Then he correlates to each number x the formalized numeral x. We will show that x can be nothing else than a variable in the formal system. In the well-known mapping ("Gödelization") he considers x as a predicate and correlates to it an expression, that is variable. This is contrary to the correlation of a fixed number to each variable, as the mapping is stated to be biunique. Gödel wanted the mapping itself not to be directly included in the proof, but in fact it is included and this is veiled by an incorrect predicate. In a further step (Corollary V) he effectively replaces x (that has a variable image) by the variable a (that has the image 17). This problem shakes the foundation of Gödel’s theorem and therefore its validity should be discussed again without any dogmatism.
