Logic and Language
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Copyright   James R Meyer    2012 - 2023 https://www.jamesrmeyer.com

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Page last updated 16 May 2023


It was so many years ago that I cannot now remember the details, but I do recall that when I first came across Gödel’s incompleteness theorem in a book about various mathematical ideas, it struck me straightaway that it seemed to be contradictory: if the proof itself were stated in a formal system, then surely, by Gödel’s proof, that system could talk about itself, and then the system would seem to be saying that it could produce a statement that the system could not prove, while at the same time the system itself would be proving it. I explain this somewhat more formally on the page Gödel’s Contradiction.


At the time I didn’t dwell on the matter, but several years later I encountered it again and I decided to investigate Gödel’s proof in more detail. I remember being frustrated by the fact that, at a crucial step of the proof, Gödel did not in fact provide a proof for this crucial step, but merely gave a very brief sketch of how he thought one might generate a fully worked out proof of that step. Having too many other conflicting matters in hand, I was unable to devote any more time to it, and I set the proof to one side and forgot about it.


Years later, completely by chance, in a local library sale, I came across the book Gödel, Escher, Bach by Douglas Hofstadter, and my interest in the proof was reignited. However, I found that Hofstadter, as is the case in so many popularizing books of its type, didn’t actually give any insight into Gödel’s proof - he didn’t give any explanation of Gödel’s crucial unproven step, instead providing his own quite different and obviously erroneous alternative. (Footnote: For a detailed analysis, see the page: Hofstadter’s ‘Gödel, Escher, Bach’.)


But the book had rekindled my interest, and more - it had made me more determined than ever to set out again on a trail to attain a full understanding of the proof. I searched for other background material that might already be available and which would provide some assistance in this trail, but there was nothing, and I was forced to conclude that I would have to travel a rather lonely path.


As I tried to completely immerse myself into the proof, I was adamant that I would not jump to any conclusions; instead I would from the outset strive to gain a complete understanding of how the proof operated. I was determined that the only way forward was by a ruthless application of logical argument, and that everything would be subjected to examination under the microscope of logic. In fact, as I delved into the proof, there were many times when I did wonder whether it might be the case that the commonly accepted interpretation of the result was incorrect and that perhaps I would discover that there was no inherent contradiction. But the only way to be absolutely sure was to pin down with complete clarity and irrefutable logic those parts of Gödel’s proof that were lacking in such clarity.


One of the tasks to be carried out was to write out in full detail that part of Gödel’s proof for which he had only furnished a sketch. Once that was done, it became apparent that the sketch overlooked important details, in that Gödel had made assumptions that not only were without any logical foundation, but which confused different levels of language, conflating a meta-language and a sub-language. These assumptions resulted in the erroneous appearance that a well-defined consistent formal language could include certain self-referential statements that in reality were logically impossible.


After discovering this confusion of levels of language in Gödel’s proof and posting the details on this site, various people, instead of finding any error in what I had written, challenged me to explain how, if Gödel’s proof was wrong, then how could there be so many other proofs of incompleteness. Armed with the knowledge that the self-reference in Gödel’s proof was due to confusion of levels of language, I was able to look at those proofs, knowing that if there was an error in any such proof, it was probably also due to an unfounded assumption that involved a confusion of levels of language. And, indeed, that turned out to be the case - in all the incompleteness proofs that I have examined, (more than twenty) they exhibit at least one illogical assumption that generates in one way or another a confusion of levels of language. (Footnote: See, for example: Errors in Incompleteness Proofs, The Diagonal Lemma, The Halting Problem and Incompleteness Proofs and Oh no ! Yet Another Flawed Incompleteness Proof.)


Having discovered that a confusion of levels of language seemed to be a common error in various Incompleteness proofs, I then began to wonder if that type of error might also be implicated in other proofs; after all, if mathematicians and logicians could be so blind to that sort of error where it occurs in incompleteness proofs, then it would not be at all surprising if they had also overlooked that same type of error in other mathematical proofs.


It did not take long to realize that there was another well-known mathematical claim that was based on a confusion of levels of language, but which, interestingly, had never been supported by anything bearing the slightest resemblance to a rigorous proof. This was the assumption that the absence of a one-to-one correspondence between the real numbers and the natural numbers somehow implies that one set with limitlessly many elements somehow can have fewer elements than another set with limitlessly many elements.


But for infinite sets a one-to-one correspondence between two infinite sets can only be given by a definition of a function that defines that one-to-one correspondence in terms of the properties of the elements of the sets, and the absence of a one-to-one correspondence between the elements of two infinite sets results from differences in the properties of their elements, not from a difference in their quantity.⁠ (Footnote: See One-to-one Correspondences and Properties.)


The illogical assumption that leads directly to a contradiction - that one limitlessly large set can have fewer elements than another limitlessly large set - has been attached to the diagonal proof, and has been a mainstay of set theory. That assumption, along with a reliance on simplistic and illogical notions, results in a hopeless confusion of levels of language, as I demonstrate on the page The Diagonal Proof. (Footnote: See also, for example: Why do people believe weird things and Proof of more Real numbers than Natural numbers?)


It was apparent that this had resulted in two quite distinct branches of mathematics - one a branch that is directly applicable to real world scenarios, as in science and technology, and the other an esoteric branch that has no such real world applications. Knowing that the notion of different levels of infinity is based on an erroneous confusion of language, I decided to investigate the area of mathematics that was based on the notion of different levels of infinity and associated matters. I recall that, at the outset of this endeavor, my initial determination was that I would make no presumption that this esoteric branch was in itself “incorrect”, and that my presumption from the outset would be that both branches were consistent within themselves.


However, it soon became evident that the assumptions that led to the notion of different levels of infinity had multiple associated problems, and which intertwined in various ways to generate numerous strange results - in fact, results that are not only strange, but contradictory. After much analysis, it became clear that the sources of these contradictions could be traced back to a small number of crucial assumptions. And it turned out that, in fact, my initial presumption that the esoteric non real world branch of mathematics was surely consistent in itself turned out to be very wide of the mark, and logical analysis shows that that branch of mathematics is replete with contradictions that should not be tolerated by any mathematician or logician.


In summary, the bulk of the work on this website is not some random collection of matters of trifling importance, but a collection of serious articles that demonstrate that a substantial chunk of mathematics is plagued by a common fundamental error: the confusion of levels of language. The result enables an elimination of a plethora of ugly deformities that have bedeviled the mathematical world over the past hundred years and more. The landscape of mathematics that remains when those mathematical deformities have been weeded out is one of serene beauty and wonder. But there are those who refuse to see that beauty, which brings to mind the old adage: there are none so blind as those who refuse to see.


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Copyright   James R Meyer   2012 - 2023