Copyright © James R Meyer 2012 - 2016 www.jamesrmeyer.com
There are several opinions on what constitutes a paradox, and you can find several differing definitions, but, in general a paradox is either:
The apparent contradiction can almost invariably be attributed to one or more of the following:
Some paradoxes raise interesting issues and some of the more interesting ones will be discussed on this site. More may be added at a later date. Currently there are the following:
Note that this is not intended to be in any way comprehensive, since there are plenty of descriptions of paradoxes available elsewhere.
The response of mathematicians and logicians to paradoxes has shifted considerably since the beginning of the twentieth century.
At that time, when Bertrand Russell discovered (in 1901) his famous paradox, (Footnote: See, for example Russell’s paradox entry in Stanford Encyclopedia.) such paradoxes were viewed as intolerable contradictions that were totally unacceptable in the world of logic. Russell’s paradox marked a crisis point in logic and mathematics, and logicians and mathematicians tried, without success, to come up with a foundation of logic and mathematics that was simple and appeared intuitively to be correct.
As time progressed, however, particularly after Gödel published his incompleteness proof, the way in which paradoxes such as Russell’s paradox were viewed underwent a subtle shift. Logicians began to believe that contradictory paradoxes were an inevitable part of any comprehensive system of logic, and that one might as well learn to live with them.
So, instead of the viewpoint that was prevalent at the time of the discovery of Russell’s paradox, when the common opinion was that the source of such contradictions should be rooted out and dealt with, we now have a situation where many people (for example, Gregory Chaitin and Raymond Smullyan) take a perverse delight in such contradictions, and embrace them as desirable notions, rather than as problems to be solved.
But it is not only possible to eliminate such contradictions from logical language and logical analysis, it is imperative that this should be done. In fact, when it is seen that the root of most of these contradictions arise from confusion of levels of language, it is seen that the elimination of such contradictions is quite straightforward.
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There is a new addition to the page Yet another flawed incompleteness proof, where Berto’s proof of incompleteness in his book There’s something about Gödel comes under scrutiny.
I found that making, adding or deleting footnotes in the traditional manner proved to be a major pain. So I developed a different system for footnotes which makes inserting or changing footnotes a doddle. You can check it out at Easy Footnotes for Web Pages (Accessibility friendly).
I have now added a new section to my paper on Russell O’Connor’s claim of a computer verified incompleteness proof. This shows that the flaw in the proof arises from a reliance on definitions that include unacceptable assumptions - assumptions that are not actually checked by the computer code. See also the new page Representability.
There is now a new page on Chaitin’s Constant (Chaitin’s Omega), which demonstrates that Chaitin has failed to prove that it is actually algorithmically irreducible.
For convenience, there are now two pages on this site with links to various material relating to Gödel and the Incompleteness Theorem
– a page with general links:
– and a page relating specifically to the Gödel mind-machine debate:
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Copyright © James R Meyer 2012 - 2016