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There are several opinions on what constitutes a paradox, and you can find several differing definitions, but, in general a paradox is either:

• a statement that appears to be correct but also appears to contradict itself, or
• a statement that appears to be correct but also appears to contradict commonly accepted opinions, or
• a statement that appears to be correct but also appears to be counter intuitive.

The apparent contradiction can almost invariably be attributed to one or more of the following:

1. The ambiguity and context dependency of natural languages such as English.
2. The incorrect application of a logic to a statement of natural languages such as English.
3. An unproven assumption.
4. Faulty intuition.

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. And today there seems to be an increase in the willingness of certain academics to follow this line, and to ignore any need for critical logical analysis.

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.

Footnotes:

Diverse opinions and criticisms are welcome, but messages that are frivolous, irrelevant or devoid of logical basis will be blocked. Difficulties in understanding the site content are usually best addressed by contacting me by e-mail. Note: you will be asked to provide an e-mail address - any address will do, it does not require verification. Your e-mail will only be used to notify you of replies to your comments - it will never be used for any other purpose and will not be displayed. If you cannot see any comments below, see Why isn’t the comment box loading?.

## NEWS

### Lebesgue Measure

There is now a new page on a contradiction in Lebesgue measure theory.

### Illogical Assumptions

There is now a new page Halbach and Zhang’s Yablo without Gödel which analyzes the illogical assumptions used by Halbach and Zhang.

### Easy Footnotes

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).

### O’Connor’s “computer checked” proof

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.

### Previous Blog Posts

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:

Gödel, Minds, and Machines

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