What is the difference between a counter-intuitive statement and a paradox?

Question

In mathematics and logic, what is the difference between a counter-intuitive statement and a paradox?

For example, what differs something like the Banach-Tarski theorem or Gabriel's horn from something like Russell's paradox?

Answer 1

A counter-intuitive statement is either true or false like a normal statement, but surprising in the sense that one would expect a different outcome.

A paradox is undefined, in the sense that if one accepts it, it would be in contradiction to some other statement (an antinomy). It gives the dilemma to drop either the paradox or those statements which are in contradiction to the paradox (in that case it would be either true or false).

Answer 2

Well, it's question of semantics. Usually you start with a theory built upon some axioms that seem reasonable enough (let's call it T). Then you build upon them, and arrive to a certain statment (let's call it S) that doesn't sound as reasonable as the axioms from which you started. Now there are various possibilities:

• You conclude that there must be some error in the reasoning that led you to S. T is still an ok theory and you just have to brush up your reasoning skills. In this case you search for the fault in your reasoning, identify it and maybe call it a fallacy.
• You conclude that your reasoning is solid, and that S in unacceptable, a contraddiction even. Therefore you conclude that T is unacceptable, despite the appearances, and you look for a better theory to work with. This is the case of Russell's paradox, with T being naive set theory and S being "There is a set R such that $ R \in R \iff R \notin R $" .
• You conclude that your reasoning is solid, and that you can accept S. It wasn't what you were expecting from T, but you can live with it, as it's not contraddictory, just strange-sounding. This is the case of the Banach-Tarski theorem and Gabriel's horn.

Now, from this you could be tempted to say that in the second case you call S a paradox and in the third case you just say it's a counter-intuitive statement, but it's not exactly like this: "paradox" can be used in the third case, and in the second case you can use something more specific depending on the nature of the problem you have with S. For example Russell's paradox is also called "Russell's antinomy", and the Banach–Tarski theorem is also called Banach–Tarski paradox.

I'll conclude by mentioning an example in physics: the hydrostatic paradox, which is just a counter-intuitive statement (any quantity of liquid, however small, may be made to support any weight, however large).