What are some paradoxes in mathematics?

[NeutronStar]

I'm looking for paradoxes in mathematics.

Things like Gabriel's Horn where the internal surface area of an object is infinite yet the volume is finite.

Or the fact that there are larger and smaller infinities yet infinity is supposed to be an endless process.

Any others?

Is there a book on mathematical paradoxes?

Thanks in advance.

 

[arildno]

Why are these paradoxes?

 

[chroot]

The term 'paradox' is inappropriate, because it (generally) means something that is inconsistent and not understood; these mathematical facts are neither inconsistent nor not undersootd. Maybe a better phrase would be 'unintuitive truths.'

- Warren

 

[NeutronStar]

The term 'paradox' is inappropriate, because it (generally) means something that is inconsistent and not understood; these mathematical facts are neither inconsistent nor not undersootd. Maybe a better phrase would be 'unintuitive truths.'

- Warren

Ok, so these are only 'truths" with respect to the definition of the limit.

That's something that I do understand.

I basically already knew that was the answer but I just needed a memory jog.

Thanks.

 

[arildno]

Ok, so these are only 'truths" with respect to the definition of the limit.

Incorrect.

These statements are only paradoxes within a basically incoherent, and usually unstated, set of assumptions.

 

[robert Ihnot]

Consider Russell's Paradox -

"A logical contradiction in set theory discovered by Bertrand Russell. If R is the set of all sets which don't contain themselves, does R contain itself? If it does then it doesn't and vice versa" http://computing-dictionary.thefreedictionary.com/Russell+paradox

Two futher comments by author are: Zermelo Fränkel set theory is one "solution" to this paradox. Another, type theory, restricts sets to contain only elements of a single type, (e.g. integers or sets of integers) and no type is allowed to refer to itself so no set can contain itself.

A message from Russell induced Frege to put a note in his life's work, just before it went to press, to the effect that he now knew it was inconsistent but he hoped it would be useful anyway.

 

[robert Ihnot]

Take the matter of the tortoise and the hare. Before the hare can catch the tortoise, it must go half the distance between them. Then it must go half again, so the tortoise can never reach the hare as Zeno suggested.

But it travels each new distance in half the time as before, thus the series is in time: Limit 1+1/2+1/4+1/8++=2. Indicating that an infinite number of things can be done in a finite amount of time.

This seems, somewhat, similar to Gabriel's Horn, does it not? The fact is the amount of paint put on the surface if we think of this as first a paint job, gets less and less as we go along, in fact, the thickness of the paint covers from wall to opposite wall, and eventually the thickness of the paint is less than any given amount. So that only a finite amount of paint would cover everything.

 

[robert Ihnot]

The other one about larger and smaller infinites is quite different, since before Cantor gave us such set theory, no one took much stock in that. But Cantor looked at the matter of cardinality, which is to put one set in one to one correspondence with another. With some infinite sets this is possible and with others it is not. For example, even Galileo pointed out the mapping sending n to n^2 shows that there are as many squares of integers as they are integers. This is referred to as Galileo's paradox. http://en.wikipedia.org/wiki/Galileo's_paradox. A further note found there is:

"Galileo concluded that the ideas of less, equal, and greater applied only to finite sets, and did not make sense when applied to infinite sets. (You see then Galileo "solved" the paradox by ruling out the situation.)

BUT, in the nineteenth century, Cantor, using the same methods, showed that while Galileo's result was correct as applied to the whole numbers and even the rational numbers, the general conclusion did not follow: some infinite sets are larger than others, in that they cannot be put into one-to-one correspondence."

Some people refuse to accept Cantor, such as the intuitionists, who argued that infinity meant only a potential situation, an end point, something never actually achieved. They argued that proofs must be constructible in a finite number of steps, and ruled out proof by contradiction.

But as a logic professor pointed out to me once, "How can you refuse to accept the set of all integers?" So that intuitionists generally accept denumerable infinity (a one to one correspondance with the integers) but not higher orders. They reject the Axiom of Choice.

 

[matt grime]

Or the fact that there are larger and smaller infinities yet infinity is supposed to be an endless process.

Find me one mathematics reference that defines 'infinity as an endless process'.

 

[kesh]

banach-tarski "paradox" (though counter-intuitive truth is better) is a nice one.

chop up a ball, put the pieces together and get two balls of equal size to the first

 

[Hurkyl]

Some people refuse to accept Cantor, such as the intuitionists, who argued that infinity meant only a potential situation, an end point, something never actually achieved. They argued that proofs must be constructible in a finite number of steps, and ruled out proof by contradiction.

That's curious; |P(X)| > |X| is a theorem of intuitionist set theory! It's true that intuitionist logic does throw out the law of the excluded middle ($P \vee \neg P = T$), but it keeps the law of noncontradiction ($P \wedge \neg P = F$), which means you can use proof by contradiction for proving negative statements, such as $|\mathcal{P}(X)| \neq |X|$.

You have to go all the way to constructivism (or something more exotic) before you can permit |X| = |P(X)|!

But as a logic professor pointed out to me once, "How can you refuse to accept the set of all integers?"

It's not that tough, actually. For many purposes, accepting the class of integers is enough; you don't have to assume they actually form a set.

 

[Gib Z]

You may be interested that the area under 1/x^2 from 0 - infinity converges, but the one for 1/x does not, even though the shapes are extremely similar. And infinities larger than others actually make sense. Some functions will reach large numbers quicker than others, both eventually reaching infinity at the limit, but the other is still larger. eg n^n will reach it much faster than say, n!.

 

[matt grime]

You may be interested that the area under 1/x^2 from 0 - infinity converges, but the one for 1/x does not,

neither of those areas is finite. you meant 1 to infinity, not 0 to infinity. Areas don't converge, by the way. They just are. I

even though the shapes are extremely similar. And infinities larger than others actually make sense. Some functions will reach large numbers quicker than others, both eventually reaching infinity at the limit, but the other is still larger. eg n^n will reach it much faster than say, n!.

that is asymptotics, (size of finite things), not infinite cardinals as was meant by 'different sizes of infinity'.

 

[CRGreathouse]

Incorrect.

These statements are only paradoxes within a basically incoherent, and usually unstated, set of assumptions.

I'm not sure I like that wording either... they're contradictions with those (unstated, intuitive, inconsistent) set of assumptions.

 

[DeadWolfe]

http://en.wikipedia.org/wiki/Skolem's_paradox

 

[Gib Z]

my bad, yep i meant for 1 to infinity, not 0. And sorry about that, I am not familiar with infinite cardinals.

 

[arildno]

I'm not sure I like that wording either... they're contradictions with those (unstated, intuitive, inconsistent) set of assumptions.

They certainly are contradictions. I regard that as an effect of the set being, basically, incoherent at the outset.