Math Is Fun Forum

Stanley_Marsh

Member

Registered: 2006-12-13

Posts: 345

infinity

the set of all rational numbers is countable , but set of rationals seems to have "greater" infinity than the set of positive integers does.
  The concept of infinity is really vague. Is the R^1 both closed and open a math contradiction ? Will it lead to some crisis ?

Last edited by Stanley_Marsh (2007-04-19 09:20:53)

---

Numbers are the essence of the Universe

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: infinity

the set of all rational numbers is countable , but set of rationals seems to have "greater" infinity than the set of positive integers does.

No, it does not! The rationals have the same cardinality as the positive integers, namely

.

Is the R^1 both closed and open a math contradiction ? Will it lead to some crisis ?

I direct you to my (ahem) intellectually stimulating thread:

http://www.mathsisfun.com/forum/viewtopic.php?id=6677

Last edited by JaneFairfax (2007-04-19 10:23:07)

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: infinity

the set of all rational numbers is countable , but set of rationals seems to have "greater" infinity than the set of positive integers does.

No, it does not! The rationals have the same cardinality as the reals, namely

.

It could be argued that Stanley is right. The rationals do "seem to" be a bigger infinity than the integers, because all integers are rationals but there are rationals that aren't integers. But just because something seems to be true doesn't mean that it is.

I assume you meant to say integers instead of reals.

---

Why did the vector cross the road?
It wanted to be normal.

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: infinity

I did mean positive integers, sorry.

Yeah, you could also say that the set of all integers “seems” to be larger than the set of all even integers. Yet there is a bijection between the integers and the even integers, just as there is a bijection between the rational numbers and the integers. In fact, this is a property of all infinite sets: a set S is infinite if and only if there is a bijection between S and a proper subset of S. This can actually be taken to be the definition of an infinite set.

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: infinity

One may be tempted to say that if A is a subset of B, then B has a larger size than A.  It certainly works for all finite sets.  Maybe we can extend the idea to infinite sets.

But think about the set of integers union {0.5}.  The integers are certainly a subset, but would you really say the integers are smaller than this set?  That's like saying infinity is smaller than infinity + 1.

So talking about set size in terms of subsets doesn't really work out.  So lets try something different.  We can only make a bijection between two finite sets of the size.  And we can expand this to infinite sets as well.  And in fact, this became the definition behind cardinality, because although it does some weird things, it doesn't have any problems that we know of.

---

"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

Stanley_Marsh

Member

Registered: 2006-12-13

Posts: 345

Re: infinity

The characteristic only exists in math lol  ~one thing with two opposite  character.

---

Numbers are the essence of the Universe