Number of elements is odd infinity

Question

If we have one continuous function F(x), and if we define f(x)=F(x) on domain from open interval (a, b), and if F(a)=F(b)
If function f(x) is monotonically increasing from point a to point M, and monotonically decreasing from point M to point b
Can we assume there is odd infinity number of numbers for which f(x) is defined?
Because for every x there is one z where x,z are from domain of f(x) except for M?
Every element from that range has it's pair except for M

We know that there is infinitely many numbers from point a to b, but can we say that we are sure that there is odd number of that from all of that given above — Hrca12, Nov 17 '15 at 18:38
The term "odd" number applies only to finite numbers. You cannot use it with "infinity". — user247327, Nov 17 '15 at 18:40
I don't understand what you are asking. What is the domain of $F$? The interval $(a,b)$ is open, not closed. How can infinity be odd? — copper.hat, Nov 17 '15 at 18:41
@Hrca12 I suggest looking at this to understand why your question is not well-posed: http://math.stackexchange.com/questions/49034/is-infinity-an-odd-or-even-number?rq=1 — Brenton, Nov 17 '15 at 18:41
@copper.hat: Just look at all the results we get from the existence of infinite sets. Infinity is odd! :-P — Asaf Karagila, Nov 17 '15 at 18:45
An interesting thing that we can prove: a continuous function defined on $(a,b)$ cannot be exactly 2-to-1. But we don't prove it by talking about "even" and "odd" infinity. — GEdgar, Nov 17 '15 at 18:49

score 2 · Accepted Answer · edited Apr 13 '17 at 12:21

If your open interval has a cardinality which is odd infinity then presumably a similar argument would suggest that a half-open interval would have a cardinality which is an even infinity.

It is possible to find a bijection between a half-open interval and an open interval, or between a half-open interval and a closed interval: see How to define a bijection between $(0,1)$ and $(0,1]$? and the questions linked form it.

The bijection shows that these two cardinalities are in fact the same, which is why infinities are not described as odd or even.

score 0 · Answer 2 · answered Nov 17 '15 at 18:41

0

Infinity cannot be even or odd. "Odd infinity" is a nonsensical phrase.

answered Nov 17 '15 at 18:41

Deusovi

2,792
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That's not true. Assuming the axiom of choice every infinite cardinality is even; and that's quite odd. More specifically in asking "how many numbers satisfy blah" we essentially talk about cardinality rather than the intentionally "less defined" notion of infinity in calculus. – Asaf Karagila Nov 17 '15 at 18:44
@Asaf: Interesting - how would the axiom of choice imply every infinite cardinality being even? – Deusovi Nov 17 '15 at 18:47
Every infinite cardinal satisfies $\kappa=\kappa+\kappa$. – Asaf Karagila Nov 17 '15 at 18:50
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@Asaf Not only that, they are also perfect squares. – Tobias Kildetoft Nov 17 '15 at 18:51
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@AsafKaragila doesn't it mean that infinite cardinals are odd too, since $\kappa + 1 = \kappa$? – lisyarus Nov 17 '15 at 18:54
@Tobias: Yes, that is also true. – Asaf Karagila Nov 17 '15 at 19:01
@lisyarus: Yes. Thus it lends even more credence to the statement that infinite sets are odd. Aren't they? – Asaf Karagila Nov 17 '15 at 19:01
@AsafKaragila they definitely are! – lisyarus Nov 17 '15 at 19:03

Number of elements is odd infinity

2 Answers2