What statements can you apply to infinity

[TheScienceOrca]

For example;

Please tell me which are false and true.

If false explain why, don't just say because so and so said so, please EXPLAIN the concept otherwise I will never learn or explain the true answer!


Can infinity odd or even? If not what state is it in? Or the fact that it is never simply a static number, it is dynamic. Whatever the highest number you can think of at the time, infinity is FOR YOU.

Infinity is relative to the observer correct?


So infinity does have a limit?

Limit of f(x)=infinity as x approaches infinity = the highest number relative to the observer

That statement is essentially true correct?

If not why?

Thank you! I am calc first year now and having some questions about limits.



2)

 

[TheScienceOrca]

Sorry wrong category I see now!

 

[Fredrik]

Infinity is neither odd, nor even. Only integers are classified as even or odd. These are the definitions: An integer n is said to be even if there's an integer m such that n=2m. An integer n is said to be odd if there's an integer m such that n=2m+1. Every integer is either even or odd, but "infinity" is not an integer.

Infinity is not relative to the observer. There's no such thing as "infinity for you". The standard way to define $+\infty$ and $-\infty$ ensures that the following statement is true: For all real numbers x, we have $x<+\infty$.

The claim $\lim_{x\to+\infty}f(x)=+\infty$ means that for each real number M, there's a real number r such that f(x)>M for all x>r. (This means that to the right of the point r on the x axis, the graph of f stays above the line y=M).

 

[HakimPhilo]

To understand more the philosophy of the infinite read Rucker - Infinity and the mind, it will clarify a lot of misconceptions you have.

1) Infinity isn't odd nor even if you try to apply the definition of an odd and an even number it will basically fail for the infinite, adding to that the fact that the set of all odd and even numbers is $\Bbb Z$ and $\infty\notin\Bbb Z$.

2) No, can you observe infinity?

 

[Stephen Tashi]

"Infinity" doesn't refer to just one single mathematical concept. Mathematical definitions actually don't define single words. For example the statement "The limit of f(x) as x approaches A is equal to L" has a definition, and people call this "the definition of limit", but it isn't the definition of the single word "limit" or the single word "approaches". And there are other statements in mathematics that use the word "limit" and have a different definition.

There are quite a few complete statements in mathematics that use the word "infinity". If you ask a question about the single word "infinity", it is ambiguous. You aren't asking about something that has a specific mathematical meaning.

 

[FactChecker]

Infinity is actually a pretty deep subject. There are different types (actually different sizes) of infinity. The types 'odd' and 'even' do not fit, but the infinity of the the size of the set of all counting numbers is smaller than the infinity of the set of all real numbers. So they are infinities of different types. When you start studying infinity, there are very precise definitions. To see that there are different sizes of infinity, check out this link: http://en.wikipedia.org/wiki/Cantor's_diagonal_argument[/PLAIN]

 

[gopher_p]

If you don't specifically (and arbitrarily) restrict the definitions of even and odd to apply only to finite numbers, then the infinite ordinal $\omega\cdot2=\omega+\omega$ is even. What's even more fun is that the ordinal $\omega\cdot2+1$ is both odd (obviously) and even, since, according to the rules of ordinal arithmetic, $(\omega+1)+(\omega+1)=\omega\cdot 2+1$. Now the first infinite ordinal $\omega$ is neither odd nor even. So some ordinals are odd, some are even, some are both, and some are neither.

Cardinal numbers are comparatively less interesting. Every infinite cardinal is both even and odd according to the rules of cardinal arithmetic; $\kappa+\kappa=\kappa+\kappa+1=\kappa$ for any infinite cardinal $\kappa$.

Of course, neither of these has anything to do with the infinity of calculus (which is really just a convenient, though misleading, notation for "arbitrarily large"). But it's interesting nonetheless.

 

[FactChecker]

If you don't specifically (and arbitrarily) restrict the definitions of even and odd to apply only to finite numbers, then the infinite ordinal $\omega\cdot2=\omega+\omega$ is even. What's even more fun is that the ordinal $\omega\cdot2+1$ is both odd (obviously) and even,

Ha! I guess that's right.