Question about infinite sets ?

[cragar]

Before i say this i think transfinite numbers and infinite sets are really cool.
But could you argue that infinite sets don't exist, I mean you couldn't show me one.
I apologize if this is in the wrong section.
I do believe in infinite sets but I was just wondering if you could argue this. Or justify how they exist.

 

[kru_]

What does it mean for something to exist? Isn't it enough that we can give a precise definition to something and utilize that definition to advance our knowledge of things that do exist?

Similar logic would have that negative numbers don't exist. After all, you can't show me -2 apples.

We have to accept the "existence" of certain things when we deal with abstract thinking, otherwise we would get nowhere.

 

[cragar]

ok I see what your saying, that's a good answer.

 

[micromass]

Well, I personally don't believe that infinite sets exist. But that's not why mathematicians use it. We use infinity because it is easier to work with than with finite things and because it's reasonable approximation.

For example, let's say I want to measure the lengths of some cars. Most likely, we first want to make a mathematical model of the lengths of the cars. That is, we want to introduce a "sample space", this is a space that contains all possible measurements.

For example, 2 meters would be a possible measurements, so 2 is in the sample space. 2.18 meters should be in the sample space, 1.97 should be in the sample space, etc.
But instead of taking all possible measurements, we make it easy on ourselfs and we simple take $\mathbb{R}^+$as sample space. Of course, we will never measure 9999 meters or $\pi$ meters, but we just include it because it's easy.

Furthermore, we can apply analysis and calculus on $\mathbb{R}^+$. If we would just take the finite set of measurements, then this was not possible...

 

[cragar]

thanks for your answer, is a sample space similar to a subspace in linear algebra, I never completely understood the concept of a subspace in linear algebra.

 

[micromass]

thanks for your answer, is a sample space similar to a subspace in linear algebra, I never completely understood the concept of a subspace in linear algebra.

Hmm, that's not how I meant it. I actually made the word sample space up, but it has nothing to do with subspaces. A subspace in linear algebra is informally just a subset that is a vector space under the usual operations. There's nothing more to it, really...

 

[cragar]

we just can't have operations that take us out of our subspace?

 

[micromass]

we just can't have operations that take us out of our subspace?

Indeed! More specifically, given a vector space (V,+,.), then a subspace is a non-empty set U such that + and . does not take vectors out of U.
Additionally, this gives us that U itself is a vector space under + and .

 

[cragar]

thanks for your answer.