1

Tao Analysis

For instance, it is possible to have an infinite set consisting of finite numbers (the set $\mathbb N$ of natural numbers is one such example), and it is also possible to have a finite set consisting of infinite objects (consider for instance the finite set $\{\Bbb {N,Z,Q,R}\}$,which has four elements, all of which are infinite.

My question:

"finite numbers" here refers to each element itself in $\mathbb N$ which is finite as infinity is not one of the natural numbers. "infinite objects" here refer to each set of $\{\Bbb {N,Z,Q,R}\}$ which is infinite.

Am I right?

Thank you!

Andrew Li
  • 431
  • 4
    Each individual natural number is finite. Example, 7824077 is finite. But the set of all natural numbers is an infinite set. We say $\mathbb N$ has "infinitely many" elements. But none of the elements is infinite, so we do not say $\mathbb N$ has "infinite" elements. – GEdgar Jul 22 '22 at 16:41
  • 2
    Each natural number is finite. – jjagmath Jul 22 '22 at 16:41
  • Basically, when talking about infinite things, you have to be really precise in what you say. In fact, that's why the (epsilon-delta) definitions of limits are so precise. – Adam Rubinson Jul 22 '22 at 16:45
  • So $\mathbb N$ has finite elements? Is there any set that has infinite elements? @GEdgar – Andrew Li Jul 22 '22 at 17:03
  • 1
    Yes, the set ${\Bbb {N,Q,R}}$ has infinite elements. Each of its three elements in infinite. We can also say that each one has infinitely many elements, which may be what you are trying to say when you say $\Bbb N$ has infinite elements. – Ross Millikan Jul 22 '22 at 17:07
  • The natural numbers start at $0$ (or $1$) and they go all the way up, but that don't go all the way up to anything. They never end. And every single natural number you ever pick no matter how big it is will be finite. "Infinity" is not a natural number and is not an element of the set. – fleablood Jul 22 '22 at 17:09
  • 2
    "So ℕ has finite elements?"

    This is an ambiguous statement, and I would avoid it. Should I read it to mean "$\mathbb{N}$ has finitely many elements" (which is wrong), or "the elements of $\mathbb{N}$ are all finite" (which is correct).

    "Is there any set that has infinite elements?"

    It depends on what you mean by "an infinite element". For example, the extended real line contains two infinite elements: $-\infty$ and $+\infty$.

     – Xander Henderson Jul 22 '22 at 17:09
  • 2
    "So N has finite elements?" $\mathbb N$ consists of finite elements but it has an infinite amount of them. There is a big difference between "A set is infinite" and "the elements of the sets are themselves infinite". As Ross Millikin points out. ${\mathbb N, \mathbb Q, \mathbb R}$ is a finite set (it only has three elements) but each of its elements are each infinite. – fleablood Jul 22 '22 at 17:12
  • "So ℕ has finite elements?" It is not true that (a) $\mathbb N$ has finitely many elements. It is true that (b) all of the elements of $\mathbb N$ are finite. But "$\mathbb N$ has finite elements" probably means (b). To be clear, state what you mean using either (a) or (b). – GEdgar Jul 22 '22 at 18:51
  • We have sets$\mathbb N,Z,Q,R$ , and each one is an infinite set consisting of finite elements. Is this right? @fleablood – Andrew Li Jul 23 '22 at 12:29
  • @GEdgar What is your definition of infinite elements? – Filippo Jul 23 '22 at 12:39
  • What I mean is the cardinality of $\mathbb N$, It's infinite, then I use "infinite number" to mean there are infinite amount of numbers in $\mathbb N$. So should I use "infinitely many" numbers to infer the cardinality of infinite sets? @RossMillikan Millikan – Andrew Li Jul 23 '22 at 13:19
  • Yes, that is correct. – Ross Millikan Jul 23 '22 at 13:37
  • Thank you for your explanation,really appreciate. @XanderHenderson – Andrew Li Jul 23 '22 at 14:03

1 Answers1

1

For instance, it is possible to have an infinite set consisting of finite numbers (the set $\mathbb N$ of natural numbers is one such example), and it is also possible to have a finite set consisting of infinite objects (consider for instance the finite set $\{\Bbb {N,Z,Q,R}\}$,which has four elements, all of which are infinite.

Clearer:

  • For instance, it is possible to have a set of infinitely many finite numbers (for example, $\mathbb N$), and it is also possible to have a set of finitely many infinite sets, that is, a set of finitely many sets of infinitely many finite numbers (for example, $\{\Bbb {N,Z,Q,R}\}$).

"finite numbers" here refers to each element itself in $\mathbb N$ which is finite as infinity is not one of the natural numbers.

Yes.

"infinite objects" here refer to each set of $\{\Bbb {N,Z,Q,R}\}$ which is infinite.

"Infinite objects" here refer to each set in $\{\Bbb {N,Z,Q,R}\},$ which is infinite (has infinitely many elements).

ryang
  • 38,879
  • 14
  • 81
  • 179
  • Is it correct to say $1 \in {\Bbb {N,Z,Q,R}} $? I think the answer is no, because ${\Bbb {N,Z,Q,R}} $ just has four elements, and $1$ is not in it. Am I right? thank you. @ryang – Andrew Li Mar 14 '23 at 16:31
  • @AndrewLi I agree with you. – ryang Mar 14 '23 at 16:34