No Limit Points in Natural Numbers: Problem with Proof?

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In summary, the conversation discusses the concept of limit points in sets of natural numbers. The sets S1, Sk, and Sk+1 are all finite and therefore have no limit points. However, the conclusion that the set of natural numbers has no limit points is not accurate, as the union of these finite sets does have a limit point. This is because the statement is trying to make a conclusion about an infinite set using only finite sets, which is not always accurate. The thread provided by HallsofIvy further explains this concept.
  • #1
Dschumanji
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Let S1 be the set that contains the natural number 1. Since S1 is finite it has no limit points.

Let Sk be the set that contains the natural numbers less than or equal to k. Sk is finite and therefore has no limit points. The set Sk+1 contains only one more element than Sk and therefore also contains no limit points.

Therefore the set of natural numbers contains no limit points.

I've been told that the conclusion does not follow. Why is that the case?
 
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  • #2
Dschumanji said:
I've been told that the conclusion does not follow. Why is that the case?

The sets [itex]S_k = \{\frac{1}{n}:1 \leq n \leq k\}[/itex] for each [itex]k \in \mathbb{N}[/itex] are finite and have no limit points, but their union does have a limit point. By your reasoning, the union should have no limit points though. Do you see the problem?
 
  • #3
jgens said:
Do you see the problem?
Is it because I am trying to make a statement about an infinite set using only finite sets?
 
  • #4

1. What are no limit points in natural numbers?

No limit points in natural numbers refer to the concept of infinity within the set of natural numbers. It means that there is no largest or smallest number in this set, and the numbers continue infinitely in both directions.

2. What is the problem with proving the existence of no limit points in natural numbers?

The main problem with proving the existence of no limit points in natural numbers is that it goes against the fundamental principles of the set. The definition of natural numbers excludes infinity, making it difficult to prove its existence within the set.

3. Can you provide an example of a no limit point in natural numbers?

Yes, an example of a no limit point in natural numbers is the concept of even and odd numbers. No matter how high or low you go in the set of natural numbers, there will always be an even and odd number present, showing the infinite nature of this set.

4. What are some implications of the existence of no limit points in natural numbers?

The existence of no limit points in natural numbers has significant implications in mathematics, as it challenges our understanding of infinity and the concept of a "largest" or "smallest" number. It also has implications in other fields, such as physics and computer science, where infinity is often used in calculations and theories.

5. How do mathematicians approach the problem of proving the existence of no limit points in natural numbers?

Mathematicians often use different approaches and techniques, such as set theory, to tackle the problem of proving the existence of no limit points in natural numbers. They also look at the implications and consequences of such a proof and continue to explore and expand our understanding of infinity and the natural numbers set.

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