Prove Set of Real Numbers Unbounded: Tips & Examples

  • Context: Undergrad 
  • Thread starter Thread starter xlalcciax
  • Start date Start date
  • Tags Tags
    Sets
Click For Summary

Discussion Overview

The discussion centers on how to prove that the set of real numbers is unbounded, exploring various definitions and approaches related to the concept of boundedness in mathematical contexts.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • Some participants inquire about the definition of "unbounded" and suggest that it may involve showing that the cardinality of the real numbers exceeds ℵ ("aleph null").
  • Others note that "unbounded" has multiple equivalent definitions, such as not being a subset of some open ball.
  • One participant suggests that proving ℝ is unbounded is easier than proving it is uncountable, contingent on having a clear definition of "bounded."
  • There is a call for clarification on the definition of "bounded" to guide the discussion further.
  • A humorous suggestion is made that one could simply reference Cantor's diagonalization process to demonstrate that the real numbers are uncountable.

Areas of Agreement / Disagreement

Participants express differing views on the definitions of "unbounded" and "bounded," indicating that multiple competing interpretations exist without a consensus on a single approach.

Contextual Notes

The discussion highlights the importance of definitions in mathematical proofs and the potential for ambiguity in terms like "bounded" and "unbounded." Specific definitions and assumptions are not fully established, which may affect the direction of the proof.

xlalcciax
Messages
12
Reaction score
0
How to prove that set of real numbers is unbounded?
 
Physics news on Phys.org
What is your definition of unbounded?
 
xlalcciax said:
How to prove that set of real numbers is unbounded?

It most likely involves showing that the cardinality of the real numbers exceeds ℵ ("aleph null").
 
reef said:
It most likely involves showing that the cardinality of the real numbers exceeds ℵ ("aleph null").
That would be "uncountable". "Unbounded" has a bunch of different but equivalent definitions, like "is not a subset of some open ball".
 
True. I figured that you would have to go about it the same type of way. Do you have to establish that a set without bound has no limit? Regardless, I'd be interested to see the proof when xlalcciax figures it out.
 
It's much easier to prove that ℝ is unbounded than to prove that ℝ is uncountable. Once you've written down a definition of "bounded" and thought about what it means, you're pretty much done.
 
I wish xlalcciax would get back to us with his definition of "bounded" so we would know in which direction to go.
 
You could be really lazy and just say that the real numbers can be proven to be uncountable by means of Georg Cantors diagonalization process.. lol.
 

Similar threads

Undergrad Proof of a fact about real numbers - transcendental and algebraic
  • · Replies 7 ·
Replies
7
Views
3K
MHB Unbounded subset of ordinals a set?
  • · Replies 1 ·
Replies
1
Views
2K
High School Unbounded Logical Trees: Constructing Natural Numbers with Logical Trees
  • · Replies 7 ·
Replies
7
Views
2K
Undergrad Countability of binary Strings
  • · Replies 18 ·
Replies
18
Views
4K
Undergrad Formalizing infinity sets
  • · Replies 3 ·
Replies
3
Views
425
Undergrad Defining a transcendental number and countability
  • · Replies 8 ·
Replies
8
Views
2K
Undergrad I don't understand Dedekind cuts
  • · Replies 14 ·
Replies
14
Views
1K
Undergrad This fact on 2-dimensional space blows my mind
  • · Replies 17 ·
Replies
17
Views
3K
Undergrad Computable reals and r.e./Diophantine
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad Countably Infinite Unions and the Real Numbers: Can They Really Be Uncountable?
  • · Replies 4 ·
Replies
4
Views
3K