Proving R is Bigger Than N Set Elements

  • Thread starter Thread starter dextercioby
  • Start date Start date
AI Thread Summary
The discussion focuses on proving that the cardinality of the set of real numbers, \mathbb{R}, is greater than that of the natural numbers, \mathbb{N}. It highlights Cantor's theorem, which states that no bijection exists between these two sets. A proof is mentioned that illustrates this concept through a simple idea, although it may not be immediately apparent to those unfamiliar with it. Participants express appreciation for the clarity and effectiveness of the proof. The conversation emphasizes the significance of understanding set cardinality in mathematics.
dextercioby
Science Advisor
Insights Author
Messages
13,390
Reaction score
4,046
that the number of elements of \mathbb{R} (seen as a set, obviously) is bigger than the number of elements of \mathbb{N} ...? :bugeye:

Daniel.
 
Mathematics news on Phys.org
Two sets have the same cardinality iff there exists a bijection between the sets. Cantor showed that there is no bijection between \mathbb{R} and \mathbb{N}. A "[URL proof[/URL] of this involves a very simple idea - simple once one has seen it, but not until then.

Regards,
George
 
Last edited by a moderator:
Thankyou for the reply.

Daniel.
 
Very nice. I've never seen that proof before.
 
Wow...that's good!
 

Thread 'Fermat's Last Theorem'

Fermat's Last Theorem has long been one of the most famous mathematical problems, and is now one of the most famous theorems. It simply states that the equation $$ a^n+b^n=c^n $$ has no solutions with positive integers if ##n>2.## It was named after Pierre de Fermat (1607-1665). The problem itself stems from the book Arithmetica by Diophantus of Alexandria. It gained popularity because Fermat noted in his copy "Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et...
View full post »

Thread 'What Exactly is Dirac’s Delta Function? - Insight'

Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
View full post »
Thread 'Imaginary Pythagorus'

Thread 'Imaginary Pythagorus'

I posted this in the Lame Math thread, but it's got me thinking. Is there any validity to this? Or is it really just a mathematical trick? Naively, I see that i2 + plus 12 does equal zero2. But does this have a meaning? I know one can treat the imaginary number line as just another axis like the reals, but does that mean this does represent a triangle in the complex plane with a hypotenuse of length zero? Ibix offered a rendering of the diagram using what I assume is matrix* notation...
View full post »

Similar threads

I (0,1) is uncountable using binary expansions?
Replies
15
Views
2K
MHB Number of multiplications and divisions for LU decomposition
Replies
22
Views
3K
I Proof of a fact about real numbers - transcendental and algebraic
Replies
7
Views
1K
Can you use Taylor Series with mathematical objects other than points?
Replies
4
Views
2K
Prove ##1 + a=s(a)=a+1## for ##a \in \mathbb{N}##
Replies
1
Views
1K
MHB Challenge problem #5 [Olinguito]
Replies
2
Views
1K
Defining Vectors: Scalars vs 1-Dimensional Vectors
Replies
18
Views
2K
I Zero raised to a positive real number
Replies
12
Views
2K
A Why the triangle inequality is greater than the 2 max{f(x),g(x)}
Replies
1
Views
1K
I SL(n,R) Lie group as submanifold of GL(n,R)
Replies
36
Views
4K

Hot Threads

Recent Insights

Back
Top