Exploring Cantor's Sweet Infinity: C vs. N

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In summary: His famous diagonal argument shows that for any set X, the set of all subsets of X has strictly larger cardinality than X. Since the real numbers are a subset of the set of all subsets of the natural numbers, it follows that |\mathbb{R}|>|\mathbb{N}|.In summary, the conversation is about the comparison between the cardinality of the continuum of numbers (with decimals) and the cardinality of all natural numbers. The question is whether there are more numbers with infinite decimals in the continuum than there are natural numbers. The answer, according to Cantor's theorem, is yes.
  • #1
Hippasos
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Cantor's s(w)et :)

Hi!

I don't know if this makes any sense, but after reading the book "Pi in the Sky"
this summer (It got really exceptionally hot here in high latitudes) I was let wondering the following:

Let C be the number of the continuum of numbers (with decimals).

Let N be the number of all natural numbers.

When counting decimals in the numbers which belongs to C,
is there numbers which have C-number decimals
more than numbers which have N-number decimals ?

Thanks!
 
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  • #2


What do you mean by C-number decimals and N-number decimals?
 
  • #3


I apologize for my mediocre English...
Thinking another way to rephrase this.
 
  • #4


Hippasos said:
Hi!

I don't know if this makes any sense, but after reading the book "Pi in the Sky"
this summer (It got really exceptionally hot here in high latitudes) I was let wondering the following:

Let C be the number of the continuum of numbers (with decimals).

Let N be the number of all natural numbers.

When counting decimals in the numbers which belongs to C,
is there numbers which have C-number decimals
more than numbers which have N-number decimals ?

Thanks!
Unfortunately, I can't quite make out what you are asking here. You say "Let C be the number of the continuum of numbers" but then you ask about "numbers which belong to C" so apparently what you really mean is "Let C be the continuum of numbers (the set of such numbers)" and "Let N be the set of all natural numbers". Now what do you mean by "counting decimals"? Are you asking "is the Cardinality of C greater than the Cardinality of N"? If so, the answer is definitely yes- that was shown by Cantor's "diagonal" method. We "count" cardinality by setting up a "one to one correspondence", that is an assignment to such that each object in one set is assigned to one and only one object in the other set. If such is possible, then the two sets have the same Cardinality. Cantor showed that, however you try to set up such an assignment, there are numbers in C that are "left over"- that is, in this sense, there are "more" numbers in C than there are in N and so C has greater cardinality than N.
 
  • #5


Hi!

I think I got it right this time... :D

Let C be the number of the continuum of numbers (with decimals).

Let N be the number of all natural numbers.

When counting decimals in the numbers which belongs to the continuum,
is there numbers which have C decimals
more than numbers which have N decimals ?
 
  • #6


Hippasos said:
Let C be the number of the continuum of numbers (with decimals).

Let N be the number of all natural numbers.

When counting decimals in the numbers which belongs to the continuum,
is there numbers which have C decimals
more than numbers which have N decimals ?

I think that C is [tex]\mathbb{R}[/tex], the set of real numbers. In that case I interpret the question as: Is [tex]|\mathbb{R}|>|\mathbb{N}|[/tex]? The answer, of course, is yes: this is Cantor's theorem.
 

FAQ: Exploring Cantor's Sweet Infinity: C vs. N

What is Cantor's Sweet Infinity?

Cantor's Sweet Infinity is a mathematical concept created by Georg Cantor in the 19th century. It refers to the infinite number of real numbers between any two numbers on a number line.

What is C vs. N in relation to Cantor's Sweet Infinity?

C vs. N refers to the comparison between the cardinality (size) of the set of real numbers (C) and the set of natural numbers (N). Cantor's work showed that the cardinality of the set of real numbers is greater than the cardinality of the set of natural numbers, even though both are infinite.

Why is Cantor's Sweet Infinity important?

Cantor's work on infinity revolutionized the field of mathematics and has applications in various areas such as set theory, calculus, and computer science. It also challenges our understanding of infinity and the concept of a "countable" set.

What is the significance of C vs. N in mathematics?

The concept of C vs. N helps us understand that there are different levels of infinity and that not all infinite sets are equal. It also has implications for other mathematical concepts, such as limits and continuity.

How can Cantor's Sweet Infinity be visualized?

Cantor's Sweet Infinity can be visualized using a Cantor set, which is a fractal created by removing middle thirds from a line segment. This set has an infinite number of points, but it is still considered "smaller" than the set of real numbers.

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