# Thread: Is the infinyty of set {1.2.3...} equal to the infinity of set {1.3.5...} ?

1. ## Is the infinyty of set {1.2.3...} equal to the infinity of set {1.3.5...} ?

Different levels of infinity, another Cantor axiom to argue to death...

Cheers
L-zr

2. ## Re: Is the infinyty of set {1.2.3...} equal to the infinity of set {1.3.5...} ?

Originally Posted by Lazer
Different levels of infinity, another Cantor axiom to argue to death...

Cheers
L-zr
Axiom means "self-evident" or "worthy". There is nothing self-evident about any of Cantor's idiotic ideas, much less worthy.

3. ## Re: Is the infinyty of set {1.2.3...} equal to the infinity of set {1.3.5...} ?

Originally Posted by Lazer
Different levels of infinity, another Cantor axiom to argue to death...
Pre-dates Cantor, not really an axiom
Is the infinyty of set {1.2.3...} equal to the infinity of set {1.3.5...} ?
Seems pretty obvious that n --> 2n-1 is an exact pairing of the first set to the second set, every member of the first set is paired with exactly one member of the second set, no duplications, no elements left out.

4. ## Re: Is the infinyty of set {1.2.3...} equal to the infinity of set {1.3.5...} ?

koeddyrbh:

invisitesimal c. postulate

invisibility action by

because

and

0.999... = 9/10 + 9/100 + 9/1000 + ... =

0.999...9 (n 9s) = 0.999...
Look:

Comprehend ?

2.7
2.97
2.997

2.999...997

Let Sn denote the nth partial sum. Clearly Sn = 1 - 1/10ⁿ
Then for every ε > 0 we have an N = log10 (1/ε) such that:
|Sn - 1| = |1/10ⁿ| < ε for all n > N.
Therefore 0.999... = 1

Let Sn denote the nth partial sum. Clearly Sn = 1 - 1/10ⁿ
Then for every ε > 0 we have an N = log10 (1/ε) such that:
|Sn - 1| = |1/10ⁿ| < ε for n=∞ (all n > N)
Therefore 0.999... = 1

Let S(∞) denote the ∞th partial sum. Clearly S(∞) = 1 - 1/10^∞
Then for every ε > 0 we have an N = log10 (1/ε) such that:
|S(∞) - 1| = |1/10^∞| < ε for n=∞ (all n > N)
Therefore 0.999... = 1

Let S(∞) denote the ∞th partial sum. Clearly S(∞) = 1 - 1/10^∞
Then for ε=inf > 0 we have an Z = log10 (1/ε) also 10^Z = 1/inf such that:
|S(∞) - 1| = |1/10^∞| < ε for n=∞ (all n > Z)
Therefore 0.999... = 1

5. ## Re: Is the infinyty of set {1.2.3...} equal to the infinity of set {1.3.5...} ?

Originally Posted by grapes
Seems pretty obvious that n --> 2n-1 is an exact pairing of the first set to the second set, every member of the first set is paired with exactly one member of the second set, no duplications, no elements left out.
Um, no. Bijective cardinality does not mean every member of a set is paired with exactly one member of another set, because not all the "members" are numbers.

So it's not only false, but stupid to say that no elements are left out. After all, most of those mythical objects called "irrational numbers" are not elements of any set. Chuckle.

6. ## Re: Is the infinyty of set {1.2.3...} equal to the infinity of set {1.3.5...} ?

A set is a well defined collection of distinct objects.

Since infinity is undefined 123...and 246.. are not sets.

7. ## Re: Is the infinyty of set {1.2.3...} equal to the infinity of set {1.3.5...} ?

Originally Posted by astrotech
A set is a well defined collection of distinct objects.
Since infinity is undefined 123...and 246.. are not sets.
Huh...

The set of all natural numbers {1,2,3...} is so well defined and important
that it has its own symbol N. as has other similar sets. It can not be more
well defined then that.

What is at question here and in the .999... thread is how to interpret infiniinty and infinitesimality.
If the set of natural number is finite or not. If line has a finite number of points or not. The answer
to this is not crystal clear and both viewpoints exist. It has however been proven (I can not rember
who did this) that it is not important for other mathematic in general. But it is an intersting subject.

Both viewpoints are attractive for different reasons.

Cheers
L-zr

8. ## Re: Is the infinyty of set {1.2.3...} equal to the infinity of set {1.3.5...} ?

Originally Posted by Lazer
If the set of natural number is finite or not.
Thanks! That makes things easy.

9. ## Re: Is the infinyty of set {1.2.3...} equal to the infinity of set {1.3.5...} ?

Originally Posted by Lazer
Huh...

The set of all natural numbers {1,2,3...} is so well defined and important
that it has its own symbol N. as has other similar sets. It can not be more
well defined then that.

What is at question here and in the .999... thread is how to interpret infiniinty and infinitesimality.
If the set of natural number is finite or not. If line has a finite number of points or not. The answer
to this is not crystal clear and both viewpoints exist. It has however been proven (I can not rember
who did this) that it is not important for other mathematic in general. But it is an intersting subject.

Both viewpoints are attractive for different reasons.

Cheers
L-zr
But if there are infinite many natural numbers then all natural numbers cannot be defined. So I think the point Astrotech was making is that a "set" is complete, all accounted for. The symbol N is used to describe a "set" that cannot be all accounted for, which really makes no sense.

I liked your post though by the way.

10. ## Re: Is the infinyty of set {1.2.3...} equal to the infinity of set {1.3.5...} ?

Originally Posted by grapes
Thanks! That makes things easy.

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