Are there an infinite amount of infinities?

[Random Variable]

There are an infinite number of real #'s between 0 and 1. There is also an infinite number of real #'s between 0 and 1,000,000. Is the infinity in the second case of greater magnitude than the infinity in the first case?

 

[mobobuff]

Nope. Infinity = Infinity.

Don't think of it as a number so much as it is a concept.

 

[DaveSimmons]

No, but IIRC the infinite set of reals is larger than the infinite set of rational numbers.

 

[her209]

Aren't real #'s the set of positive integers?

 

[Goosemaster]

Originally posted by: mobobuff


Don't think of it as a number so much as it is a concept.

Aye..think of infinity as an irrational number such as pi....which is an approximation of an approximation..etc...

 

[jagec]

*sigh*

infinity is not a number.

 

[Goosemaster]

Originally posted by: her209
Aren't real #'s the set of positive integers?

from mathworld

Number Theory > Real Numbers v

Real Number


The field of all rational and irrational numbers is called the real numbers

 

[Goosemaster]

Originally posted by: jagec
*sigh*

infinity is not a number.

Don;t worry he'll get it

 

[DaveSimmons]

Originally posted by: her209
Aren't real #'s the set of positive integers?

Nope, those are the Natural numbers (N). There are also the integers (Z), Rational numbers (Q) m/n where n,m are elements of Z and n != 0, and finally the Real numbers (R) that include all of the above plus the "in between" numbers.

 

[TheLonelyPhoenix]

*Comes in holding cross before himself*

I declare everything you believe about infinity to be wrong!!!

*Leaves with flowers springing up at his heels*

 

[ArmenK]

Yes, there are an infinite number of infinities. See Cantor's work. Text

 

[Random Variable]

Originally posted by: Goosemaster

Originally posted by: mobobuff


Don't think of it as a number so much as it is a concept.

Aye..think of infinity as an irrational number such as pi....which is an approximation of an approximation..etc...

An irrational number is not an approximation, even though it goes on forever.

 

[element]

Originally posted by: DaveSimmons
No, but IIRC the infinite set of reals is larger than the infinite set of rational numbers.

Is there a proof for this?

 

[KevinF]

The two infinites you list have the same cardinality (basically the same order of infinite-ness), aleph-1. The set of integers or natural numbers or even numbers or odd numbers and believe it or not, rational numbers, are all aleph-0. You can then start adding higher orders of infinite by getting a power set, basically calling something the set of all possible sets of real numbers, and get aleph-2 and beyond.

Cantor is the man to read.

I'd recommending exploring Wikipedia for more info.

 

[DaFinn]

huh?

 

[KevinF]

Originally posted by: element

Originally posted by: DaveSimmons
No, but IIRC the infinite set of reals is larger than the infinite set of rational numbers.

Is there a proof for this?

Yes (also a good read for more info about infinite). The real question, is whether there is anything between the reals and integers/rationals.

 

[Random Variable]

So sets that are countably infinite (e.x., 1,2,3...) are of one type of infinity, and sets that are uncountably infinite (e.g. (0,1)) are of a different type of infinity?

 

[Goosemaster]

Originally posted by: Vespasian

Originally posted by: Goosemaster

Originally posted by: mobobuff


Don't think of it as a number so much as it is a concept.

Aye..think of infinity as an irrational number such as pi....which is an approximation of an approximation..etc...

An irrational number is not an approximation, even though it goes on forever.

my bad...I mean to say that we use approximations to understand infinite quanitties, or in the case of pi, infinite length (exactness) such as we use approximations for a circular diameter that is infinite inspecific length according to what we usualyl to to find a diameter


I was tryign to get him to at least thing of infinity as an irrational idea because it can't be quantified jsut to start him off becausesometimes if you say that inifinity is a concept outright it jsut confuses them....

 

[Goosemaster]

Originally posted by: Vespasian
So sets that are countably infinite (e.x., 1,2,3...) are of one type of infinity, and sets that are uncountably infinite (e.g. (0,1)) are of a different type of infinity?

think of infinity as a concept only then...such as color...

is the blue of your sweater different than the blue of my car?

infinity just mean continuous and exact in those cases, respectively.

 

[Random Variable]

Originally posted by: Goosemaster

Originally posted by: Vespasian
So sets that are countably infinite (e.x., 1,2,3...) are of one type of infinity, and sets that are uncountably infinite (e.g. (0,1)) are of a different type of infinity?

think of infinity as a concept only then...such as color...

is the blue of your sweater different than the blue of my car?

infinity just mean continuous and exact in those cases, respectively.

Outside of set theory, you're correct. But set theory is like a world in itself.