Is Infinity Equal to Infinity According to Wolfram Alpha and Friends?

[Alt_Nim]

The program Wolfram Alpha say "True" but my friends want the Proof.
Could you help me please?

 

[mathman]

What is the "Probation"? In any case, there are many different "infinities".

 

[Alt_Nim]

I'm so sorry I mean " proof "

 

[Alt_Nim]

http://www.wolframalpha.com/input/?i=infinity+=infinity

 

[Alt_Nim]

many different "infinities".

So we can't find the answer for this question right?

 

[Hurkyl]

The answer is either "obviously yes", "obviously no", "that's obviously nonsense" depending on what you mean by the word "infinity".


Normally, mathematicians aren't so obfuscating that they would opt to use the same word in two different ways in such a short time. And so the answer would be "obviously yes" in pretty much any instance it would come up.


In your instance, I think Mathematica uses \infty to mean the positive infinite number in the extended real numbers.

 

[Alt_Nim]

This is my proof ..please tell me if I wrong
We know that
inf+1 = inf-----(1)
(inf+1) +1 = inf+1
from (1) replace the left (inf+1) with inf
inf+1 = inf+1
and
inf +1 - 1 = inf +1 -1
inf = inf ###

 

[Alt_Nim]

my friend said there are not equal because
he think about Aleph numbers in the set theory
what do you think about it?

 

[Alt_Nim]

#6 Intelligent!
nice to meet you .^^

 

[DrRocket]

my friend said there are not equal because
he think about Aleph numbers in the set theory
what do you think about it?

There is a theory of cardinal number and a theory of ordinal numbers as well.

 

[Dragonfall]

Yes, infinity = infinity. This follows from the logical definition of equality. Whatever you mean by "infinity" it is equal to itself.

 

[Hurkyl]

(moderator's note: I've removed some posts that have derailed the thread from the original question)

There is a theory of cardinal number and a theory of ordinal numbers as well.

For the opening poster's sake, I want to point out that while there are many infinite cardinal and ordinal numbers, none of them are called "infinity". The occasional use of the noun "infinity" in that context is a historical artifact of the time before people realized those number systems had more than one infinite element -- and I don't recall ever hearing a mathematician use the word "infinity" in that way.

 

[DR13]

Depends on what infinities we are talking about. For instance, let us define group A as the amount of real numbers divisible by 2. Obviously, the size of group A is infinity. Now let us define group B as the amount of real numbers divisible by 10. Again, the size of group B is clearly infinity. However, our own logic tells us that there are five numbers divisible by 2 for every one number divisible by 10. Therefore, these infinities are not equal. Another instance of this would be have group A be the amount of real numbers between 1 and 2 and have group B be the number of real numbers between 1 and 3. Just another situation with different infinities.

 

[Dragonfall]

Depends on what infinities we are talking about. For instance, let us define group A as the amount of real numbers divisible by 2. Obviously, the size of group A is infinity. Now let us define group B as the amount of real numbers divisible by 10. Again, the size of group B is clearly infinity. However, our own logic tells us that there are five numbers divisible by 2 for every one number divisible by 10. Therefore, these infinities are not equal. Another instance of this would be have group A be the amount of real numbers between 1 and 2 and have group B be the number of real numbers between 1 and 3. Just another situation with different infinities.

You'd have to define divisibility for real numbers first.

 

[DR13]

sorry, i meant integers

 

[Hurkyl]

Therefore, these infinities are not equal.

What do you mean by size? Normally in that setting it's used to refer to cardinality -- and the two infinite numbers you defined really are equal.

 

[Dragonfall]

sorry, i meant integers

For integers those sets are actually equal in size.

Regardless, in set theory you don't say things like size "infinity", you would say the specific cardinal. Like 2^{\aleph_0}.

 

[DR13]

What do you mean by size? Normally in that setting it's used to refer to cardinality -- and the two infinite numbers you defined really are equal.

I am just referring to how we would logially interpret it. Obviosly the gap between 1 and 3 is twice the size of the gap between 1 and 2

 

[Hurkyl]

I am just referring to how we would logially interpret it. Obviosly the gap between 1 and 3 is twice the size of the gap between 1 and 2

You don't mean "logically interpret" you mean "intuit".

The problem is that you are intuiting something that is not cardinality -- but then demanding that cardinality adhere to that intuition.

In the latter case -- the size of the intervals [1,2] and [1,3] -- your intuition is that you want to measure these subset of the line with the notion of "length". Length has very little to do with cardinality -- the only relationship between the two ideas is that only sets of cardinality c can have positive length.

(where c is the cardinality of the set of real numbers)

(I may be assuming the continuum hypothesis)

 

[DR13]

To be honest, this conversation is starting to go over my head. I was just trying to make a point (apparently I failed but o well). Hopefully you can kinda see what I was trying to say.

 

[Alt_Nim]

Hurkyl you are genuis .!
^^