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

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In summary, there is a program called Wolfram Alpha that claims "True" but the user's friends want proof. They discuss the concept of "infinity" and whether or not it is equal to itself. The conversation also touches on the theories of cardinal and ordinal numbers. Ultimately, the user's question is answered and they are pleased to meet the respondents.
  • #1
Alt_Nim
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The program Wolfram Alpha say "True" but my friends want the Proof.
Could you help me please?
 
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  • #2
What is the "Probation"? In any case, there are many different "infinities".
 
  • #3
I'm so sorry I mean " proof "
 
  • #5
many different "infinities".

So we can't find the answer for this question right?
 
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  • #6
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 [itex]\infty[/itex] to mean the positive infinite number in the extended real numbers.
 
  • #7
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 ###
 
  • #8
my friend said there are not equal because
he think about Aleph numbers in the set theory
what do you think about it?
 
  • #9
#6 Intelligent!
nice to meet you .^^
 
  • #10
Alt_Nim said:
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.
 
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  • #11
Yes, infinity = infinity. This follows from the logical definition of equality. Whatever you mean by "infinity" it is equal to itself.
 
  • #12
(moderator's note: I've removed some posts that have derailed the thread from the original question)

DrRocket said:
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.
 
  • #13
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.
 
  • #14
DR13 said:
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.
 
  • #15
sorry, i meant integers
 
  • #16
DR13 said:
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.
 
  • #17
DR13 said:
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 [itex]2^{\aleph_0}[/itex].
 
  • #18
Hurkyl said:
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
 
  • #19
DR13 said:
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)
 
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  • #20
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.
 
  • #21
Hurkyl you are genuis .!
^^
 

1. Is infinity equal to infinity?

Yes, infinity is equal to infinity. This is because infinity is not a specific number, but rather a concept that represents something without an upper limit.

2. Does infinity have a numerical value?

No, infinity does not have a numerical value. It is a concept that represents something without an upper limit, and cannot be expressed as a specific number.

3. Can infinity be measured or calculated?

No, infinity cannot be measured or calculated in the traditional sense. It is a concept that represents something without an upper limit, and cannot be expressed as a specific number.

4. Can infinity be divided or multiplied?

No, infinity cannot be divided or multiplied in the traditional sense. It is a concept that represents something without an upper limit, and cannot be expressed as a specific number. However, certain mathematical operations involving infinity can be defined and used in certain contexts.

5. Is infinity a real number?

No, infinity is not a real number. It is a concept that represents something without an upper limit, and cannot be expressed as a specific number. In mathematics, infinity is often treated as an "ideal" concept rather than a real number.

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