A Paradox?

[{LAW} Gamer_2k4]

So we were discussing sets and recursion in math class today, and my mind started going.  I came up with this:


Fact: If a number is finite, its value plus one is also finite.  (Ex. 6 (finite) + 1 = 7 (finite))
Fact: A set of integers starting a 1 and increasing continuously to an arbitrary integer n will always have a size equal to n.  (The set {1, 2, 3} has a size of 3, and so on.)

Claim: There is no limit to the amount of times a finite number can be increased by one.  Therefore, a set containing all finite integers {0, 1, 2, ... , n} has an infinite size.

Paradox: Since the size of a set is equal to the magnitude of the largest number in that set, the largest number must be equal to infinity.  However, the largest number is finite, as it is simply the previous number plus one (which is equal to the number before that plus one, and so on).  To put it another way, there must be an infinite number of finite numbers (which seems logically impossible).


Does anyone want to tell me where the flaw is? I know I'm doing something wrong somewhere, but I can't put my finger on it.

[Kagesha]

This looks like a job for! Kazuki Man!

[Wraithlike]

Unless I'm mistaken, the problem is in the way you're thinking about. Infinity is just used to show that a set of numbers doesn't stop, and it doesn;t represent a number itself. Thats why in set notation, you'll always use a parenthesis for infinity instead of a bracket ex: (-∞, ∞); [0, ∞); (-∞, 1].

If you haven't learned set notation, a parenthesis essentially means > or <, instead of ≤ or ≥.

Edit: It's probably easire to think of in terms of a graph. Assume you're graphing the function f(x)=csc(x)



The vertical blue lines aren't part of the graph, they're asymptotes, which the seperate parts of the graph converge on, but never reach. Essentially, infinity is the amount x would have to be to touch the vertical asymptote, but beccause a function can't be vertical, x can never reach infinity.

That was badly explained, I know, but hopefully it helps/is correct.

[bja888]

Yes: ∞ + ∞ = ∞
No: ∞ + ∞ = 2∞

∞ is not a number, standard calculations don't apply.

[{LAW} Gamer_2k4]

I think I've found the solution; at any rate, I've found a way to rewrite the problem so as to simplify it:

Assuming an infinite amount of finite numbers (which is hard to refute), what's the value of the largest number?

The answer, of course, is that there is no largest number.  The nature of the set of finite numbers (defined recursively as F(n - 1) + 1) means that no matter how large a number gets, it will remain finite.  Since the sum of any finite numbers is always finite, one will never reach infinity, even if the set is extended indefinitely.

[Mangled*]

The way to look at it is that infinity is the number of numbers, but the numbers themselves are of finite value.

There are infinite finite numbers.

Infinity doesn't even come into it anyway, it's totally irrelevant.

[VijchtiDoodah]

Your flaw is that you're looking for the largest number in an infinite set -- the idea of infinity is that there is no limit and infinity itself is a concept, not a number.

However, bja888 is wrong when he says that standard calculations do not apply to infinite.  Most standard calculations simply don't make sense, but I could, for example, find that ∞²/∞ = ∞ whereas ∞/∞² = 0.  In some instances, there are actually different magnitudes of infinite and calculations can be made to tease certain results out of equations.

[a-4-year-old]

you are allowed to square infinity?

[excruciator]

how can you claim that the largest number is both infinite and finite. That doesn't make sense.
I don't think there was a paradox from the very beginning. Since the largest number would go on indefinitely, it is considered to be ∞. and that # is not finite.

[Espadon]

As countless [haha?] people have said, infinity is not a number. So therefore it's not the largest number...since it represents infinity, which...doesn't have a "largest" cap, you see?

[excruciator]

True, ∞ is not even a number.

[bja888]

However, bja888 is wrong when he says that standard calculations do not apply to infinite.  Most standard calculations simply don't make sense, but I could, for example, find that ∞²/∞ = ∞ whereas ∞/∞² = 0.  In some instances, there are actually different magnitudes of infinite and calculations can be made to tease certain results out of equations.

This is true. Nothing in math is questionable, it is all rules and facts. However, I'm sure that in the end 2∞ has the same effect as 1∞.

[{LAW} Gamer_2k4]

...whereas ∞/∞² = 0.

I'm almost positive that's incorrect.  Infinity doesn't cancel out in fractions; if it did, there would be no need for L'Hôpital's rule.  Besides, infinity multiplied by anything, finite or otherwise, results in infinity.  Therefore, ∞/∞² simplifies to ∞/∞, which is indeterminate.

[Smegma]

...whereas ∞/∞² = 0.

I'm almost positive that's incorrect.  Infinity doesn't cancel out in fractions; if it did, there would be no need for L'Hôpital's rule.  Besides, infinity multiplied by anything, finite or otherwise, results in infinity.  Therefore, ∞/∞² simplifies to ∞/∞, which is indeterminate.

Well, he's just using the incorrect notation I believe. In fact, you two are on the same page, as you've pretty much stated the method he used to solve it.

[{LAW} Gamer_2k4]

...whereas ∞/∞² = 0.

I'm almost positive that's incorrect.  Infinity doesn't cancel out in fractions; if it did, there would be no need for L'Hôpital's rule.  Besides, infinity multiplied by anything, finite or otherwise, results in infinity.  Therefore, ∞/∞² simplifies to ∞/∞, which is indeterminate.

Well, he's just using the incorrect notation I believe. In fact, you two are on the same page, as you've pretty much stated the method he used to solve it.

If I understand him correctly, he's saying that ∞/∞² reduces to 1/∞, which would be zero.  I'm saying it reduces instead to ∞/∞, which is a much different page.

[Smegma]

I think he meant to write it a bit more like

lim_(x->∞) x/x²

I could be wrong, but I just assumed her forgot to phrase it correctly. Yes, they are two different things, and make a huge difference. So, I'll just wait for him to post.

If he did, however...it ruins the initial point of his post. At least, it may have been the path of thinking he followed to arrive to his statement.

[{LAW} Gamer_2k4]

I think he meant to write it a bit more like

lim_(x->∞) x/x²

Oh, right...because you end up getting 1 / 2x, which does become zero as x goes to infinity.

[Smegma]

I think he meant to write it a bit more like

lim_(x->∞) x/x²

Oh, right...because you end up getting 1 / 2x, which does become zero as x goes to infinity.

Still, that's quite different than ∞/∞²

[VijchtiDoodah]

∞/∞² = 0 is a shorthand way of writing lim_x->∞ x/x².  If you want to be technical, this is an improper way of writing a limit and, as Smegma pointed out, attempting to use an equation with a limit ruins the initial point of my post.  Looking back at what bja888 said, though, I realize that I just misunderstood him (in which case, even the shorthand ∞/∞² = 0 and the alternative a/∞ = 0, which doesn't strictly require a limit, agree with what he was saying).  Standard calculations do not apply.

[PapaSurf]

So in conclusion, there ARE, in fact, an infinite quantity of finite integers.

Essentially, it's saying that there is no such thing as a the largest integer, which when put that way, makes the whole thing a damn lot easier, eh?