In real numbers, you can't have infinitely small. It's a very simple proof by contradiction that there is no smallest real number.
Assume that x is the smallest (positive) real number. However, properties of real numbers dictate that if x is real, then x/2 is also real (and positive). Obviously x/2 < x if x is positive, a contradiction. (We can change x/2 to 2x to prove that there is no largest real number, neither most positive nor most negative).
This is what I meant by infinite divisibility and accuracy.
@Exclamation Mark That's contradictory. If it's exactly equal in theory, then it's definitely "equal" in real world. If we treat 1.01 as "basically" 1 and 3.14 as basically pi, then why shouldn't 1 = 1?
And calculators are designed to be finite. You can't always rely on them to do proofs.
@Swish
Math is not bound by real world. A natural example showing the concept of infinity is the set of integers. No matter how far you count, you're going to have more integers, making the set infinite.
Now using Trivialis handle.Well yeah, you can't have an infinite set with smaller cardinality than N.
It makes more sense when you throw out the "numbers" idea and treat the elements as just objects.
Which expression?
edited 10th Sep '11 9:02:15 AM by abstractematics
Now using Trivialis handle.I don't know. This is almost as bad as that ill-formed arithmetic expression that went memetic a few weeks ago.
By the way. I think this thread has lead me to a greater understanding of why Cantor went in and out of mental hospitals.
[1] This facsimile operated in part by synAC."Infinitely small" is meaningless in the real numbers. (And expressions like 0.000...01 are nonsensical. You can't have an infinite sequence of digits with 1 at the "end". It'd be like a 4-sided triangle: a contradiction in terms.) If you want to work with weird concepts like that, you have to work with a system that has infinitesimals, such as the hyperreals
or another nonstandard number system. At that point, you're not really working with "numbers" in the usual sense.
On the topic of infinity, as I've said before, mathematicians work with things that are actually infinite all the time. Sure, that means you can't represent it physically in certain ways, but you don't need to. If it bothers you that we work with the infinite, too bad; it's quite useful to do so. (The real fun starts when we use infinite sets that are larger than the usual infinity
.)
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Technically, if you don't use the axiom of choice, you can (probably) consistently assert that certain uncountable sets have subcountable cardinality
. But if you accept the axiom of choice, you can prove that such subcountable sets don't exist.
edited 10th Sep '11 9:41:40 AM by Enthryn
@Yej
"No, "infinitely small" is not zero, just as close as can be."
The problem here is that "as close as can be" to zero is infinitely close, which, accepting Raven's reasoning, is zero. The statement that "the number that is infinitely small is as close as can be to zero" is true irrespective of what you believe, and is therefore not contradictory to Raven's statement. What you need to do is independently establish why the statement "the number as close as can be to zero is zero" is false.
And I don't think you can.
Because it pushes human ability to percieve the world with instincts and "gut feelings" over the limit? Things just get so strange that even with proofs people still just refuse to believe it...or just incapable of imagine it...
edited 10th Sep '11 10:42:39 AM by onyhow
Give me cute or give me...something?You can construct ε as ω-1, but I think you can also construct arbitrary powers of ε.
[1] This facsimile operated in part by synAC.Two numbers are considered distinct if and only if a number between them exists.
No number between .999 repeating and one exists. Therefore, they are not distinct numbers.
Infinite Tree: an experimental storyIt was something like 1+1+1+1+1+1+1*0=?
remember the order of operators, people!
Very big Daydream Believer. "That's not knowledge, that's a crapshoot!" -Al Murray "Welcome to QI" -Stephen Fryits ludicrously easy things like that. No clue how many 1s I put there, but I have seen that on facebook, and a significant amount of people put 0.
>Headdesk<
And the 2=1 thing makes errors because it divides by 0.
edited 10th Sep '11 4:59:53 PM by Enkufka
Very big Daydream Believer. "That's not knowledge, that's a crapshoot!" -Al Murray "Welcome to QI" -Stephen FryThere are a few others "proofs" that 2 = 1. One trick is pretending that the complex exponential is injective (which it isn't), and concluding that a^b = a^c (where a > 0 and a ≠ 1) implies b = c. This works for real-valued b and c because the real exponential function actually is injective, but it doesn't work for complex numbers.
edited 10th Sep '11 5:06:22 PM by Enthryn

I'm gonna repeat myself:
Since the nines go on forever, the difference between .999... and 1 is infinitely small. And if "infinitely large" encompasses everying, than "infinitely small" encompasses nothing. So "nothing" is the difference between those two numbers.