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October 17th, 2014, 03:54 AM   #21
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Originally Posted by topsquark View Post
Well...Newton was one of the ones that first developed Calculus and applied it to central force problems. In these problems he did use the concept of infinity. I don't know if Leibnitz did or not.

-Dan
There was someone who introduced money in India and he did bring the concept of 25 paise. But is it still used today ?
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October 17th, 2014, 04:45 AM   #22
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This concept of 'a completed infinity' is very unclear to me. As CRG is fond of pointing out, there are relatively few areas of mathematics in which it is correct to use the word 'infinity'. Either we use the concept to mean 'unbounded' or the area has a moe refined few and 'infinity' iss too general a term to have meaning in the. context.

You talk about the real line going from $-\infty$ to $+\infty$, but neither of them are points on the line. (They are on the extended real line, but that's different). In mainstream calculus $\pm\infty$ are notational devices depicting the unbounded growth of a quantity.

Set theory does have infinite sets, but the word infinity is meaningless in that context. You must talk about more refined concepts to make any sense. Any argument that infinite sets don't exist seems flawed, because clearly the natural numbers exist, and all of them have shared properties. The same goes for the rationals and the algebraic numbers. So we can't claim that these sets don't exist because we can define what these numbers are, and given any number we can tell what type it is.
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October 17th, 2014, 04:55 AM   #23
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Originally Posted by Karma Peny View Post
I am throwing it out on the basis that it has no sound logic behind it (as explained on my website), not on the basis that it is tricky or difficult.
No. Just because you don't understand the logic doesn't mean that it isn't sound. It means that you don't understand it.

This isn't a problem, there are plenty of areas of mathematics you can study without needing a concept of the infinite, but you can't make everybody else discard it just because you don't follow the reasoning.

The main problem is that your argument is not mathematical, it is philosophical. And therefore your thinking is more akin to religion than to mathematics. The Catholic church spent many centuries trying to exclude the infinite (and zero) from mathematics because of a perceived incompatibility with their beliefs. But they failed because logical thought makes the concept necessary in many branches of scientific endeavour. Once the concept is found to be necessary, it can be said to exist and can therefore be studied as Cantor did, rigorously.

You therefore must argue against the mathematical rigour of proven results. This is by definition rather difficult to do because reason is not on your side. It has already been deployed against you.
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October 17th, 2014, 05:38 AM   #24
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For the mathematics I have encountered, infinity appears to add mysticism and it creates problems that do not exist without it. I have always found it less than useful.
Read better math. The purpose of mathematical exposition is to simplify concepts, not add mysticism.

I've only ever seen this in writings by non-mathematicians, personally.

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There are many processes that appear to use infinity, such as calculus, but the word is often used because people think of infinity as being an unimaginably large number. Calculus was not devised using infinity
Calculus was most assuredly developed using infinite and infinitesimal numbers. It was not until Cauchy, I believe, that a rigorous version of calculus avoiding both was developed. (And then Robinson gave a rigorous version of calculus using infinite and infinitesimal numbers in the 60s, taking it full circle.)

As taught today calculus (following Cauchy) does not use infinite or infinitesimal numbers anywhere, so you don't need to worry about it. It does use the symbol $\infty$ but this does not refer to a number (within calculus), it's a pure formalism.

If you want to actually use infinite numbers you need to go beyond basic calculus.

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the proof of the fundamental theorem does not require infinity, the use of differential and integral calculus does not require infinity.
All thanks to Cauchy, yes. The original proofs, such as they were, did use fluxions and other sorts of infinite/infinitesimal numbers.

At the risk of repeating myself: basic calculus does not use infinite or infinitesimal numbers at all. The first exposure to infinite numbers in that field would probably be complex analysis with the Riemann sphere. (But it's arguable whether that's infinite in any meaningful sense anyway -- it's just the top point of a sphere, right?)

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Gauss’s views on the subject can be paraphrased as: 'Infinity is nothing more than a figure of speech which helps us to talk about limits. The notion of a completed infinity doesn’t belong in mathematics'.
This point of view was typical of pre-Cantorian mathematicians. They didn't have any rigorous and useful way to deal with infinite sets and numbers.

If you'd like to be stuck in the 18th century you could adopt his view on the matter. If you want to be a finitist in the modern world -- and that's possible! -- you'll need better reasons.

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Imaginary numbers make perfect sense and provide a good example for comparison.
I thought you wanted to reject real numbers, why would you accept pairs of real numbers?

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It soon becomes apparent that we can form a logically consistent set of mathematical rules that include the square root of -1.
It might clarify your thinking to make this reasoning explicit. How did it become apparent that this could be done consistently?

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In a similar fashion, it is perfectly acceptable to assume a completed collection of an endless sequence can exist. But in this case it is not possible to form a logically consistent set of rules.
Again, you need to clarify your reasons here. Are you saying you can derive a contradiction from ZF?
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October 17th, 2014, 06:18 AM   #25
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I don't understand why so many people argue about the infinite, infinity, endlessness, limits, etc... It's been studied to death already in the literature, so read it and learn from it.

Limits apply to functions, so $\displaystyle \lim(0.\dot{3})$ makes no sense to me. $\displaystyle 0.\dot{3}$ is just a constant. The whole point of a limit is to investigate what value a function converges to as the dependent parameters vary to some limit. You might not like the idea that a limit can be infinity, but this is not a problem in mathematics.

For example, you can define a function, in the domain of positive x, such as $\displaystyle f(x) = \frac{1}{x}$ and then say $\displaystyle \lim_{x\rightarrow 0} f(x) = \infty$ or $\displaystyle \lim_{x\rightarrow \infty} f(x) = 0$. As your number gets closer and closer to 0, f(x) gets larger and larger without bound, or as your number gets larger and larger, f(x) converges to zero. Mathematics deals with any attempted evaluation of the function at zero by saying that the function is "undefined" at $\displaystyle x=0$. A proper definition of the the above function is in fact $\displaystyle f(x) = \frac{1}{x}, x \in \mathbb{R}^+, x \neq 0$. After all, division of a finite number must occur with another finite number (at least in the most common number systems, like the naturals or the reals). I don't see what's so problematic about understanding this and what is unsatisfactory about it. The statements are clear, the concept is clear and anything worked out in a real physical situation is compatible with this. Is infinity required to describe reality? Possibly not. Is infinity (and other related concepts such as cardinality) required to describe abstract mathematics? Yes, depending on what field you're studying.

Also, the set of real numbers contains irrational numbers, but this doesn't mean those numbers aren't exact, they just cannot be represented exactly using a decimal number system and are instead specified exactly using algebraic letters or surds. The decimal number system can represent rational numbers using the recurring notation, for example

$\displaystyle 0.\dot{3} = \frac{1}{3}$
$\displaystyle 0.\dot{0}7692\dot{3} = \frac{1}{13}$

There is a quirk where a number can have two decimal representations, which needs taking into account in very few circumstances (such as Cantor's proof):
$\displaystyle 0.\dot{9} = 1$
but again, this is just a quirk with the decimal number system; it is a number system where some numbers have more than one representation.

The above case for decimal numbers is true based on the definitions of recurring numbers. Because it is true as a matter of definition, it cannot be disproved, just changed or redefined depending on applications.
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October 17th, 2014, 06:22 AM   #26
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Originally Posted by Karma Peny View Post
I am not claiming there is a restriction of any kind. We can make x as large as we like, but we cannot make it non-finite.
If you think this is different from $$\lim_{x\to\infty}$$ then you misunderstand that statement.

A similar statement in set theory:
$$
\forall\nu\in\omega,|\nu|<\infty
$$
(where the symbols "$<\infty$" mean "is finite").
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October 17th, 2014, 06:27 AM   #27
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The real number line supposedly stretches from -infinity to +infinity and any section of it, however small, contains an infinite number of numbers. As you might have guessed, I reject the whole basis of a continuum as it is based on the concept of infinity, which is nonsensical.
"any section of [the real numbers], however small, contains an infinite number of numbers".

Does this bother you? Does that mean you reject the rational numbers, too?
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October 17th, 2014, 07:25 AM   #28
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Originally Posted by Benit13 View Post
decimal number system
It's important to draw a distinction between a number system, such as the Real Numbers, the Rational Numbers, etc.. and a numeral system which is merely a representation of numbers that exist within a number system. Numeral systems include the decimal system.

I realise that you were trying to do just that.

You are correct in saying that limits apply to functions, but we can consider the decimal system to be a function that maps a sequence of numbers $\cdots a_3 a_2 a_1 a_0 . b_1 b_2 b_3 \cdots$ to the function $$f(x)=\sum_{i=0}^\infty a_i {10}^i + \sum_{j=1}^\infty \frac{b_j}{10^j}$$
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October 17th, 2014, 07:28 AM   #29
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Originally Posted by v8archie View Post
we can consider the decimal system to be a function that maps a sequence of numbers $\cdots a_3 a_2 a_1 a_0 . b_1 b_2 b_3 \cdots$ to the function $$f(x)=\sum_{i=0}^\infty a_i {10}^i + \sum_{j=1}^\infty \frac{b_j}{10^j}$$
Note that the sequence a_i must be finite (i.e., there is some N such that a_n = 0 for all n > N).
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October 17th, 2014, 07:36 AM   #30
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Yes. I was considering your bracketed case.
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