New answers tagged

1

Hmm, after trying a squinty-eyes interpretation of the Dutch, it looks like the question is asking for both the maximal and minimal value that $\frac{a+b}{ab}$ can take for any $a\in[0.15,\infty)$ when $b=0.017$. In other words you're interested in the behavior of $$ \frac{a+0.017}{a\cdot 0.017}$$ for large $a$. We can rewrite this as $$ ...


0

In modern mathematics we have several ways of formalizing infinity. The one that is most relevant to your question was provided by Abraham Robinson; see here. Following his framework, there are both infinitesimals and infinite numbers. Thus if $H$ denotes an infinite number, one can indeed divide a unit interval into $H$ parts. Each part would have ...


0

Calculate the probability that you lose every time, using an infinite product. For example, for starting at $1/2$ probability and cutting in half every time, the probability that you have not won after playing $n$ times is $$\prod_{i=1}^{n} \left(\frac{2^{i}-1}{2^{i}}\right) = \prod_{i=1}^{n}(1-2^{-i}).$$ I'm not motivated work out the exact formula, but ...


0

On the first flip, you have a $\frac12$ chance of winning. Assuming you lost (with probability $\frac12$) you win with probability $\frac14$, which gives you an additional $\frac18$ probability of winning, for a total of $\frac58$. Assuming you lost again (probability $\frac38$) you get another shot at probability $\frac18$, for an additional $\frac3{64}$, ...


0

$f:(\color\red{-1},\color\green{1})\rightarrow(\color\orange{0},\color\purple{4})$ $f(x)=(x-(\color\red{-1}))\cdot\frac{\color\purple{4}-\color\orange{0}}{\color\green{1}-(\color\red{-1})}+\color\orange{0}=2x+2$ $g:(\color\orange{0},\color\purple{4})\rightarrow(\color\red{-1},\color\green{1})$ ...


1

From the things you say it seems clear that when you talk about splitting $AB$ into infinitely many parts, you mean infinitely many parts of equal length. You simply can't do that. You seem to realize what the problem is, but you're not drawing the right conclusion: The problem is if the parts have positive length that would mean $AB$ had infinite length, ...


1

Length is formalized by the mathematical concept of Lebesgue measure, which assumes real number values. A subset of the real number line cannot have infinitessimal measure, it is either zero or positive. Some subsets are nonmeasurable though. Also, there is no "value that comes immediately after zero." One is free to create their own partially ordered sets, ...


2

The conversation seems almost equivalent to this hypothetical one: ME: Suppose there are positive integers $m$, $n$ such that $m^2=2n^2$. Then we can generate a list of all prime factors and read off the exponent of $2$. This exponent should be even (because it is in the unique factorization of $m^2$) but also odd (because it is in the unique factorization ...


-5

The key here is to let it go. If you let it bother you then you yourself risk becoming the "pro-Cantor crank". Brouwer, Poincare and Wittgenstein were all opponents of Cantor's work and many respectable, published, logicians have disputed Cantor's diagonal argument. Since it's a proof by contradiction it relies heavily upon the law of the excluded middle, ...


5

One natural use for smallish infinite ordinals, which makes clear that different infinite ordinals have distinct properties, is the length of games. I'm playing a chess-like (turn-taking, deterministic, complete information) board game of some kind against an opponent. My teammate is also playing a separate game against my opponent's teammate. Moves are ...


0

Summing over an uncountable index results in an infinite sum if uncountably many terms are non-zero. To see this, we prove the contrapositive: that if a sum over an uncountable index is finite implies that at most countably many terms are non-zero. Proof. Let $\sum_{\alpha \in A} x_\alpha = L$. Let $S_n = \{\alpha \in A \mid x_\alpha > 1/n\}$. Then $$L ...


1

Here's an intuitive way to think of it. Consider a more general case:$$y=\sin(\frac{1}{x})$$ When $x$ gets closer and closer to $0$, there will be an infinite amount of times it crosses the x-axis. This is because $\frac{1}{x}$ approaches $-\infty$ from the left and $\infty$ from the right. When $\frac{1}{x}$ gets very close to the y-axis it will grow ...


1

Since $\sin x$ is zero at $\pi,2\pi,\ldots$, $(\sin x)(\sin x^{-2}) $ is zero at $\dfrac{1}{\sqrt \pi},\dfrac{1}{\sqrt {2\pi}},\ldots $, so the answer to your question is "yes". $\sin\left(\dfrac1x\right)$ is a simpler example with the same property.


0

First, this function $f$ can be defined at $0$ by continuity, with value $0$. Now take $x_k = \frac{1}{\sqrt{2\pi k}}$, $k >0$. All $x_k$ are in $[0,1]$, and $f(x_k)=0$, an infinity of times in the interval, and increasingly denser near $0$. A lot of them are quite easy to build, and motivate of lot of exercices for students to study continuity, ...


6

One issue not yet addressed is why we have both $\aleph$s and $\omega$s. These both exist because Cantor introduced two separate concepts about infinite sets - their sizes ($\aleph$) and their order-types ($\omega$). Everyone else explained sizes, so I won't go over that. For order-types, begin by considering the following set: { 1/2, 2/3, 3/4, ..., ...


4

It is an indeterminate form and as such cannot be assigned any value. It is better expressed as $\lim_\limits{{x\to \infty}\\,\\{y\to 0}}x^y$. As commented by Did, it is true that $x^y$ has no limit when $x\to \infty$ and $y\to 0$. And $\infty^0$ has no definite meaning in mathematics. It is basically some kind of a meaningless statement where the ...


4

First of all: you cannot just substract infinity from infinity. Infinity is not a real number so you can't simply use the basic operations as you're used to do with (real) real numbers. However, in the context of limits, there are things you can do. As pointed out, the limit in your question is $0$ because $x-x=0$ for all $x$, so the 'calculation' would be ...


0

The answer is $0$ as first you simplify limit expression and then plug in value to check the limit if it's not indeterminate. So the limit is$0$


0

For a beginner, I would simply introduce the idea of one-to-one correspondence and not blow him away with Cantor and Hilbert. Thus, two sets A and B have the same number of elements if each element of A can be paired with exactly one element of B, and that pairing covers every element of B. Thus, 2 is paired with 3, 4 with 6, 6 with 9, 8 with 12, etc, as ...


2

Your question touches on issues both mathematical and philosophical; in fact, you might enjoy an introductory text in philosophy of mathematics and logic, or on the foundations of mathematics. Truth is, an infinity is not just an infinity. One reason for this peculiarity is due to the seminal work of Georg Cantor. Cantor proved that the set of real numbers ...


25

As an insight, think of the size of the usual sets of numbers. Think of the set of all positive integers, and the set of all positive multiples of 5. It's a bit strange to the "uninitiated", but it's ultimately not hard to wrap your head around the fact that there's the same amount of each, because we can list them side by side without any problem: ...


3

If it's any consolation, in practice you won't encounter more than two different types of infinity: that corresponding to the natural numbers, and that corresponding to the real numbers (cardinality of the continuum). The reason why it's necessary to differentiate between those two is that the "size" (cardinality) of the real numbers corresponds to the ...


5

The major motivation for classifying infinities (other then the intrinsic enjoyment of mathematics) is that different infinities permit different properties. Countable infinities can be reasoned about using inductive proofs. On the other hand many of the properties analysis makes use of requires uncountable sets. Having different infinities often makes ...


8

I won't comment on your more philosophical questions, but I will give what I think is one of the more important applications of different sizes of infinity. There is a rigorous mathematical way of thinking about a computer program, called a Turing machine. One can show that the cardinality of the set of Turing machines is $\aleph_0$, however the set of all ...


1

I wish to propose another simple, perhaps, less mathematical but more intuitive understanding of 'infinities'. It also happens to be by an author that you may possibly be familiar with: “There are infinite numbers between 0 and 1. There's .1 and .12 and .112 and an infinite collection of others. Of course, there is a bigger infinite set of numbers between ...


1

Neither argument is complete. In each case, it's not enough to provide a listing of some of the set you want to show is countable; you also need to show that this list contains all the elements of that set. In the former case (the rationals), this isn't too hard; but technically it takes an argument. In the latter case (sets of naturals), this is in fact ...


2

The first argument is perfectly valid. The second ignores infinite subsets of the natural numbers: any infinite subset (e.g., the set of all odd natural numbers) doesn't correspond to the binary expansion of any natural number, so all the argument shows is that there are countably infinitely many finite subsets of the natural numbers.


4

There have been many answers already, but I'd thought maybe I could try to provide an easier explanation of how cardinality works. Consider the three infinite sets $A = \{1,2,3,...\}$, and $B = \{1,2,3,...\}$ and $C = \{1,2,3,...\}$, that is to say, each of them are just all natural numbers. Surely you agree that they have the same size, regardless of how ...


42

This is a wonderful question; unfortunately many of the answers jump directly to explaining cardinality as if that is somehow THE only valid response to your musings. Now, there's nothing wrong with cardinality -- it is an interesting and important concept in mathematics, and often useful. But there's no rule thats says it is how you must think about the ...


2

The following holds: Theorem: let $(X,<)$ be an ordered space in the order topology. Then $X$ is compact iff for every $A \subseteq X$, $\sup(A) \in X$ exists. For a proof, see this post. Note that this implies that $X$ is Dedekind-complete (i.e. every non-empty subset $A \subseteq X$ that has an upper bound in $X$ has a supremum in $X$). But ...


20

Here's the classic story: the Hilbert's Hotel Paradox. Once upon a time in Hilbertland there was a Hotel which was an infinite Hotel. Indeed Mr Hilbert, the owner, explicitly asked for an infinite number of rooms since booking a room during the high season was a really big issue in Hilbertland and it was common not to find a place. So what did Mr ...


5

It turns out that there are actually lots of meaningful ways to define infinity. One good example of this is the infinite hotel paradox which explains very well. There is also an excellent and very accesible chapter on this and related matters in Science of Discworld III by Terry Pratchett and Jack Cohen (a book which is well worth reading for anybody ...


10

If we count up to some large number $N$, about half the numbers will be even (the proportion is exactly half if $N$ is even, slightly less if $N$ is odd). Because of this, people say the even numbers have density $\frac{1}{2}$. The odd numbers also have density $\frac{1}{2}$, so we could say there are as many even numbers as odd numbers. The density of the ...


2

The trick to determining if two infinities are equal often come through whether or not you can connect one object to another for every object in the set. For example, which has more numbers: $$\lbrace1,2,3,\dots\rbrace\text{ or }\lbrace2,3,4,\dots\rbrace$$ Both have an infinite amount of numbers, but which has more? Since every number or the left ...


3

In order to carry out a one-point compactification, you start with a topological space. So you have to decide what topology you're going to give $[-\infty, \infty]$ and $[0, \infty]$. One possible topology you could give $[0, \infty]$ is to give a subbasis consisting of stuff like $[0,a)$ and $(a,\infty]$. That would give you the normal topology when you ...


6

So, no matter which number you stop at, the multiples of 2 will have more numbers. The problem of thinking in infinities is that you must force yourself to not think about stopping. If you stop counting, you have a finite sequence and in that case, the number of odd and even numbers will depend on where you began and where you stopped. In infinity it ...


57

In response to the side question: it alerts the system that you want them to be rendered as symbols rather than left alone. You don't want the system to try to render everything as math because then it looks like $this which is really hard to read$ (unless you force it to insert spaces etc. by hand). In this question it wasn't really all that necessary. As ...


3

To see if two collections have the same number of elements, you put them in pairs. If there's no element left alone, the numbers are equal. For instance $\{ 1, 2, 3\}$ vs. $\{ a, b, c \}$ yields three pairs $(1,a),(2,b),(3,c)$ and nothing left. This reasoning is used to compare infinite sets. If you can pair the elements, the "infinities" are equal. As ...


22

In mathematics, we often (but not always) compare the "size" of two collections $A$ and $B$ by associating elements between them. For example, if $A=\{1,3,5,\ldots\}$ and $B=\{2,4,6,\ldots\}$, we might (as you pointed out yourself) make the following association: $$ \begin{array}{cccc} 1 & 3 & 5 & \cdots\\ \updownarrow & \updownarrow & ...


1

"Is it sufficient to say that because some set is equinumerous to an infinite set, it must be infinite as well?" Yes, of course. Assume the opposite and use the definition of "equinumerous". You easily get a contradiction. And yes, the proof is easy since the statement is quite trivial.


1

You are looking for: $$\begin{eqnarray*} \prod_{n\geq 1}\left(1-\frac{k^2}{(n+k)^2}\right)&=&\lim_{m\to +\infty}\frac{m! (2k+1)(2k+2)\cdot\ldots\cdot(2k+m)}{(k+1)^2 (k+2)^2\cdot\ldots\cdot(k+m)^2}\\&=&\lim_{m\to+\infty}\frac{m!(2k+m)!k!^2}{(2k)!(k+m)!^2}\\&=&\frac{1}{\binom{2k}{k}}\lim_{m\to ...


0

This is the Holder inequality with $p=1$ and $q=\infty$. In addition, we can prove this inequality using the following way: $$\sum_{i=1}^n \vert x_i\vert^2=\sum_{i=1}^n \vert x_i\vert \vert x_i\vert\leq \max_{i}\vert x_i\vert\sum_{i=1}^n \vert x_i\vert=\Arrowvert x \Arrowvert_1 \Arrowvert x\Arrowvert_\infty.$$


3

There is no such thing as a product of uncountably many numbers (where not most of them are $=1$ or some are $=0$). Compare to sums: Even with countably many summnds, we do not speak of sums, but of series (even though we suggestively use the same symbol $\sum$ for both). Those have very different properties from sums: A sum of rationals is always defined, ...


2

Consider the Riemann integral $$\int_a^bx\,dx=\lim_{n\to\infty}\sum_{k=1}^nx_k\cdot\Delta x_k$$ where the limit is taken in such a way so that the maximum $\Delta x_i$ approaches $0$. You are adding up all real numbers between $a$ and $b$, but weighted by the infinitesimally small $dx$. You could try to so the same with a product. Try to multiply all numbers ...


0

How would one calculate, or attach a value to the infinite product of negative numbers? For example, in this case: $(-1)\times(-1)\times(-1)\times(-1)\ldots=$? The value of $\lim\limits_{n\to\infty}(-1)^n$ is undefined.


-2

You could attach some the value of a limit of the average of the values if such a limit of averages would converge. For example arithmetic average of +1-1+1-1... would be 0 as when dividing by the number n of terms so far the oscillations would get below any $\epsilon\in \mathbb{R}$.


2

The challenge is that the theory has to remain effective. You are asking about adding new axioms in stages. For the first few stages things are clear. We can make $F_2 = F + G_1$, $F_3 = F_2 + G_2$, etc. So we can make the theory $F + \{ G_i : i \in\mathbb{N}\}$. Let's call that $F_\omega$. That will be an effective theory as well. This means that, just ...


1

Cantor's proof is not about numbers at all (there is a problem with trying to use it on numbers), it is about infinite strings of two characters. See http://www.logicmuseum.com/cantor/diagarg.htm. Call them Cantor Strings. Cantor used "M" and "W" which, while they are visually interesting, are hard to distinguish. You can use "0" and "1" and claim they ...


-2

The above answer is a tad misleading, the sum of the sequence 1+2+3+... does not ever ' = infinity' it only tends to infinity as n tends to infinity in a strict sense. The series does however, strictly and within the same base, equal -1/12.


-1

Let me know if i understood somethings wrongly. You did. When you write $x \to \infty$ (as opposed to $x \to +\infty$), for $x \in \mathbb{R}$ it roughly means "when absolute value of $x$ is arbitrarily large" ($|x| > R$, not $x > R$) There is a universal definition of limit using filters. A filter $\mathcal{F}$ is a collection of sets such that ...



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