Somewhere beyond the numbers lies the concept of Infinity. But what exactly does "infinity" mean? What rules does it obey? What interesting properties does it have?

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5answers
425 views

Does this expression represent the largest real number?

I'm not very good at this, so hopefully I'm not making a silly mistake here... Assuming that $\infty$ is larger than any real number, we can then assume that: $\dfrac{1}{\infty}$ is the smallest ...
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5answers
1k views

A non-mathematician’s (programmer’s) question on infinity?

I apologize for my total ignorance in the sphere of mathematics and the possibly very silly question I'm about to ask. My mathematical knowledge level is quite limited (pretty much finished with some ...
2
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5answers
450 views

Prove the set of functions $f : \mathbb{Q} \rightarrow \{1,2,3\}$ uncountably infinite

Prove that the set of functions $f: \mathbb{Q} \rightarrow \{1,2,3\}$ is uncountably infinite. I'm totally stuck on this one. We have just been shown Georg Cantor's diagonalization argument in class ...
2
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4answers
821 views

What is the limit as $x\to\infty$ of $\cos x$?

What is the limit as $x\to\infty$ of $\cos x$? Thanks in advance.
187
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13answers
24k views

Given an infinite number of monkeys and an infinite amount of time, would one of them write Hamlet?

Of course, we've all heard the colloquialism "If a bunch of monkeys pound on a typewriter, eventually one of them will write Hamlet." I have a (not very mathematically intelligent) friend who ...
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1answer
179 views

Proof whether or not 1/k by 1/(k+1) rectangles fit inside a unit square

I am reading Concrete Mathematics and came across an interesting problem, number 37 of chapter 2. The answers to exercises lists no known answer to this problem: Will all the 1/k by 1/(k+1) ...
2
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6answers
654 views

Cardinality of $P(\mathbb{R})$ and $P(P(\mathbb{R}))$

What is cardinality of $P(\mathbb{R})$? And $P(P(\mathbb{R}))$? P is a Power Set, $\mathbb{R}$ is set of real numbers. In general - how can find cardinality of different sets? Is/are there a good ...
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6answers
560 views

Book/article/tutorial as an introduction to Cardinality

I study CS, but on the first semester I have a lot of mathematics. Of course, there is an introduction to set theory and logic. Recently, we had lectures about cardinality, different kinds of ...
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10answers
7k views

Do infinity and zero really exist?

I'm not going to prove something, this is just a question. From the first day which I went to University until now I had some root problems in some basic mathematical assumptions and concepts. Please ...
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5answers
5k views

How can a structure have infinite length and infinite surface area, but have finite volume?

Consider the curve $\frac{1}{x}$ where $x \geq 1$. Rotate this curve around the x-axis. One Dimension - Clearly this structure is infinitely long. Two Dimensions - Surface Area = ...
5
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2answers
374 views

Is the number of circles in the Apollonian gasket countable?

Is it correct to say that the number of circles in an Apollonian gasket is countable becuase we can form a correspondence with a Cantor set, as their methods of construction are similar? What about ...
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4answers
1k views

Partitioning an infinite set

Can you partition an infinite set, into an infinite number of infinite sets?
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1answer
889 views

Cardinality of a set that consists of all existing cardinalities

I have taken a look at the following topics: number of infinite sets with different cardinalities Cardinality of all cardinalities Are there uncountably infinite orders of infinity? Types of ...
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6answers
3k views

Why is $\omega$ the smallest $\infty$?

I am comfortable with the different sizes of infinities and Cantor's "diagonal argument" to prove that the set of all subsets of an infinite set has cardinality strictly greater than the set itself. ...
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2answers
2k views

Are there uncountably infinite orders of infinity?

Given a set $S$, one can easily find a set with greater cardinality -- just take the power set of $S$. In this way, one can construct a sequence of sets, each with greater cardinality than the last. ...
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4answers
1k views

You are standing at the origin of an “infinite forest” holding an “infinite bb-gun”

I use stories like these to develop intuition... or perhaps to destroy it. I have my own answers in mind, but I want to see if I have made any mistakes... You are standing at the origin of an ...
2
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2answers
586 views

Understanding of convergence of intersections of sets

If you start with an infinite set, you can have a sequence of nested sets which converge to a single point. (ie Intersection of $\left(\large\frac{-1}{n}, \frac{1}{n}\right)$ as $n\to \infty$) ...
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4answers
9k views

Types of infinity

I understand that there are different types of infinity: one can (even intuitively) understand that the infinity of the reals is different from the infinity of the natural numbers. Or that the ...
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3answers
2k views

Probability and Infinity

If the probability of an event is $\frac{1}{\infty}$ and $\infty$ trials are conducted, how many times will the event occur — $0$, $1$, or $\infty$?
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5answers
626 views

Induction versus Natural Numbers

$0$ is finite. If $n$ is finite, then $n+1$ is finite. Hence, by induction, all numbers are finite. What is the catch?
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3answers
2k views

On applying the quadratic formula to a first-degree equation

You're probably thinking, "Why?" Please let me explain... It is (very) well-known that $$ \forall (a,b,c,x) \in \mathbb{C}^* \times \mathbb{C}^3: ax^2 + bx + c = 0 \Leftrightarrow x = \frac{-b \pm ...
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7answers
7k views

Are there more rational numbers than integers?

I've been told that there are precisely the same number of rationals as there are of integers. The set of rationals is countably infinite, therefore every rational can be associated with a positive ...
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8answers
3k views

What is larger — the set of all positive even numbers, or the set of all positive integers?

We will call the set of all positive even numbers E and the set of all positive integers N. At first glance, it seems obvious ...