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Why is compactness in logic called compactness?

In logic, a semantics is said to be compact iff if every finite subset of a set of sentences has a model, then so to does the entire set. Most logic texts either don't explain the terminology, or allude to the topological property of compactness. I…
vanden
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15 answers

Prove that $\lim \limits_{n \to \infty} \frac{x^n}{n!} = 0$, $x \in \Bbb R$.

Why is $$\lim_{n \to \infty} \frac{2^n}{n!}=0\text{ ?}$$ Can we generalize it to any exponent $x \in \Bbb R$? This is to say, is $$\lim_{n \to \infty} \frac{x^n}{n!}=0\text{ ?}$$ This is being repurposed in an effort to cut down on duplicates,…
Matt Nashra
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9 answers

Why do mathematicians sometimes assume famous conjectures in their research?

I will use a specific example, but I mean in general. I went to a number theory conference and I saw one thing that surprised me: Nearly half the talks began with "Assuming the generalized Riemann Hypothesis..." Almost always, the crux of their…
Joseph DiNatale
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3 answers

What is the importance of the infinitesimal generator of Brownian motion?

I have read that the infinitesimal generator of Brownian motion is $\frac{1}{2}\small\triangle$. Unfortunately, I have no background in semigroup theory, and the expositions of semigroup theory I have found lack any motivation or intuition. What is…
Potato
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8 answers

Prove that the union of countably many countable sets is countable.

I am doing some homework exercises and stumbled upon this question. I don't know where to start. Prove that the union of countably many countable sets is countable. Just reading it confuses me. Any hints or help is greatly appreciated! Cheers!
wonggr
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90
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6 answers

Cardinality of set of real continuous functions

I believe that the set of all $\mathbb{R\to R}$ continuous functions is $\mathfrak c$, the cardinality of the continuum. However, I read in the book "Metric spaces" by Ó Searcóid that set of all $[0, 1]\to\mathbb{R}$ continuous functions is greater…
kennytm
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2 answers

Difference between metric and norm made concrete: The case of Euclid

This is a follow-up question on this one. The answers to my questions made things a lot clearer to me (Thank you for that!), yet there is some point that still bothers me. This time I am making things more concrete: I am esp. interested in the…
vonjd
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2 answers

Is Lagrange's theorem the most basic result in finite group theory?

Motivated by this question, can one prove that the order of an element in a finite group divides the order of the group without using Lagrange's theorem? (Or, equivalently, that the order of the group is an exponent for every element in the…
lhf
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3 answers

Paul Erdos's Two-Line Functional Analysis Proof

Legends hold that once upon a time, some mathematicians were rather pleased about a 30-ish page result in functional analysis. Paul Erdos, upon learning of the problem, spent ten or so minutes thinking about the original problem, and came up with a…
Emily
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Can a row of five equilateral triangles tile a big equilateral triangle?

Can rotations and translations of this shape perfectly tile some equilateral triangle? I've now also asked this question on mathoverflow. Notes: Obviously I'm ignoring the triangle of side $0$. Because the area of the triangle has to be a…
Oscar Cunningham
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6 answers

Not every metric is induced from a norm

I have studied that every normed space $(V, \lVert\cdot \lVert)$ is a metric space with respect to distance function $d(u,v) = \lVert u - v \rVert$, $u,v \in V$. My question is whether every metric on a linear space can be induced by norm? I know…
Srijan
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Combinatorial proof of summation of $\sum\limits_{k = 0}^n {n \choose k}^2= {2n \choose n}$

I was hoping to find a more "mathematical" proof, instead of proving logically $\displaystyle \sum_{k = 0}^n {n \choose k}^2= {2n \choose n}$. I already know the logical Proof: $${n \choose k}^2 = {n \choose k}{ n \choose n-k}$$ Hence summation can…
Lance C
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Would you ever stop rolling the die?

You have a six-sided die. You keep a cumulative total of your dice rolls. (E.g. if you roll a 3, then a 5, then a 2, your cumulative total is 10.) If your cumulative total is ever equal to a perfect square, then you lose, and you go home with…
Newb
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8 answers

Is there a size of rectangle that retains its ratio when it's folded in half?

A hypothetical (and maybe practical) question has been nagging at me. If you had a piece of paper with dimensions 4 and 3 (4:3), folding it in half along the long side (once) would result in 2 inches and 3 inches (2:3), which wouldn't retain its…
Pyraminx
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2 answers

The Duals of $l^\infty$ and $L^{\infty}$

Can we identify the dual space of $l^\infty$ with another "natural space"? If the answer is yes, what can we say about $L^\infty$? By the dual space I mean the space of all continuous linear functionals.
omar
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