# Tagged Questions

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### There is no smallest infinity in calculus?

Somewhat of a basic question, but I tried mixing set theory and calculus and the result is a giant mess. From set theory (assume ZFC) we know there is a smallest infinite cardinal, $\aleph_0$, and ...
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### Circles and the continuum hypothesis

I was trying to understand the undecidable nature of the continuum hypothesis and came up with the following question: The set of circles with a rational diameter is countably infinite (with ...
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### Help explain the set being constructed in this Cantor-Schroder-Berstein proof

The Cantor-Schroder-Bernstein theorem states that: Suppose $A$ and $B$ are sets. If $|A|\le |B|$ and $|B|\le |A|$, then $|A|=|B|$ Proof: So, $|A|\le|B|$ implies we can choose an injection ...
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### Sizes of infinity

I was just thinking about infinity (as you do) and thought the following. "There are infinitely many reals in the interval $x\in[0,1]$ and an 'equal number of reals' $x\in[1,2]$, so there are 'double ...
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### One set of functions larger than another set of functions?

This summer I've been slowly working through Halmos's Naive Set Theory. I'm not that far, but I know what lies ahead, which is proving that one infinite set is larger than another (the reals larger ...
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### Is there a mathematical concept of fractions using transfinite numbers as numerators and denominators?

http://de.wikipedia.org/wiki/Cantors_erstes_Diagonalargument (German) http://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument (English) While looking at Cantors method of proof, which he used to ...
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### Question about the cardinality of sets and infinity

Let's say we have $\mathbb{N}$, the set of natural numbers: $\{1, 2, 3, 4, 5...\}$ ...which has a cardinality of infinity, and the set $A_x$ which consists of the variable "$x$" (so $\{x\}$). If I ...
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### Cardinality of infinite between the set of rationals and set of reals

I remember learning that whether or not there is a cardinality of infinity between the set of rational numbers and the set of real numbers is unprovable. Is this true, and if so, how do we know it to ...
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### A strange puzzle having two possible solutions

A friend of mine asked me the following question: Suppose you have a basket in which there is a coin. The coin is marked with the number one. At noon less one minute, someone takes the coin ...
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### Which of these sets is bigger?

I am a fourth year computer science student and I am taking second year level maths because they are very useful for computer stuff. At the end of the linear algebra lecture the Prof left us with a ...
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### What is your intuitive understanding of infinity? [duplicate]

What is your intuitive understanding of infinity? Mine is the following, I prepared it as image: Those were the main points I got to after thinking by myself about what infinity is, without ...
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### Is there a highest order of infinity?

Does there exist an infinite set of cardinality such that it can never be reached by taking power sets of a set with cardinality aleph-null. Please prove your answer, or include a link to a proof. I ...
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### Having trouble showing the cardinality of two infinite sets is the same

We just learned about Aleph-naught today and I read about it on wikipedia but I do not know how to go about solving this problem in my homework: Prove that N(natural numbers) has the same ...
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### Cardinality of the set of at most countable subsets of the real line?

I'm exploring an unrelated question about power series with complex coefficients. While exploring this question, I wondered: What is the cardinality of the set of all such power series? Or with ...
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### Are all uncountable infinities greater than all countable infinities? Are some uncountable infinities greater than other uncountable infinities? [duplicate]

I recently finished a discrete mathematics class, and near the end of the semester, the prof (very superficially) touched on countable and uncountable infinities. His explanation of countable ...
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### Is $\lim\limits_{n \to \infty} n$ “equal” to $\mathbb{N}$?

In set theory, the natural numbers are defined by means of inductive sets and the successor operation $S(n+1) = n \cup \{n\}$ As such, we have $1 = \{0\}$, $2 = \{0, 1\}$, $3 = \{0, 1, 2\}$, ...
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### What is the cardinality of the set of infinite cardinalities?

I am currently aware of only two infinite cardinalities: $\aleph_0 = |\Bbb N|$ $\aleph_1 = |\Bbb R|$ Questions: Is there an infinite number of infinite cardinalities? If yes, is this set of ...
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### Is there, for every set $X$, a set $Y$ for which $|Y| < |X|$ but $|\mathcal{P}(Y)| \geq |X|$?

As the title says, my question is: Is there, for every set $X$, a set $Y$ for which $|Y| < |X|$ but $|\mathcal{P}(Y)| \geq |X|$? I'm fairly certain this is true for finite sets but maybe ...
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### Creating the set of natural numbers

I am not a mathematician but an engineer, so I can read some basics of the language proofs are written in. Second I am bad in probability and infinity and my question covers both. So I like to ...
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### Is there an infinite sequence AB, BC, CD, DX, …, YZ

Is it possible to construct an infinite set of ordered pairs of form S = {(A, B), (B, C), (C, D), (D, x), ..., (y, Z)}? Every element (B, C...) must appear only once as the first object in one of the ...
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### Elaboration of infinite, finite and enumerable definition

I am starting to learn some of the basic concepts of math. The concept I am learning now is infinite, finite, and denumerable. I am having trouble understanding the book's definiton. I am hoping if ...
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### How many different sizes of infinity are there?

It's pretty straightforward to say that there is an infinite number of different sizes of infinity, but then I thought, "What size of infinity is that?" My thoughts are that the number of unique ...
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### Is it viable to ask in an infinite set about the Cardinality?

Can you ask given an infinite set about its cardinality? Does an infinite set have a cardinality? So, for example, what would be the cardinality of $+\infty$?