0
votes
1answer
53 views

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 ...
3
votes
2answers
83 views

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 $1 = \{0\}$, $2 = \{0, 1\}$, $3 = \{0, 1, 2\}$, ...
6
votes
3answers
451 views

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 ...
3
votes
2answers
84 views

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 ...
0
votes
1answer
35 views

What is the cardinality of the equivalence class

Consider this relation: $$R = \left\{ {\left\langle {f,g} \right\rangle \in {{\left\{ {0,1} \right\}}^N} \times {{\left\{ {0,1} \right\}}^N}|\exists k \in N\left| {\left\{ {i \in N|f(i) \ne g(i)} ...
0
votes
2answers
26 views

Are all infinite sets equivalent by an indexing function?

Would it be true to say that two infinite sets would always be equivalent since you could always match the index of their elements? For instance, match the first element in $A$ to the first element ...
0
votes
1answer
56 views

Countable and Uncountable sets

Is $\mathbb{N}\cup\{a\}$, for some $a\not\in\mathbb{N}$ countable or uncountable? $\mathbf{Attempt: }$ It is true that a set is countable if there exists an injective function $f : S → N$ from $S$ to ...
1
vote
1answer
63 views

A Monkey Choosing Real Numbers for an Infinite Time

A common illustration of the nature of infinity is that, given an infinite amount of time, a monkey on a typewriter will, with probability $1$, produce the complete works of Shakespeare. Consider now ...
0
votes
2answers
39 views

What this statement is really saying to prove one Real number has missed the bijection with Integers?

In a Combinatorics text, I find this: Not all infinite sets have the same cardinality. Consider the set of all integers and the set of all reals. Assume that the set of reals can be put in ...
0
votes
1answer
91 views

Are there any infinites not from a powerset of the natural numbers?

With the cardinality of the natural numbers as $|\mathbb{N}| = \aleph_0$ and its powerset as $|\mathcal{P}(\mathbb{N})| = 2^{\aleph_0}$, the continuum hypothesis and the axiom of choice says that ...
7
votes
5answers
1k views

how do we assume there is infinity?

Definition of infinite: A set is infinite iff it is equivalent to one of its proper subsets. We know that our universe doesn't contain infinite number of elements, so how do we assume there is ...
1
vote
2answers
75 views

If the Kleene star of countable sets is countable, how are the real numbers uncountable?

The formal languages we use to represent number systems are interchangeable, which is why we don't hesitate to use different notations, e.g. hexadecimal, octal, binary, etc... to represent the reals. ...
3
votes
1answer
68 views

Set theory, show a set is countable, homework. check my answer

I solved this question but there is something strange going on and I am unsure of myself. Would like someone to review it. We are given a total order (or linear order) $<^{*}$on group $A$ such ...
2
votes
1answer
96 views

Largest infinite cardinal used in a proof

I've heard before that Knuth holds the record for the largest constant used in a mathematical proof. I was wondering what is the largest cardinal ever explicitly considered in set theory. I presume ...
3
votes
4answers
206 views

Hilbert's Hotel and Infinities for Pre-university Students

Hilbert's paradox of the grand hotel is a fun and exciting ground to base a talk on the set theoretic concept of infinity for interested students - even in middle- and high school. However, it does ...
1
vote
1answer
40 views

Show A is countable infinity

One more question about set theory: $A\subseteq R$ is an infinite set of positive numbers. Assume there is a value $k \in Z$ such that for any $B \subseteq A$: $\sum_{i=0}^\infty b(i) \le k$ where ...
0
votes
1answer
56 views

Is the set of all sums-of-rationals-that-give-one countable?

Some (but not all) sums of rational numbers gives us 1 as a result. For instance: $$\frac12 + \frac12 = 1$$ $$\frac13 + \frac23 = 1$$ $$\frac37 + \frac{3}{14} + \frac{5}{14} = 1$$ Is the set of all ...
2
votes
3answers
225 views

Why can't you count real numbers this way?

Sorry but this is probably a naive question. Why can't you generate real numbers by a*10^b, the same way as rational numbers by a/b? a and b could be integers so that you would start counting real ...
30
votes
7answers
1k views

Refuting the Anti-Cantor Cranks

I occasionally have the opportunity to argue with anti-Cantor cranks, people who for some reason or the other attack the validity of Cantor's diagonalization proof of the uncountability of the real ...
0
votes
1answer
61 views

Are there countably many infinities?

$\aleph_0$, $\aleph_1, \aleph_2$ and so on are indexed by a natural number so shouldn't there be countably many infinities?
1
vote
2answers
355 views

Must an infinite intersection of infinite sets be infinite?

If $A_2$ is a subset of $A_1$, $A_3$ is a subset of $A_2$, and this goes on infinitely and all contain an infinite number of elements, then is the intersection from $n=1$ to infinity, infinite as ...
0
votes
1answer
515 views

Proper Subset of an Infinite Set is Equinumerous to the Set Containing It

I noticed that there is a question about $S$ being denumerable, which implies $S$ is equinumerous with a proper subset of itself, but what about an infinite set? That is, how to do I prove that every ...
4
votes
2answers
87 views

Which texts do you recommend to learn about the work of Cantor?

A friend asked me if I know texts which talk about Cantor's Work on infinity and Set theory . And this is the reason Which pushed me to ask the question here . I prefer if you recommend texts for ...
1
vote
2answers
297 views

Comparing infinite binary fractions to infinite decimal fractions

I'm trying to understand the cardinality between the set of all infinite binary (base-2) fractions and the set of all infinite decimal (base-10) fractions. I can easily think of infinite binary ...
0
votes
3answers
190 views

Equinumerosity of infinite sets

Key issue: For infinite sets, does the existence of a bijection mean they have the same number of elements? For example, does the set of natural numbers N = {1,2,3,4...} have the same number of ...
4
votes
1answer
50 views

Generlized Büchi Games and Closed under superset Muller Games

For a unique infinite play $p$ in a 2-Player game $G=(V_0,V_1,E)$. Let $$ \inf(p) \subseteq V_0 \cup V_1 $$ be the set of vertices which occur infinitly often in $p$. Generlized Büchi (GB) Games ...
3
votes
2answers
126 views

how to know the set is finite, countable or uncountable

I am trying to understand whether the set is finite, countable, or uncountable. $$\{x \in\Bbb Q \mid 1<x<2 \} \qquad\text{is countable. }$$ but i dont understand why though. is it countable ...
4
votes
4answers
149 views

Dividing Two Infinities

I am Curious if the following is mathematically correct: Let $a$ be the infinite set of all nonnegative integers $0,1,2,3...$. Let $b$ be the infinite set of all nonnegative EVEN integers ...
0
votes
3answers
180 views

Can countability coexist with infinity?

This question concerns the countability of the real numbers. First I will show how I count the numbers between 0 and 1 on the real line. It is done by reversing digits behind the coma, so that e.g. ...
5
votes
2answers
247 views

Subtracting two infinities

I am Curious if the following is mathematically correct: Let $a$ be the infinite set of all nonnegative integers $0,1,2,3...$. I take from $a$ some of its elements, say integers $10$, $11$, and $12$ ...
2
votes
1answer
73 views

What are the factors of $\aleph_0$?

Extend the system of positive natural numbers with $\aleph_0$. Then we have: $$\aleph_0 = \aleph_0\cdot n,\quad \forall n \in \mathbb{N}^+$$ Does it make sense to talk about factors of $\aleph_0$? ...
1
vote
3answers
179 views

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 ...
2
votes
2answers
88 views

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 ...
2
votes
2answers
138 views

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 ...
8
votes
2answers
1k views

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 ...
6
votes
2answers
186 views

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 $+\infty$?
1
vote
1answer
49 views

$A+\alpha\sim A$ when $\omega\le\alpha<h(A)$

I have been considering about $A+\alpha\sim A$ when $\omega\le\alpha<h(A)$, in which $h(A)$ is the Hartogs number of $A$. If there is $\alpha$ s.t. $\omega\le\alpha<h(A)$, we can get $\alpha ...
1
vote
2answers
107 views

Is there any Dedekind-infinite set can be split to two smaller Dedekind-infinite sets?

I have been considering about $A+\alpha\sim A$ when $\omega\le\alpha<h(A)$, in which $h(A)$ is the Hartogs number of $A$. If there is $\alpha$ s.t. $\omega\le\alpha<h(A)$, we can get $\alpha ...
4
votes
1answer
109 views

Is $\aleph_0$ the minimum infinite cardinal number in $ZF$?

$\aleph_0$ is the least infinite cardinal number in ZFC. However, without AC, not every set is well-ordered. So is it consistent that a set is infinite but not $\ge \aleph_0$? In other words, is it ...
1
vote
2answers
91 views

Prove that if a set is Peano finite, then it is Dedekind finite.

I understand that this should be done by induction, but I have very limited knowledge on proof by induction. Could someone explain it in a way which also makes clear exactly what each stage of ...
3
votes
1answer
165 views

Prove that a formal language is infinite

I'm having trouble with the following exercise: Let $\Sigma = \{a,b,c\}$ and $L$ be a formal language, that consists of all words which contain all three letters at least once. Show that $L$ is ...
3
votes
1answer
182 views

“Real” cardinality, say, $\aleph_\pi$?

Is there any meaningful definition to afford for $\aleph_r$ (as in cardinality) where $r\in\mathbb{R}^+$? $r\in\mathbb{C}$? What about $\aleph_{\aleph_0}$? Can we iterate this? ...
2
votes
1answer
53 views

Is this set finite?

Let's say you are given a function $\mu:S\rightarrow(0,1]$ and you can additionally assume $$\sum_{s\in S}\mu(s)=1$$ Does this imply that $S$ is finite?
0
votes
0answers
73 views

Size of infinite sets [duplicate]

Possible Duplicate: Different kinds of infinities? I heard from a lecture at university, many years ago now, that some groups of infinite sets are bigger than other groups of infinite sets. ...
0
votes
2answers
107 views

Is it possible to iterate through an infinite set?

Is it coherent to suggest that it is possible to iterate, one-by-one, through every single item in an infinite set? Some have suggested that it is possible to iterate (or count) completely through an ...
1
vote
4answers
143 views

Does every sequentially ordered infinite set contain sequentially ordered infinite subsets?

I am not very familiar with mathematical proofs, or the notation involved, so if it is possible to explain in 8th grade English (or thereabouts), I would really appreciate it. Since I may even be ...
3
votes
4answers
585 views

Cardinality of the set of all functions from blank to blank

I'm trying to determine if a bunch of examples constitute countable or uncountable sets. One of which is the set of all functions from the positive integers to the positive integers. Word on the ...
3
votes
3answers
689 views

Comparison of two infinity [duplicate]

Possible Duplicate: Different kinds of infinities? Today I got to know that two infinity can be compared, But I want to know how is this possible? infinity will be infinity. If it doesn't ...
6
votes
2answers
1k views

Why do the rationals, integers and naturals all have the same cardinality?

So I answered this question: Are all infinities equal? I believe my answer is correct, however one thing I couldn't explain fully, and which is bugging me, is why the rationals $\mathbb Q$, integers ...
4
votes
2answers
315 views

Proof that the Irrationals are Countable

Proof: Between any two irrationals lies a rational, by the Density of the rationals in the real number system. There are only countably many rationals; therefore, there are only countably many pairs ...