All Questions
7
questions
0
votes
1
answer
125
views
What is the cardinality of non-singleton subsets of $\mathbb{N}$?
I am studying a course on ZF Set Theory (without the Axiom of Choice) and am currently looking at the cardinalities of infinite sets. One question that I came across is the following:
Determine the ...
- 4,331
0
votes
1
answer
80
views
Is this proof using Cantor's Diagonal Argument correct?
We use $\sim$ to indicate to sets being bijective to eachother, i.e. having the same cardinality in this context. There exists $\psi:\mathbb{N}^2\mapsto\mathbb{Q}$ ,via $\left(a,b\right)\mapsto\frac{a}...
- 650
2
votes
4
answers
365
views
$\mathbb{N} ⊇ A_1 \supset A_2 \supset A_3 \supset \cdots$ but $\bigcap_{n=1}^∞ A_n$ is infinite?
What is an example of an infinite intersection of infinite sets is infinite?
I know that the intersection of infinite sets does not need to be infinite. However, I am seeking for an explicit example ...
- 51
1
vote
2
answers
1k
views
Arranging $\mathbb N$ into a two-dimensional array to prove a countably infinite collection of countable sets is countable.
If $A_n$ is a countable set for each $n \in\mathbb N,$ then $$\bigcup_{n=1}^\infty A_n$$ is countable. How does arranging the natural numbers in a two-dimensional array allow one to show the statement ...
- 1,324
0
votes
1
answer
225
views
Is $\mathbb{N}$ a well-founded set?
I was reading about Von Neumann's construction of $\mathbb{N}$, I understood that $\mathbb{N}=\{\emptyset, \{\emptyset\}, \{\emptyset, \{\emptyset\}\},...\} $.
I see that, with this construction, $\...
- 215
3
votes
2
answers
111
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
$1 = \{0\}$, $2 = \{0, 1\}$, $3 = \{0, 1, 2\}$, etc.
...
- 711
1
vote
2
answers
573
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 ...
