All Questions
Tagged with natural-numbers elementary-set-theory
21
questions
127
votes
9
answers
76k
views
Produce an explicit bijection between rationals and naturals
I remember my professor in college challenging me with this question, which I failed to answer satisfactorily: I know there exists a bijection between the rational numbers and the natural numbers, but ...
- 4,251
8
votes
6
answers
6k
views
Proving there is no natural number which is both even and odd
I've run into a small problem while working through Enderton's Elements of Set Theory. I'm doing the following problem:
Call a natural number even if it has the form $2\cdot m$ for some $m$. Call ...
- 22.4k
1
vote
2
answers
1k
views
Construction of uncountably many non-isomorphic linear (total) orderings of natural numbers
I would like to find a way to construct uncountably many non-isomorphic linear (total) orderings of natural numbers (as stated in the title).
I've already constructed two non-isomorphic total ...
- 31
11
votes
5
answers
3k
views
What is the meaning of set-theoretic notation {}=0 and {{}}=1?
I'm told by very intelligent set-theorists that 0={} and 1={{}}. First and foremost I'm not saying that this is false, I'm just a pretty dumb and stupid fellow who can't handle this concept in his ...
- 111
17
votes
2
answers
4k
views
Is aleph-$0$ a natural number?
Would I be right in saying that $\aleph_0 \in \mathbb N$?
Or would it be a wrong thing to do?
user108343
14
votes
8
answers
2k
views
Why can't we define arbitrarily large sets yet after defining these axioms? (Tao's Analysis I)
In Tao's Analysis I I am very confused why he says we do not have the rigor to define arbitrarily large sets after defining the below 2 axioms:
Axiom 3.4 If $a$ is an object, then there exists a set
$...
- 3,019
5
votes
1
answer
618
views
Explicit bijection between $\mathbb N$ and $\mathbb N \times \mathbb N$ [duplicate]
We can consider the quadratic scheme above for a possible explicit bijection between $\mathbb N$ and $\mathbb N \times \mathbb N$.
The part $\mathbb N \times \mathbb N \to \mathbb N$ is easy via $(m,...
- 251
3
votes
0
answers
119
views
Defining Addition of Natural Numbers as the Algebra of 'Push-Along' Functions
Let $N$ be a set containing an element $1$ and $\sigma: N \to N$ an injective function satisfying the following two properties:
$\tag 1 1 \notin \sigma(N)$
$\tag 2 (\forall M \subset N) \;\text{If } ...
- 11.5k
2
votes
1
answer
467
views
Intersection of a non-empty set of natural numbers (set-theoretic definition) gives an element of that set?
Consider the following set-theoretic definition of natural numbers:
$0$ is defined as $\emptyset$
If $n$ is defined, then the successor of $n$ is defined as $n^+ = \{n\} \cup n$
Thus $1 = \{0\}$, $2 ...
- 4,154
0
votes
1
answer
133
views
How to prove that $N\setminus A$ is finite? [closed]
$A \subseteq \mathcal{R}(N)$ and given that (by inductive definition):
$N ∈ S$.
If $a \in R$, then $R \setminus \{a\} \in A$.
I need to prove that for each $A \in S$, $N\setminus A$ is finite. How ...
user567507
9
votes
2
answers
1k
views
Is it necessary to use the axiom of Regularity to prove the successor function being injective?
Basically the problem is that given an inductive set $X$ we can define the successor function on $X$ such that $S:X\longrightarrow X$ and for all $x\in X$, $S(x)=x\cup \{x\}$. So, one of Peano axioms ...
- 3,988
7
votes
3
answers
2k
views
Can the natural number have an uncountable set of subsets?
Let $\mathbb{N}$ be the set of natural numbers. Let $X_{i},i\in I$ be an uncountable sequence of subsets such that
$$
\bigcup_{i\in I}X_{i}=\mathbb{N}
$$
and
$$
\bigcup_{i\in J}X_{i}\subsetneq \...
- 19.7k
3
votes
2
answers
2k
views
Intersection of a non-empty set of natural numbers (set-theoretic definition) is a natural number?
This question is very similar to Intersection of a non-empty set of natural numbers (set-theoretic definition) gives an element of that set?
Consider the following set-theoretic definition of natural ...
- 4,154
3
votes
1
answer
169
views
For $m,n\in \omega, m \leq n$ imply $\exists ! p\in \omega\ s.t\ m+p=n$
For a set $A$, we define $A^+:=A\cup\{A\}$
When we define,
$$0=\emptyset,\ 1=0^+,\ 2=1^+,\ \cdots$$
set of natural number $\omega$ is defined as
$$\omega=\{0,1,2,\cdots\}$$
The order $"\leq"$ is ...
- 801
2
votes
1
answer
50
views
How can I define $A$ in according to these demands?
I'm given $M\subseteq P(\mathbb{N}\times\mathbb{N})$, such that $M$ is countable set.
So: $M=\{A_0 , A_1 ,A_2 , \ldots\}$ .
How can I define $A \in P(\mathbb{N}\times\mathbb{N})$ so that, $\forall i\...
- 257