All Questions
Tagged with natural-numbers elementary-set-theory
152
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
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
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
9
votes
7
answers
8k
views
If the order in a set doesn’t matter, can we change order of, say, $\Bbb{N}$?
I’m given to understand that the order of the elements of a set doesn’t matter. So can I change the order of the set of natural numbers or any set of numbers ( $\mathbb{W,Z,Q,R}$ for that matter) as ...
- 4,953
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
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
7
votes
1
answer
2k
views
Can the natural numbers be uncountable?
Definition of a countable set, from Stanford, as I didn't want to quote Wikipedia:
Definition. A set S is countable if |S| = |N|.
Thus a set S is countable if there is a one-to-one mapping of Num ...
- 253
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
$$
\bigcup_{i\in J}X_{i}\subsetneq \...
- 19.7k
6
votes
0
answers
652
views
Crititism of the set-theoretic definition of natural numbers
A while ago I read in a book (or a paper?) that a very well-known mathematician (Saunders Maclane?) in his lectures used to mock the classical set-theoretical definition of natural numbers:
0 = {}, 1 ...
- 617
5
votes
1
answer
155
views
Cardinality of subsets of $\mathbb{N}$ with fixed asymptotic density
For a set $S\subset \mathbb{N}$, let
$$a(S)=\lim_{n\rightarrow\infty}\frac{\#\{s\in S\>|\>s\le n\}}{n}$$
be the limiting asymptotic density of $S$ in the natural numbers if the limit exists, ...
user139000
5
votes
4
answers
3k
views
Natural numbers in set theory is $\{0,1,2,...\}?$
The set of natural numbers $\mathbb{N}$ in set theory is defined with the axiom of infinity as the smallest inductive set and then it is usually proven that $\mathbb{N}$ satisfies the Peano axioms and ...
- 2,015
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
4
votes
3
answers
3k
views
Is there a natural number between $0$ and $1$?
Is there a natural number between $0$ and $1$?
A proof, s'il vous plaît, not your personal opinion. (Assume the Peano Postulates.)
- 43
4
votes
2
answers
176
views
Is my proof that the Sharkovsky Ordering is a total ordering, correct?
The Sharkovsky ordering is an ordering of the natural numbers $\mathbb{N}$, where
$3$ $\prec$ $5 $ $\prec$ $7 $ $\prec$ $9$ $\prec$ ...
$2*3$ $\prec$ $2*5$ $\prec$ $...
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