All Questions
Tagged with natural-numbers elementary-set-theory
21
questions with no upvoted or accepted answers
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
4
votes
0
answers
115
views
Show that $f(a,b,c)=(a+b+c)^3+(a+b)^2+a$ is injective.
For a function $f : \mathbb{N} \times \mathbb{N} \times \mathbb{N} \rightarrow \mathbb{N}$ defined by
$f(a,b,c)=(a+b+c)^3+(a+b)^2+a$
I want to show that $f$ is injective.
How can I show this?
I ...
- 41
3
votes
0
answers
119
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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
0
answers
86
views
Is there a name for this set-theoretical definition of natural numbers, or has it been invented?
I'll call it the binary encoding with sets. I think it's nice and trivial, should have been discovered by many genius brains, but i can't find it by searching with efforts.
Prior arts are Zermelo's ...
- 109
2
votes
0
answers
128
views
Statement of Well-ordering principle
The statement of well ordering principle appears in different mode - on subsets of natural numbers, or well-ordering of every (non-empty) set. For the question below, I am considering it w.r.t. non-...
- 3,047
1
vote
0
answers
106
views
Can every bijection of natural numbers be defined with a closed form formula
I am interested in the existance of closed form formulas for bijections on natural numbers.
With the term closed form is lose. Any information on formulas that represent permutations on N are welcomed....
- 47
1
vote
0
answers
515
views
Adding a Fixed Value to Each Element in a Set (How to Denote)
To denote a set such as, for example, the set of every natural number that is 3 greater than a multiple of 5, would $5\mathbb{N}+3$ be generally understood as $\{8,13,18,23,28,33,\dots\}$? If not, how ...
- 11
1
vote
0
answers
91
views
Show that: The set of all finite subsets of $\mathbb{N}$ is a countable set
Show that:
The set of all finite subsets of $\mathbb{N}$ is a countable set
My attempt:
Lets define the set $M:=\lbrace K : K \subset \mathbb{N} \wedge |K|<\infty \rbrace$
We now show that $|M|=|...
- 1,011
1
vote
0
answers
120
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Prove that $\mathbb{Z}_m$ and $\mathbb{Z}_n$ have the same cardinality iff m=n
In the following proof, $\mathbb{Z}_n= \{ 1,2,....n \} $
After $g$ is defined , they defined the composition $gof$ and then they defined $h$ by removing the last ordered pair $(n+1,m)$ from the ...
- 3,162
1
vote
0
answers
820
views
What are the finite and co-finite subsets of the set of Natural Numbers?
Let W be the set of Natural Numbers. What are all of the finite and co-finite subsets of W? It seems that all of the finite subsets of W will be inside of the power set of W, which we know will be ...
- 179
1
vote
1
answer
42
views
Natural Number Inductive Proof
Prove the following statement:
For every $\lambda$>1, there exists a number a∈N and b∈[0,1) such that $\lambda$=a+b.
I first defined a = sup{n ∈ N | n ≤ x}, so m is the integer part of x or the ...
- 1,913
1
vote
1
answer
76
views
how many linear orderings on $\omega$ the are and how can we identify when 2 of them are in fact isomorphic.
how many linear orderings on $\omega$ the are and how can we identify when 2 of them are in fact isomorphic. I think that by instability argument there are $2^{\aleph_0}$ of them, but I do not know ...
- 3,978
0
votes
0
answers
40
views
Induction principle in its set formulation and in its property formulation: which one to use in a well-redacted Induction Step of an induction?
I have read this answer about the well ordering principle and the induction principle. It especially says that "any proper axiomatization of $\mathbb N$ in modern logic does not involve set-...
- 1,231
0
votes
0
answers
30
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Elementary question regarding various definitions of the integers and naturals
So the question is as follows
We need to prove that $\mathbb{\mathbb{Z}}$ is the same set as the following three sets
(i) $\{x\in\mathbb{R}\hspace{0.1cm}|\hspace{0.1cm}x\in\mathbb{\mathbb{N}}\hspace{...
- 85
0
votes
0
answers
24
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How to count a number of elements in a set with subsequent natural numbers and why substracting the lowest number from the highest is wrong??
I just started learning math from 0 and looking for explanation
I have a set {18,...,32} and the task is to find its cardinality. I was thinking that substracting the lowest (18) from the highest (32) ...
