All Questions
Tagged with natural-numbers elementary-set-theory
152
questions
13
votes
8
answers
2k
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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
$...
- 2,979
1
vote
2
answers
142
views
proving the set of natural numbers is infinite (Tao Ex 2.6.3)
Tao's Analysis I 4th ed has the following exercise 3.6.3:
Let $n$ be a natural number, and let $f:\{i \in \mathbb{N}:i \leq i \leq n\} \to \mathbb{N}$ be a function. Show that there exists a natural ...
- 3,325
-1
votes
1
answer
90
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What is the intuitive meaning of natural numbers as constructed in ZF set theory?
My understanding is that in elementary set theory, the natural numbers are defined so that $0 = \emptyset$ and $n+1 = n \cup \{ n \}$. I understand that this gives us some very pleasant properties ...
- 483
2
votes
1
answer
33
views
Equivalent characterizations of finite sets
How can we show that the following notions of finiteness for a nonempty set $X$ are equivalent?
There exists $n \in \mathbb{N}$ such that there is an injection $X \hookrightarrow \{1, \ldots, n\}$
...
- 1,647
2
votes
0
answers
85
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
0
votes
1
answer
135
views
Does Cantor’s theorem rely on the Empty Set being in the power set of a set?
As I understand, Cantor’s diagonal set can be empty, that is, there could be a mapping from the the Natural Numbers to the Power Set of the Natural Numbers in which the empty set is not mapped. The ...
4
votes
0
answers
114
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
0
votes
1
answer
163
views
Proving (rigorously) that the number of $m$ element subsets of an $n$ element set is ${n \choose m}$
I am trying to solve the following problem (Amann & Escher Analysis I, Exercise I.6.3):
Show that the number of $m$ element subsets of an $n$ element set is ${n \choose m}$.
I emphasize that the ...
- 1,143
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
32
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Partitioning $\mathbb{N}$ into infinitely many infinite pairwise disjoint sets [duplicate]
Find an infinite collection of infinite sets $A_1,A_2,A_3\ldots$ such that $A_i\cap A_j=\emptyset$ for $i≠j$ and $$\bigcup_{i=1}^{\infty}A_i=\mathbb{N}.$$
My Attempt:
Let $p_i$ be the $i^{\textrm{th}}...
- 2,145
3
votes
1
answer
103
views
Bijection between $\mathbb{N}$ and the set of finite parts of $\mathbb{N}$
Let $\mathbb{N} = \{0,1,2,3,...\}$ be the set of natural numbers (with $0$) and $\mathbb{F}$ the set of finite parts of $\mathbb{N}$. I want to find a bijection, as simple as possible, between $\...
- 576
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
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
2
votes
1
answer
185
views
Is the set of all linear orders on $\mathbb{N}$ linearly orderable?
In studying the issue of linear orders and well ordering in the context of ZF Set Theory (without the Axiom of Choice), I have recently been thinking about the following question:
Is the set of all ...
- 4,345
1
vote
1
answer
76
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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