All Questions
8
questions
2
votes
1
answer
186
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 ...
- 4,331
1
vote
1
answer
75
views
How do you prove that there exists a highest element of any finite, nonempty subset of Natural Numbers? Is the following algorithmic proof valid?
Since the given set, $C \subset \mathbb{N}$ is non empty, hence by well ordering principle there exists $\alpha \in C$ which is the lowest element in C. Also, since the set $C$ is finite, $\quad \...
- 194
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
0
votes
0
answers
118
views
A Doubt about Well Ordering Principle and Principle of Mathematical Induction
I have had this lingering doubt in my mind for a very long time: One of the standard constructions of N starts by assuming the 5 Peano Axioms, proving that every non-zero is a successor and s(n) is ...
- 21
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 $ $\prec$ $9$ $\prec$ ...
$2*3$ $\prec$ $2*5$ $\prec$ $...
- 85
0
votes
2
answers
147
views
Can the well-ordering principle of the natural numbers replace the principle of mathematical induction in Peano axioms?
The well-ordering principle of the natural numbers states that the natural numbers are well-ordered through it's usual ordering.
I've already seen a demonstration of the principle of mathematical ...
- 3,881
0
votes
2
answers
592
views
Proof of the well-ordering principle
I tried to prove Well-Ordering Principle by myself, and I finally did it. However, I'm not sure if this proof is correct. Can anyone evaluate my proof?
Proof:
Since the set of natural numbers, $\...
- 123
2
votes
1
answer
89
views
Why is $\mathbb{N}$ well-ordered?
Define
$$0:= \emptyset$$
$$1:= \{\emptyset\} =\{0\}$$
$$2:= \{\emptyset, \{\emptyset\}\}=\{0,1\}$$
$$\vdots$$
$$n:= \{0,1, \dots, n-1\}$$
And put $\mathbb{N}:= \{0,1, \dots\}$.
Questions:
(1) ...
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