All Questions
Tagged with natural-numbers elementary-set-theory
152
questions
0
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2
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105
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Is there a specific symbol for $\mathbb{N}\cup\lbrace 0\rbrace$? [duplicate]
It is well known that natural numbers start in 1.
However, sometimes people work with a "widened set" of natural numeres plus zero, $\mathbb{N}\cup\lbrace 0\rbrace$. That is, all non-...
- 164
1
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2
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96
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Is this the right way to view infinity in real analysis?
So, I've lately been having confusion on how to understand infinity, but I think I have progress in my intuition. So, I'd appreciate if someone would let me know if I'm on the right track, and which ...
- 1,093
0
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1
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79
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Set of all finite subsets of $\mathbb{N}$ not equal to the to set of subsets of $\mathbb{N}$
I can kind of grasp why this is the case as if we take the union of all finite subsets of cardinality $i$ as $i$ runs through every natural number, we are listing finitely many elements each time.
...
- 459
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2
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231
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Is the set of all natural numbers acctually a proper class?
I have been searching about the difference between a set and a class. The main definitions I found can be resumed in “all sets are classes, but not all classes are sets. If a class is not a set, then ...
user784856
4
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2
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176
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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
4
votes
2
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211
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Why is mathematical induction necessary to prove results (eg, commutativity) for natural numbers but not for real numbers?
I've been studying the construction of the natural numbers, and I can't solve my own question, namely
Why is it necessary to use mathematical induction?
Let me clarify this. For example, we know ...
- 133
1
vote
2
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266
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is there any one one correspondence between an empty set and set of natural number $\Bbb N$
Actually, I wants to know that how an empty set is finite.
If it's finite then it must have one to one correspondence to the segment of natural number.
- 37
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2
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147
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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
3
votes
2
answers
140
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How do I prove injective property of $(x + y)^2 + y: \mathbb{N}×\mathbb{N} \to \mathbb{N}$
Given this function: $(x + y)^2 + y$, how do I go about proving it's injective property of mapping $\mathbb{N}×\mathbb{N} \to \mathbb{N}$ ? Surjection is not required. My current attempts include ...
- 59
3
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2
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158
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Does $\{ 0, \{ 0 \} \}$ contradict the axiom of regularity?
I'm trying to understand the ZFC axioms, and I understand most of them except the axiom of regularity.
$$\forall x[\exists a(a\in x) \Rightarrow \exists y(y\in x \wedge \neg\exists z(z\in y \wedge z\...
- 310
1
vote
1
answer
216
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How to show that a triple $(P, S, 1)$ constitutes a Peano System?
Mendelson (in Number Systems & the Foundations of Analysis) defines a Peano System as a triple $(P, S, 1)$ consisting of a set $P$, a distinguished element $1 \in P$, and a singulary operation $S :...
- 260
2
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1
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220
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Determine whether or not the following structure $(P,S,1)$ is a Peano System
First this is how the book define as a Peano System.
By a Peano System we mean a set $P$, a particular element $1$ in $P$, and a singulary operation $S$ on $P$ such that the following axioms are ...
3
votes
1
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258
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Alternative formulations of the natural numbers
Do there exist other ways to create the natural numbers, other than the definition given by
$$0=\emptyset \\
x^+ = x \cup \{x\}$$
For example, one could also define the succesor operation as
$$x^+ = \{...
- 1,093
1
vote
0
answers
91
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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
-2
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1
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702
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"Proof" of $0=1$ in set theory [closed]
Ok, so here is a proof of "$0 = 1$" I came up with today. You can do in set-theory, where natural numbers are defined in the usual way.
Proof: Let $\mathsf{Succ}$ be the function that takes ...
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