# Questions tagged [natural-numbers]

For question about natural numbers $\Bbb N$, their properties and applications

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### what a number really is? [closed]

https://youtu.be/dKtsjQtigag?si=-7F6L380uJhs2dd2**strong text** in this video this guy took set theory as the foundation and my doubt is that in this video this guy tells that an empty set exists but ...
1 vote
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### Do all finite-cycle-free permutations of $\mathbb{N}$ have square roots?

This answer nicely categorises which permutations of a finite set have square roots. This prompts the following question: Does every permutation $\sigma:\mathbb{N}\to\mathbb{N}$ that contains no ...
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### Is there a continuous function that is worth -1 in the negative naturals and 1 in the naturals? [closed]

Is there a continuous function that is worth -1 in: -1,-2,-3,... and 1 in: 1,2,3,...?
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### A question about the overall characteristics of GCD statistical series

Problem Define a sequence $a$ with $a_i=\sum_{j=1}^{n}\sum_{k=1}^{n}[gcd (j,k)=i]$. We define another sequence $b$ about $a$, $b_i=\frac{a_i}{\sum_{j=1}^{n}a_i}$. In $n \to \infty$, what ...
1 vote
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### Large sets and Erdős-discrepancy

Large Sets Erdos conjecture I have a conjecture that is stronger than the Erdos discrepancy conjecture, can someone think of a counter example? Let $S$ be any large set and let $(x_1,x_2,...)$ be any ...
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### Is this proof of the well-ordering principle using induction correct?

This question is about proving the well-ordering principle of the natural numbers using the principle of induction. This other question uses a different induction argument than the one below. I'd like ...
1 vote
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### listing all finite subsets of natural numbers [duplicate]

is there a computable algorithm which lists all the finite subsets of natural numbers ?... i know that such a set is atleast countable... but can't determine if we can list every such subset in a ...
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### How are numbers assigned to a group of real life objects?

I hope I don't come off as dense but suppose I constructed (informally) the decimal number system assuming the existence of symbols, 1-9 and defining the operation of addition on it with usual ...
1 vote
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### N is infinite (a proof)

In the book Introduction to Set Theory (Third Edition, CRC Press, 1999), by Karel Hrbacek and Thomas Jeck, the following appears on page 70: 2.2 Lemma. -- If n ∈ N , then there is no one-to-one ...
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### How do I solve this velocity equation? [closed]

$$t-t_0=m\int\frac{\mathrm dv}{F}=-\frac{m}{b}\int\frac{\mathrm dv}{v}$$ $$t-t_0=-\frac{m}{b}\ln\frac{v}{v_0}$$ $$v=v_0 \cdot e^{-\frac{b}{m}(t-t_0)}$$ Can you please help me? How do I convert it from ...
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### Are there nonzero natural numbers such that $\sqrt{4n+5}+\sqrt{5n+1}+\sqrt{9n+4}= \frac{nx}{y}$?

Check if there are nonzero natural numbers $n,x,y$ such that: $$\sqrt{4n+5}+\sqrt{5n+1}+\sqrt{9n+4}= \frac{nx}{y}.$$Thank you in advance! My ideas So we can simply show that $4n+5,5n+1,9n+4$ are ...
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### How to prove natural number addition using induction? [duplicate]

I am a self learner so excuse me if I am asking a seemingly easy question , But I ve been stuck at this point for couple of days , I think I understand mathematical induction and what the author is ...
1 vote
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### Why does ZF natural number construction not simply use n+1={n} instead of n+1=n $\cup$ {n}?

From Set-theoretic definition of natural numbers n+1=n $\cup$ {n} i.e. 0 = {} 1 = {{}} 2 = {{},{{}}} etc It seems to me that a simpler, equally valid definition would be n+1={n} i.e. 0 = {} 1 = ...
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### Difference between N & N+ domain

$\{0,1,2,...\}$ is $\mathbb{N}$, and $\{1,2,3,...\}$ is $\mathbb{N}^+$ if I'm not wrong. Does that mean this author does not want to include $0$ in the function? Why some authors include $0$ in ...
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### What is the proper definition of a "Factor"?

The definition I found on most websites was "A natural number $x$ is a factor of a natural number $y$ if $\frac{y}x$ leaves no remainder." This definition seemed correct until I searched &...
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### Peter winkler's "Numbers" puzzle "Zeroes, Ones, and Twos"

I have a problem with the solution for the (b) part of the problem. The problem is as follows: Let $n$ be a natural number. Prove that $2^n$ has a multiple whose representation contains only ones and ...
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### 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-...
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### Confusion regarding definition of Natural Numbers (from book Numbers, english version of Zahlen)

In the book "Numbers" by Ebbinghaus et. al, the Natural numbers are defined as: The natural numbers form a set $\mathbb N$, containing a distinguished element $0$, called zero, together with ...
1 vote
find the inf/sup of the set A= { n $\in \mathbb{N}$ | $n^2-3n +1$} before finding the inf and sup i checked for the first terms of this set, for n $\in$ {0, . . . 6} we have {1 , -1 , -7 ,...