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I have two questions about arithmetic that I've been thinking about, but I'm stuck without any answers. Your help would be greatly appreciated.

How would you define a number in a simple way? I'm looking for an abstract definition. So far, I’ve come up with something like this: $3$ is a number that indicates a set contains an element, an element, and another element.

Lets say I have one green, red and yellow balls. If I start counting from the green, to the red, to the yellow I will get $3$. If I start from the red, green, yellow - I will get $3$ and it will be so in every other way. Why is that? Why is counting not affected by the order? It may be trivial, but I can't really 'see' it.

Thanks!

Red Five
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David123Q
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    Your first idea is called the cardinality of a set. You could define $3$ as card{"element", "another element", "yet another element"}. The elements could be whatever you want. For your second question, suppose you always count the balls in the order left to right. Now suppose someone swaps the order of two of the balls before you count. Surely just moving two of the balls around without adding or removing any balls will not change your count, but it will lead to you counting them in a different order. – user10478 Aug 22 '24 at 07:04
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    How many fingers do you have on your hand, David? Does it matter whether you start counting from the left, or from the right? Why not? – Gerry Myerson Aug 22 '24 at 07:31
  • You say it yourself: an element, an element, an element. Your definition is color-blind. –  Aug 22 '24 at 07:38
  • One way to see that $\vert {R,B,G} \vert = \vert {B,R,G} \vert = \vert {G,R,B} \vert = ...$ is to note that $\vert {R,B,G} \vert = 1+1+1,\ \vert {B,R,G} \vert = 1+1+1,\ \vert {G,R,B} \vert = 1+1+1,\ $ and so on. Then note that, if we change the order of the $1's$ in the expression $1+1+1,$ the expression remains the same. I'm not sure if this reasoning/explanation is valid/sound (because the question doesn't seem very well-formed), but it feels like it might satisfy for the last question. – Adam Rubinson Aug 22 '24 at 08:35
  • Hmm I think I came across an intuitive solution for my second question. Lets say for example I have green, red and yellow balls. I can draw a line and tag 1,2,3 on the line. Counting is just a Bijection from {1,2,..n} to the set we count. So for example, if I count green red and yellow all I really do is place green on the point I tagged as 1 and so on. It doesn't matter if I switch between items, so It doesnt matter if I count in diffrent order. – David123Q Aug 22 '24 at 11:22

2 Answers2

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If you want to build numbers from the ground up, you generally will want to do it in a Set-Theoretical way. Meaning, you start with the notion of sets, and then you can build numbers recursively. $$0\equiv\emptyset$$ $$1\equiv\{\emptyset\}$$ $$2\equiv\{\emptyset,\{\emptyset\}\}$$ $$3\equiv\{\emptyset,\{\emptyset,\{\emptyset\}\}\}$$ And so on. Basically, you start with the empty set, and to get to the next number, you take the union of sets of the previous numbers.

LunaticLuna
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By definition, the act of counting applies to elements whatever they are. From this standpoint, they are undistinguishable and can't have an order.