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I am now studying set theory, but I am so curious about it.

1.Why we need to define number rather than treat it as something naturally exist?

2.Do we need to define point or line in Euclidean geometry by set theory? If so, how to define it?

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These two questions are too unrelated to be posted together, in my opinion. –  Git Gud Jun 21 at 9:19
    
As for 1., Why wouldn't you want to define something in the most elementary way possible? I mean, if you can define natural number with respect to even simpler concepts, why not do it? –  Git Gud Jun 21 at 9:21
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As for 2., you certainly don't need it. You can formalize euclidean geometry in first order logic. You just need a universe in which some objects you decide to call 'points', other's you call 'lines' and if nothing is escaping me, that's all you need. I think Epstein does this in Classical Mathematical Logic, I'm not sure though. –  Git Gud Jun 21 at 9:25
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@GitGud: I don't think I'd count it as Euclidean geometry unless it at least had some incidence axioms :P –  Eric Stucky Jun 21 at 9:29
    
@EricStucky Ah! I was just enumerating the objects. Should have been clearer. –  Git Gud Jun 21 at 9:30
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4 Answers 4

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It is true, that when we start with mathematics we usually treat the natural numbers as something which just exists out there. We have a good idea what it means to be a natural number, so we can work with that without worrying too much.

However set theory is a foundational theory. It can serve as a mathematical bedrock on which the rest of mathematics can be built. One of the most fundamental things in mathematics is the natural numbers. So naturally we want to show that we can define something which looks and behaves like the natural numbers within the confines of set theory.

If we simply continue with the natural numbers that we assumed to exist, then we can't work within set theory. Because the language of set theory only has one symbol, $\in$. But if we interpret the natural numbers within set theory, and then working within set theory to construct other things, like algebraic closures of fields, and function spaces, etc. then we have access to something which looks like the natural numbers, and behaves like the natural numbers, and lives within the universe of set theory.

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what about the second question –  89085731 Jun 21 at 12:27
    
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If we define natural numbers in terms of sets, then we:

  • have proven Peano arithmetic is at least as consistent as ZFC,
  • can use our knowledge of the natural numbers to work conveniently with sets,
  • can make set theoretic constructions and arguments involving the natural numbers
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"something naturally exist": this is an open philosophical question. Do quaternions or Banach algebras naturally exist ? And what about the smallest element of the set of uninteresting integers ?

Whatever the answer, mathematical rigor requires that you unambiguously identify the entities you are working with and this is done by defining them (with axioms) in such a way that their properties are undisputable.

Just think of "natural" numbers: do they start at zero or at one ?

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My $0$ could be your $1$. I agree with your answer, but not with what you intend with the last question. –  Git Gud Jun 21 at 9:32
    
No, my meaning was "does the set of natural numbers include zero, the neutral element for addition" ? Unless you have not stated that clearly, you don't know what is "the set of natural numbers", even though this is a very natural concept. –  Yves Daoust Jun 21 at 9:34
    
I see now. Up voted. –  Git Gud Jun 21 at 9:35
    
+1, btw aren't all philosophical questions open? –  Servaes Jun 21 at 10:19
    
I don't know, that's open :) –  Yves Daoust Jun 21 at 10:26
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1.The question is: what does "naturally exist" means?

Let me put an example: "what is the minimum natural number not definible with less than twelve words?". If you can define this number then the number has the following property: "The minimum natural number not definible with less than twelve words." You could even take this property as a definition of that number (because if a number satisfies the property, it is the only natural number that can satisfy such property).

But, wait a moment! "The minimum natural number not definible with less than twelve words" is a definition for that number with 11 words! So, what is happening there?

That is the kind of things against the mathematicians and logicians at the end of the XIX century and the beginning of the XX strived. They are called paradoxes, and they find a way to avoid them: working inside a formal theory that could catch enough of the natural numbers to describe their behaviour.

What we define as "natural numbers" are not exactly what we informally understand as natural numbers. Gödel show us that, whatever theory that we build in order to try to define the natural numbers, we will always find a proposition about them impossible to be proven inside the theory. We can't catch inside a definition what we understand in a natural way as "natural numbers". But if we don't work so, we can find us dealing with paradoxes like the one I set some lines avobe.

2.For the second question, we have a quite similar (but optimal) situation. There are different ways to define points, lines and so on inside the set theory. One of them is following the methodology of the sinthetic geometry, the way followed by Felix Klein in his Foundations of Geometry. This work uses the set theory to build the different kinds of geometry.

But there is another way to do it. Is a longer way, but it has the property of being attached to the same paradigm that let you define the natural numbers inside the set theory: You start stablishing the natural numbers and their arithmetic. Then, build the whole numbers and the rational numbers, and at the end you can stablish the real numbers, with their arithmetic and topology. Using linear álgebra and affine geometry you can then define the points and lines in a formal way.

The good news with this way to work is that there is a branch of abstract algebra (Galois' theory) that claims that whatever we can do with a circle and a line (i.e., whatever we can do in euclidean geometry) is equivalent to certain algebraic manipulations on the objects you will build.

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