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I think there's circularity in how the natural numbers (and so numbers; since they are all built from naturals) are defined in terms of sets, since it seems like the undefined notion of sets is intuitively about "collecting a NUMBER of things and considering them as a single whole". That's how the notion of cardinality makes sense for each set.

So when you examine the definition of the natural number 6, you see that it is a set, and it's cardinality is 6, which we obtain from COUNTING its elements.

So once we defined 6 in terms of these sets. 6 is now a specific object : "the set containing the empty set , a set containing the empty set, and...".

However we construct it in such a way that the cardinality it 6. So we already have to know about counting and numbers to define the number 6. We're using the idea of numbers that we're trying to define, to define numbers.

So is this set theoretic construction of the Natural Numbers is circular as a definition right? Is it a useful construction nonetheless? Does it help us understand the counting numbers that intuitively motivated the definition?

To be clear, I'm referring to the "Definition as Von-Neumann Ordinals" in the following webpage: https://en.wikipedia.org/wiki/Set-theoretic_definition_of_natural_numbers

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    $\begingroup$ In set theory we define a "structure" (i.e. a set with some specific properties) that proxy the natural numbers, i.e. such that mathematically behave exactly as numbers do. This is very useful to study for example the meta-mathematical properties of the theory of numbers (i.e. its models). No "sane" person can assert taht baby learn how to count starting from the set-theoretic def on natural number. $\endgroup$ – Mauro ALLEGRANZA Feb 14 '18 at 10:38
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    $\begingroup$ If you speak about a definition, quote the definition, please, not some vague sentences not exactly belonging to mathematics. $\endgroup$ – Professor Vector Feb 14 '18 at 10:41
  • $\begingroup$ @ProfessorVector Definition quoted. $\endgroup$ – trynalearn Feb 14 '18 at 10:48
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    $\begingroup$ What counts is counting. From that came the notion of number. Subsequently it was foramlized to the ZF numbers you know or NF numbers where 6 is the class of all sets that are equinuerous to some six element set. $\endgroup$ – William Elliot Feb 14 '18 at 10:53
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    $\begingroup$ You can think this way: we think we know what numbers are, but what we really know is how they behave. Thus, the set-theoretic definition of natural numbers gives us a simple abstract structure that behevaes as the numbers do. $\endgroup$ – Mauro ALLEGRANZA Feb 14 '18 at 13:10
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I think you are confusing the logic you are building your naturals with, with the meta-logical level of the human who constructs naturals.

When you are constructing the naturals, you have no concept of finite cardinality of a set. You can say that, if there is a bijection between two sets, then they have the same cardinality, but, still, you don't have the concept of numbers at all because you have not built them yet.

It is true that, when constructing natural numbers recursively, you will end up to have that the definition of the number 3 (i.e. $\{\emptyset, \{\emptyset\}, \{\emptyset,\{\emptyset\}\}\}$) contains three element, but you are saying that it contains three elements because you know what is three (and of course the definition of equinumerosity and cardinality of sets). Before defining numbers, you don't have numbers, therefore, you cannot say anything about things measured, or in any ways, related to numbers.

You can count the number of objects inside a natural number because you, as a human, know natural numbers and know how to operate with them. When you say that the definition is circular, you are saying that, starting from a concept inside your theory, you get to an equivalent one, staying inside your theory. But it's not the case: you have the concept of naturals (outside your theory) and you managed to represent it inside your theory. This is not circular: this is the expression of an idea inside a theory. That is: you have the concept of the number three in your head. If somebody asks you what is three, what would your answer be?

If you have ever programmed, in whatever programming language, you surely had to define some new data type, right? There is a form, in Computer Science, which is for defining new data types (Backus–Naur form). Naturals could be expressed in this form with the syntax. $$\mathbb{N} ::= 0 | S\,\mathbb{N}$$ Which is similar to the set-theoretical definition of natural numbers. Why have we defined it? Because computers cannot work with data types that are not defined. I think that's the same in set theory: before, you had only sets, axioms that speak about sets and, maybe, relations. Then you need to express the idea of natural number to be able to count. You don't have numbers, so you create them. The definition is not circular because this would imply that you already know, in your own theory, what is a natural number, what is counting, what is, for instance, a set of three elements. But you don't! You know them at a "meta-theory level", and, in order to use them in your theory, you need to construct them with the tools you have. Counting is not a tool because, before counting the elements of a set, you need some way to say how many are they. And for that, you need naturals.

I hope I was clear, the answer is long, but this is a delicate topic: Herbert Enderton's Elements of Set Theory, has a paragraph dedicated to this problem.

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  • $\begingroup$ Great answer. Do we have agreed upon definition for naturals "outside our theory" which we arrive at by counting. I think i'm asking is the answer to the question "what is three?" at the meta-theory level agreed upon/proposed by the community? $\endgroup$ – trynalearn Feb 14 '18 at 11:13

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