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I found that many of us (mathematicians) try to construct natural numbers defined from the intuitive concept 'size of the set'. They take $\emptyset$, the empty set as the starting, then define and represent the size of the set $A_1=\{\emptyset\}$ as $1$, now define $2$ as the size of the set $A_2=\{\emptyset,A_1\}$ and so on.

My doubt: Can we take an absolute empty set $\emptyset$ without relating to a well defined universal set $U$?

Mauro ALLEGRANZA
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Messi Lio
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    In formal set theory, there is no "universal set". The empty set exists because in formal set theory there is an assumption that sets exist, and there is an axiom that says that if $A$ is a set, and $P$ is a property (which is a well-defined notion), then the collection of all $x\in A$ for which $P(x)$ is true is also a set. Given that a set $A$ exists, $\varnothing$ can be defined as ${x\in A\mid x\neq x}$. Another axiom says that two sets are equal if and only if they have the exact same elements, which proves $\varnothing$ is unique, and well-defined. – Arturo Magidin May 09 '22 at 18:06
  • @Arturo Magidin So without assuming the existence of a set $A$, one cannot have $\phi$, right? – Messi Lio May 09 '22 at 18:11
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    If your set theory does not posit the existence of at least one set, then you can have no sets at all and your theory admits an empty model, which means it is useless. That's why all formal set theories include an axiom asserting the existence of some set. In ZFC, this is the Axiom of Infinity, which asserts the existence of an inductive set. – Arturo Magidin May 09 '22 at 18:13
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    Don't use the Greek letter \phi for the empty set. Use either \emptyset, which produces $\emptyset$, or \varnothing, which yields $\varnothing$. – Arturo Magidin May 09 '22 at 18:14
  • So, we have already some natural 'size', right? – Messi Lio May 09 '22 at 18:18
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    I do not understand what you mean by "size". The numbers are not defined in terms of "size". – Arturo Magidin May 09 '22 at 18:18
  • So one cannot define natural numbers in this way (in the question), right? – Messi Lio May 09 '22 at 18:20
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    I don't understand what you are trying to argue. "Size" does not enter into it, and nobody I knows actually defines the natural numbers "in terms of size". So your premise is incorrect. – Arturo Magidin May 09 '22 at 18:22
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    The natural numbers are not defined in terms of "size". They are defined in terms of "successor": start with the empty set; given any set $x$, the "successor of $x$" is defined to be $x^+ = x\cup{x}$. A set $S$ is inductive if (i) it contains $\varnothing$ as an element; and (ii) whenever $x\in S$, $x^+\in S$ as well. The natural numbers are defined as the least inductive set. E.g., here and here. – Arturo Magidin May 09 '22 at 18:22
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    In set theory, the natural number $0$ is not defined as the "size" of $\varnothing$. Rather, by definition, $0=\varnothing$, and further \begin{align} 1 &= {0}={\varnothing} \ 2 &= {0,1} = {\varnothing,{\varnothing}} \ 3 &= {0,1,2}=\big{\varnothing,{\varnothing},{\varnothing,{\varnothing}}\big} , . \end{align} – Joe May 09 '22 at 18:30
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    It seems to strange to define numbers as sets, but since set theory only deals with sets (and no other kind of object), we are forced to take this approach. Even though the working mathematician certainly does not conceive of the number $3$ as $\big{\varnothing,{\varnothing},{\varnothing,{\varnothing}}\big}$, the above definition is accepted because from it we can derive the usual laws of arithmetic governing natural numbers. These properties of natural numbers are much more important than what they actually "are". – Joe May 09 '22 at 18:30
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    Put differently, we have a prior notion of what natural numbers and how they behave. However unnatural the set-theoretic definition of natural numbers might seem, the fact of the matter is that this set-theoretic definition can be used to demonstrate the artithmetic truths that we learnt as children. Therefore, the fact that the definition seems artificial turns out not to matter. – Joe May 09 '22 at 18:34
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    Don;t think that defining the natural numbers in terms of sets is meant to provides us with any deeper understanding of what the natural numbers are. Indeed, think of what is going on here as a way to encode natural numbers in terms of sets ... with the purpose of being able to say: "hey look, we can reduce a lot, if not all, of mathematics to sets .. that's cool!" – Bram28 May 10 '22 at 02:02
  • For complimentation, we require a universal set, right? – Messi Lio May 10 '22 at 04:02
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    No complement (not compliment!) is taken in this problem, as far as I can see. But when set theorists take complements, it's with respect to some specific context-dependent set. – J.G. May 10 '22 at 07:12

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