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In naive set theory we can consider sets like:

$$ A = \{ x\mid x\ \text{is a letter of the English alphabet} \} $$ or $$ B = \{ x \mid x\ \text{is a painting of Picasso} \} $$ (Is this ↑ true?)

And maybe say that $$ \{``a",``b",``d"\} \subseteq A $$

Can you do the same in ZFC?

If any of the two above questions is false, how can you talk about these "informal entities"?

Asaf Karagila
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Traincopter
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  • Way is to choose some well known monograph/author in set theory and follow it. – zkutch Sep 26 '22 at 21:40
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    Such as...? This isn't very helpful – Traincopter Sep 26 '22 at 21:58
  • Also, I asked this question because reading about zfc I'm only encountering sets constructed upon the empty set – Traincopter Sep 26 '22 at 22:02
  • In ZFC you need to code everything as a set. – Zhen Lin Sep 26 '22 at 22:07
  • Concrete books https://math.stackexchange.com/questions/1460690/recommendations-for-intermediate-level-logics-set-theory-books, https://math.stackexchange.com/questions/3696462/logic-and-set-theory-books, https://math.stackexchange.com/questions/724805/recommended-books-articles-for-learning-set-theory, – zkutch Sep 26 '22 at 22:23
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    ZFC doesn't try to model informal sets of objects found in the real world. Rather, it models a minimal collection of sets built up from the empty set which is sufficient to define the objects of mathematics ... e.g. natural numbers, functions, real numbers, etc. – Ned Sep 26 '22 at 23:46
  • In $\mathsf {ZFC}$ set theory the way to define a set as $B = { x \mid \varphi(x) }$ where $\varphi(x)$ specifies a condition (like the "x is a painting of Picasso" above) does not work in general and we have to use Specification axiom. – Mauro ALLEGRANZA Sep 27 '22 at 06:44
  • For a list of already existing objects $a,b,c$, we can use the definition: $B = { x \mid x=a \text { or } x=b \text { or } x=c }$. – Mauro ALLEGRANZA Sep 27 '22 at 06:45
  • @Ned so the answer is no? In this case is there any other way to do "modeling" of these informal sets (like different theories)? – Traincopter Sep 27 '22 at 08:16
  • @MauroALLEGRANZA I'm sorry but the specification axiom, as you have just shown and stated, builds new sets using an existing set, $A$ for example. What I'm asking is, how can you construct an $A$ with something that isn't just a nesting of sets containig the empty set. – Traincopter Sep 27 '22 at 08:17
  • Exactly, but in $\mathsf {ZFC}$ there are no paintings nor painters.... Having said that, if you have already proved that the sets $a,b,c$ exist, then the above formula defines the set ${ a,b,c }$. – Mauro ALLEGRANZA Sep 27 '22 at 08:21
  • And similar, having proved thats et $\mathbb N$ exists, you can use Specification to prove that $\text {Even}= { n \mid n \in \mathbb N \text { and } n \text { is Even} }$. – Mauro ALLEGRANZA Sep 27 '22 at 08:22
  • Yes yes, now you answer me, what I wasn't asking though is "how can you construct ${a,b,c}$ knowing $a,b,c$ exist", I was asking "how you prove that certain $a,b,c$ exist". So thanks for the answer. – Traincopter Sep 27 '22 at 08:28

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