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I know there were some issues with set theory that involved self-reference and famous example being that of Russell's example. Here, I have a set that I use a somewhat vague term "use" to construct but eventually answer leads to a contradiction with the property itself. What I want to ask is, can someone explain what happens wrong here formally (e.g which axiom I am using falsely)

$$A= \{n\in \mathbb{Z} :n \text{ was used in real life by someone in any setting}\}$$

Where by any setting I mean, $n$ could be number of days, it could be highest number a child counted when they were $10$ (if they did) and also when they were $12$ (if they did), but other numbers that were never acknowledged by anyone such as my heart rate yesterday at 2:04 PM (which I never really counted) (*)

My problem is that , I think we can somehow agree all numbers that will ever be used, acknowledged (acknowledgement being a use case) is bounded somehow because people are finite objects and there will be finitely many people until end of universe.

But now let $N$ be the upper bound. Now that I mentioned it, I know $N+1$ is larger. But $N$ was the biggest one, what happened?

(*) I used asterisk because, the moment I wrote that text, I acknowledged that number (albeit abstractly).

Further, if you use something you must acknowledge it and if you acknowledge , that is its use case, so they are interchangeable.

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    $\begingroup$ Are you asking if this set is ill-defined? This reminds me of "the longest number you can't write down", which I just wrote down. $\endgroup$ Feb 23 at 17:18
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    $\begingroup$ Richard paradox $\endgroup$ Feb 23 at 17:34
  • $\begingroup$ I think Mauro's comment is what I wrote most likely to be. $\endgroup$ Feb 23 at 17:57
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    $\begingroup$ Assuming you are living in ZFC, I suppose the relevant axiom would be en.wikipedia.org/wiki/Axiom_schema_of_specification , and the way you are using it falsely is that your predicate "n was used in real life by someone in any setting" is not written as a grammatical formula $\phi$ in the language of ZFC (and cannot be, since it is ill-defined.) $\endgroup$ Feb 23 at 19:23

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I am no expert in this matters, just an enthusiast, so would be happy to be corrected if I am wrong. However, as I understand it, the problem here is the precise meaning of the words used in real life. The moment you pin down the meaning of this, it becomes a matter of logic to deduce what comes after. Let's assume you can somehow formalise all you said in a theory. That would mean that the existence of this set leads to a contradiction (which would also be formalised, of course). You can live with that, but in most logical systems contradictions are avoided at all costs as they lead to the provability of every single statement, which is not very helpful. Rather, you can somehow “ban” this set from existing in your theory, meaning you choose your axioms such that this construction is no longer possible. That is reminiscent of the Russell's paradox and the later development of Zermelo-Frenkel set theory.

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