# Subset of integers called by a somewhat ill-defined property

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.

• 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. Feb 23 at 17:18
• Richard paradox Feb 23 at 17:34
• I think Mauro's comment is what I wrote most likely to be. Feb 23 at 17:57
• 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.) Feb 23 at 19:23