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I am having a hard time understanding why it won't be sigma ideal. It is an ideal, that means closure under finite unions holds. Then it should be trivial to see that closure under countable unions hold too. For instance for an ideal J, if $E_1 \in J, \text{and}, E_2 \in J$, then based on the principle of finite unions, $E_1 \cup E_2 \in J$. Now, the sigma-ideals state that if $ E_n \in J, \forall n \in N, then \cup_{n=0}^{n=\infty}E_n \in J$.

I know my reasoning is wrong, I just don't understand why. Perhaps, I am unable to grasp the difference between finite and countable unions.

desert_ranger
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  • What reason are you giving? It seems to me that you just say you think it should a sigma ideal, so it must be a sigma ideal. That's not how the force works! – Asaf Karagila Mar 05 '22 at 23:58
  • @AsafKaragila Oh no. Perhaps my question wasn't clear. Based on my math notes, I know that FIN should not be sigma-ideal. But I don't understand why. Here is my (incorrect reason). If E1, E2 ... En \in J, then it should be that E1 U E2 U E3 ... U En \in J.

    I hope that was clear. I apologize for confusion as I am not from a math background.

    Finally, I hope the force is going strong.

     – desert_ranger Mar 06 '22 at 00:01
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    Unions of infinitely many sets are much bigger than just the union of finitely many near the start, even if it's the first $1000000000000$. It doesn't follow that $\bigcup_{n = 1}^\infty E_n \in J$ from the fact that $\bigcup_{n = 1}^N E_n \in J$ for all $N$ (in fact, that's more or less what you're supposed to be showing). Can you think of any way to write an infinite set as a union of a sequence of finite sets? – Izaak van Dongen Mar 06 '22 at 00:06
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    You've fallen into the trap of "if true for all finite $n$, then true for countable infinity". But that is obviously wrong. It is true that for every finite set of natural numbers there is an upper bound. Is it true for countably infinite sets of natural numbers as well? – Asaf Karagila Mar 06 '22 at 00:12
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    See also my answer here. – Asaf Karagila Mar 06 '22 at 00:13
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    A difference between finite and infinite unions is that a finite union of finite sets is a finite set, whereas an infinite union of finite sets can be an infinite set. – Gerry Myerson Mar 06 '22 at 00:53

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