The axiom of choice as a consequence of a stronger semantics?

First we show that $\in$ is not a semantic symbol. Suppose that $\in$ is a semantic symbol.

Contradiction: We know that $x\in \left \{ y \right \}$ iff $x=y$ (syntactic definition) is true and $x\in \left \{ y \right \}$ iff $x\Leftrightarrow y$ (semantic definition) is not true, for example: $\emptyset\in \left \{ 0 \right \}$ iff $\emptyset \Leftrightarrow 0$ is not true.

Note: $x=y$ implies $x\Leftrightarrow y$, but not always vice versa.

Until we select $x$ (syntactically) in the expression $\forall y\exists x(x\in y)$, it can be semantically true and $\in$ will be a semantic symbol, but as we showed above this is not true, so we can't do not choose $x$. If we choose $x$, then we can take two different singletons of $y$ and both of them can not simultaneously be contain $x$ without semantics.

I want to notice remove and close questions the same people: Andrés E. Caicedo, Todd Trimble.

Is there an administrator on this site?

  • $\begingroup$ Because (contrary to your proof attempt) $\in$ is a semantic symbol. Also, your post does not address the issue raised in the question. It attempts to circumvent the question by considering a related issue, but this related issue has insufficient ties to the question. Gerhard "In Other Words, Rather Irrelevant" Paseman, 2017.10.24. $\endgroup$ Oct 25, 2017 at 6:50
  • $\begingroup$ @GerhardPaseman Can you write that question about which you speak? And how can it be semantic? $\endgroup$
    – user111966
    Oct 25, 2017 at 6:53
  • $\begingroup$ The question is written in the post, and is given an interesting and relevant answer by Prof. Hamkins. I suggest you work on understanding that answer to determine on what may be semantic. Gerhard "In Has Meaning To Me" Paseman, 2017.10.25. $\endgroup$ Oct 25, 2017 at 7:20
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    $\begingroup$ The answer is total gibberish. $\endgroup$ Oct 25, 2017 at 7:26
  • $\begingroup$ @EmilJeřábek Do you see this partially or in general? Just trying to understand the course of your thoughts. $\endgroup$
    – user111966
    Oct 25, 2017 at 7:31
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    $\begingroup$ To the extent of "administrators" are on the site, yes there are a few, and Todd Trimble is indeed one of them. $\endgroup$
    – Asaf Karagila Mod
    Oct 25, 2017 at 9:39
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    $\begingroup$ (Also, to the less-engaged, this is not the first time that this exact topic comes up. The OP has posted an answer a few months ago, which was deleted by a moderator, and the OP kept editing it until it ended up being locked to prevent further bumping of that question.) $\endgroup$
    – Asaf Karagila Mod
    Oct 25, 2017 at 9:39
  • $\begingroup$ @AsafKaragila Do you want to convince everyone in advance that I'm wrong? For two of your identical comments, I can assume that it is, do not overdo it. $\endgroup$
    – user111966
    Oct 25, 2017 at 10:13
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    $\begingroup$ Where did I say that your answer is wrong or not? I am just pointing out that this is not the first time you've posted an answer on that thread, and it is not the first time it was deleted by a moderator, and that it is not the first time that you have complained about this on meta. It's called transparency. $\endgroup$
    – Asaf Karagila Mod
    Oct 25, 2017 at 10:28
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    $\begingroup$ Possible duplicate of Why was my answer deleted? $\endgroup$
    – jeq
    Oct 26, 2017 at 14:36

1 Answer 1


I cannot speak for the people who voted to delete, but from my point of view, your answer had two problems. First, the English is written in a way that is quite difficult to parse. Phrases like "we can't do not choose $x$" do not occur naturally in the English-speaking part of the world. Second, it seems that the mathematical content in your answer (as far as anyone can understand it) does not seem to be relevant to the question. While I am not an expert in this field, it seems that several people with expertise in logic strongly agree. People usually vote to delete an answer when it is clear that the answer does not contribute substantially to understanding or resolving the question.

If you are looking for people with administrative duties, you will find the names of the moderators on this page. For example, Todd Trimble is a moderator (as am I). Please do not try to retaliate for a deletion.

  • $\begingroup$ English is not my native language as you can see. "we can't do not choose $x$". The phrase means: We should always choose $x$. Because if we do not choose $x$ there is semantics inside the expression, then the symbol $\in$ will be semantic, but this is not true. $\endgroup$
    – user111966
    Oct 25, 2017 at 8:47
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    $\begingroup$ In case you haven't seen that one before, meta.mathoverflow.net/questions/3311/why-was-my-answer-deleted is about the other, deleted and locked answer of the OP. $\endgroup$
    – Asaf Karagila Mod
    Oct 25, 2017 at 9:41

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