The axiom of choice as a consequence of a stronger semantics? On what grounds?

"It is not people who break ethical standards who are regarded as aliens. It is people like me who are isolated." (Grigori Perelman)

  • $\begingroup$ Perhaps it is worth mentioning that edits to deleted answers bumps a question. (You can see that the question you answered is among recently active questions if you choose "active" tab rather than "newest" when viewing questions.) $\endgroup$ Commented Jul 23, 2017 at 15:19
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    $\begingroup$ Yes, please stop the edits, as requested by the moderators. $\endgroup$ Commented Jul 23, 2017 at 16:28
  • $\begingroup$ @AndrésE.Caicedo Ok. Perhaps. Can you tell me, what is not clear in my answer? $\endgroup$
    – user111966
    Commented Jul 23, 2017 at 16:55

2 Answers 2


Here is the post in question:

Any set has no "semantic structure". In sense: $\emptyset \notin \left \{ 0 \right \}$, because $\left \{ \emptyset \right \} \neq \left \{ 0 \right \}$. I think this statement contradicts the notion of Interpretability (https://en.wikipedia.org/wiki/Interpretability), because this notion is informally based on some part of set theory - more precisely, any formal theory $\mathbb{T}$ is a set. It follows from the definition of Interpretability, that any formal theory (set) has a "semantic structure". In sense: $\left \{ 0 \right \}\Leftrightarrow \left \{ \emptyset \right \}$; $\mathbb{T}\cup \left \{ 0 \right \}\Leftrightarrow ZFC^{-}\cup \left \{ \emptyset \right \}$, where $ZFC^{-}\cup \left \{ \emptyset \right \}$ is equivalently to $ZFC$.

We have $x \in \left \{ y \right \} \Leftrightarrow \left \{ x \right \}=\left \{ y \right \}$, but $\forall y\exists x(x\in y)$ is not true, because $\left \{ x_{y} \right \}\subseteq y$ and $\left \{ x \right \}\neq \left \{ x_{y} \right \}$, on the other hand, based on Interpretability, we have $\left \{ x \right \}\Leftrightarrow \left \{ x_{y} \right \}$. I saw that $X\Leftrightarrow Y$ is equivalently to $X= Y$, but i'm not shure that $\left \{ x \right \}=\left \{ y \right \}$ is equivalently to $\left \{ x \right \}\Leftrightarrow \left \{ y \right \}$.

The answer was flagged by a logician and set theorist as "not an answer". And indeed it seemed to me that (putting aside for the moment any deficiencies in presentation or coherence) this does not answer the question that was asked. If others feel that this is an answer relevant to the question (it doesn't have to be a correct answer), please say so.

  • $\begingroup$ If $\left \{ \emptyset \right \}\Leftrightarrow \left \{ 0 \right \}$ (this is a consequence of Interpretability), then $\left \{ \emptyset \right \}\in \left \{ 0 \right \}$ is true, because $\left \{ \emptyset \right \}\Leftrightarrow \left \{ 0 \right \}$ is equivalient to $\left \{ \emptyset \right \}= \left \{ 0 \right \}$. Only in the question it is a bit more complicated formulated. Is it relevant? $\endgroup$
    – user111966
    Commented Jul 19, 2017 at 15:11
  • $\begingroup$ It's pretty clear what I added? $\endgroup$
    – user111966
    Commented Jul 21, 2017 at 7:22
  • $\begingroup$ I and others saw that you added more text to the deleted post, yes. And that stuff has been flagged as well. $\endgroup$
    – Todd Trimble Mod
    Commented Jul 21, 2017 at 10:15
  • $\begingroup$ I knowingly wrote a special case in the form of an answer. Is this case understandable? How do I ask this? Perhaps I need to describe in more detail what the notion of Interpretability. Can not this be understood? Or is the overall picture not visible, at least in particular? $\endgroup$
    – user111966
    Commented Jul 21, 2017 at 16:27
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    $\begingroup$ It's not understandable to me and it looks like gibberish. However, I do not call myself a logician. Please do not edit any further. If a logician or set theorist (with sufficient reputation to see the deleted answer) comes along and sees the point you are making and that it is relevant to the question of the OP, then we can reopen. However, professional logicians have so far informed me they consider your post as being without merit. This discussion should now end. $\endgroup$
    – Todd Trimble Mod
    Commented Jul 21, 2017 at 18:29

Post A: Users who have "Galucie" as part of their user name are more likely to be welcomed on foreign language forums on StackExchange than as entrants to The Westchester Kennel Club Annual show.

I do not know whether Post A has true or false content, nor how Post A will impact readers. I was looking for an example to underscore a point being made. In my view, Post A is as relevant to addressing your meta question ("why was my answer deleted") as your deleted post was to addressing Pace's question regarding the axiom of choice and semantics.

The problem is that spammers do something that looks similar to call attention to something away from the purposes of the forum. Regardless of your intentions, your (then undeleted) post seemed in the wrong place to me and to others. If you think this will recur, you can try asking in meta about the relevance of a response. Some of us will address one or two honest efforts of this type with guidance on how to make good posts to MathOverflow.

Gerhard "This Response May Be Deleted" Paseman, 2017.07.19.


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