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On pp. 6-7 of his Naive Set Theory, Paul Halmos proves a result that I know very well, but I have a hard time following the argument he gives here. This question is about Halmos' argument (and not about the result itself, which I understand well enough).

I'll quote Halmos at length to make sure I don't miss anything. (I'll be as faithful as possible, but I will change his notation slightly, since he uses $\epsilon$ for $\in$, and $\epsilon^\prime$ for $\notin$; I will stick to $\in$ and $\notin$). The excerpt below begins right after he has given the Axiom of specification. (In the excerpt, I'll use boldface to indicate what I'm having trouble with.)

...To indicate the way $B$ is obtained from $A$ and from $S(x)$ it is customary to write $$B = \{x \in A:S(x)\}$$

To obtain an amusing and instructive application of the axiom of specification, consider, in the role of $S(x)$, the sentence $$\mathrm{not}\;(x \in x).$$

It will be convenient, here and throughout, to write $``x \notin A"$ ... instead of $``\mathrm{not}\;(x \in A)"$; in this notation, the role of $S(x)$ is now played by $$x \notin x.$$

It follows that, whatever the set $A$ may be, if $B = \{x\in A:x\notin x\}$, then for all $y$, $$(*)\;\;\;\;\;y\in B\text{ if and only if } (y\in A\text{ and } y\notin y).$$

Can it be that $B \in A$? We proceed to prove that the answer is no. Indeed, if $B\in A$, then either $B\in B$ also (unlikely, but not obviously imposssible), or else $B\notin B$. If $B\in B$, then, by $(*)$, the assumption $B\in A$ yields $B\notin B$—a contradiction. If $B\notin B$, then, by $(*)$ again, the assumption $B\in A$ yields $B\in B$—a contradiction again. This completes the proof that $B\in A$ is impossible, so we must have $B\notin A$.

(In the subsequent discussion, Halmos argues that this result means that "there is no universe [of discourse]", etc., etc.)


Now, this is what I don't get. If I replace $y$ with $B$ in $(*)$, I get

$$B\in B \text{ if and only if } (B\in A\text{ and } B\notin B).$$

This means that $B\in B$ and $(*)$ together imply $B\notin B$. Contrary to what Halmos writes, "the assumption $B\in A$" is not needed to draw this conclusion.

What am I missing?


(By the way, I would have worded the second part of the proof a bit differently, namely:

If $B\notin B$, then, by $(*)$ again, the assumptions $B\in A$ and $B\notin B$ jointly imply that $B\in B$.

I mention this only in case this rewording gives a clue about where my confusion lies.)

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  • $\begingroup$ There is rigor and clarity lost when you write $x\in x$ as $x$ is both a set and an element of the set, so there is an ambiguity of what $x$ is describing in any instance. This is a linguistic ambiguity, and an equivocation of two completely different notions. In my view. $\endgroup$ – CogitoErgoCogitoSum May 18 '18 at 17:37
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    $\begingroup$ @CogitoErgoCogitoSum: The very point of this argument is that $x$ is the same thing on both sides -- we're asking whether the set presently being called $x$ is an element of itself. $\endgroup$ – Henning Makholm May 18 '18 at 17:45
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You are right that the appeal to $B\in A$ in the bolded part of your quote is not necessary for the conclusion it's trying to draw.

I'd chalk it up to sloppy editing and move on.

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