Set Theory – Inconsistent Use of Subset Notation with Inequality of Numbers

Question

$A$, $B$ are sets, $x, y$ are real numbers.

(Often) in mathematical texts, the symbols $\subset$, $\subseteq$, $\subsetneq$, $<$, $\leq$ have the following meanings:

$A \subset B$ and $A \subseteq B$ mean the same thing – that $A$ is a subset of $B$, possibly with $A=B$. For $A$ a proper subset of $B$, we often write $A \subsetneq B$.

$x<y$ is a strict inequality, $x \leq y$ means that $x$ is less than or equal to $y$.

The line underneath is required to say there is equality in the numerical case, whereas in the case of sets that is not the case. Indeed, often in elementary/primary school, children would be taught this way of remembering the symbols. When reaching set theory, this is thrown out.

Is there any reason for this inconsistency? Is it simply because of tradition and changing it to be more consistent would therefore be confusing? Or is there a justification for this difference?

Answer 1

The use of $\subset$ to allow equality is due to Bourbaki, who also wanted to use $<$ to allow equality and introduced the use of "positif" to mean $\geq 0$ rather than $> 0$. They had partial success: $\subset$ and positif have the Bourbaki meanings, but $<$ still has its traditional meaning. See the comments to Eremenko's answer to the stackexchange post here.

Bourbaki's interest in allowing $\subset$ to include equality is almost certainly due to it being the more commonly encountered type of subset condition compared to strict containment. I'm not trying to justify the change in meaning, just explain why such a change would have been proposed at all.

While it may be unfortunate that we are now stuck with a clash of meanings between $<$ on numbers (and posets) and $\subset$ on subsets, I didn't even notice the inconsistency until it was pointed out to me a couple of years ago. If I had ever noticed it before, I'd long since forgotten about it and simply got used to the standard usage of $\subset$.

Haussdorff's 1914 book Grundzüge der Mengenlehre, which was the first book on set theory, can be read on archive.org here. Subset notation is introduced on page 3, where he writes

Wenn alle Elemente der Menge $A$ auch Elemente der Menge $B$ sind, so sagen wir: $A$ ist in $B$ enthalten, $A$ ist eine Teilmenge von $B$, eine Menge unter $B$, oder $B$ enthält $A$, $B$ ist eine Menge über $A$. Wir bringen dies durch eine der beiden Formeln $$ A \subseteq B \ {\rm oder} \ B \supseteq A $$ zum Ausdruck; wobei die Zeichen $\subset$ $\supset$ an die üblichen Zeichen $<$ $>$ für kleiner und größer erinnern, aber doch von ihnen unterschieden werden sollen.

Translation:

If all elements of the set $A$ are also elements of the set $B$, then we say: $A$ is contained in $B$, $A$ is a subset of $B$, a set under $B$, or $B$ contains $A$, $B$ is a set over $A$. We express this by one of the two formulas $$ A \subseteq B \ {\rm or} \ B \supseteq A, $$ where the characters $\subset$ $\supset$ are reminiscent of the usual characters $<$ $>$ that mean less than and greater than, but should be distinguished from them.

On page 4 he writes

Wenn $A$ Teilmenge von $B$, nicht aber $B$ Teilmenge von $A$, die Gleichheit beider Mengen also ausgeschlossen ist, so schreiben wir auch $$ A \subset B \ {\rm oder} \ B \subset A; $$ in diesem Falle wird $A$ gelegentlich als echte Teilmenge von $B$ bezeichnet werden.

Translation:

If $A$ is a subset of $B$, but $B$ is not a subset of $A$, so the sets can't be equal, we write this as $$ A \subset B \ {\rm or} \ B \subset A $$ and in this case $A$ will occasionally be referred to as a proper subset of $B$.