"The set is empty and [is] ordered"

Which is correct?

• The set is empty and is ordered.
• The set is empty and ordered.
• Both are correct and both mean the same thing. In that sentence. But only by arbitrary definition; since empty sets have no elements, any ordering relation on them is superfluous. Still, one could define an empty ordered set easily: the set of all integers between 1 and 2 requires an ordering relation to define at all, and it is clearly empty. Oct 9, 2014 at 17:49
• In my understanding, ordered here means the elements are arranged in some specific sequence. I can't get my head around how an empty set (containing no elements) can fit that description. Oct 9, 2014 at 18:03
• Is this a mathematical question or a grammatical one? Oct 9, 2014 at 20:42

Consider these options;

'The set is empty and it is ordered.'

'The set is empty, it is ordered and my work here is finished.'

'The set is empty, ordered, and soulless.'

'The set is empty and ordered but I'm still not happy'.

• It is good to know we can consider "white car", "black car", orange car", and "purple car". However, the question is actually about "red car" vs. "blue car". Oct 9, 2014 at 20:10

Both are equally grammatical and mean the same thing. The latter is ellipsis of the former. Which in turn is ellipsis of "The set is empty and the set is ordered".

• @RegDwigHt, Perfect. "red car" vs. "blue car" answered. Oct 9, 2014 at 20:31

*The set is empty and ordered.

These statements are illogical

• The set is empty and is ordered.
• The set is empty and ordered.

The following is also illogical

• The empty set is ordered.

Because, if a set is empty, there would be nothing to order or sort.

Mathematics say that the logical way to say it is

• The ordered set is empty.

You can have a frame defined to contain ordered list of things, and such a frame could be empty.

Further illustration of the logic. Compare

• The empty wok is cooking.
• The cooking wok is empty.
• The empty pot is boiling
• The boiling pot is empty.
• The pot is empty and boiling.
• The pot is empty and is boiling.

This is when the differences between a participle and a gerund is crucial: Gerund ending in -ings?

• I'm sorry, you remind me of an editor that insisted that the author (a family member) change the "pH" in the title of a biology paper to "Ph" because "the capitalisation is wrong", demonstrating a profound lack of domain knowledge. This math SE question discusses just the matter relating to the OP's sentence, and the answers give a proof that the empty set is, in fact, totally ordered. Oct 10, 2014 at 5:49
• Depending on the assumed maxim, we could also question if zero is an even number, except by stating that an even number results in zero residual when divided by two. If we presume that a set is ordered due to the absence of disorder then an empty set is "ordered". However if we ask if any effort had occurred to create order, as a means to qualify the use of the participle "ordered", then the empty set is not "ordered". Just as an empty pot is not and cannot be boiled. Oct 10, 2014 at 6:24
• Yes, that is intuitive, casual interpretation. OP's sentence is in mathematical jargon, where "ordered" and "empty set" have very precise definitions - no presumptions required. A set is not ordered if someone ordered it, it is ordered (under a certain relation) if it satisfies certain properties, and empty set (trivially) does (under the only relation it has), as explained in the link I provided. Thus I strongly disagree with telling the OP his statements are illogical: he likely knows his domain better than you do, and is only asking about syntax. Oct 10, 2014 at 7:28