In supermarkets in Germany I often hear "Mit Karte" as an answer to how I want to pay. And the question I think usually sounds like "Wie wollen Sie zahlen? Mit Karte oder bar?"
So the question is why do people have no article there? In English I think it should be "I pay with a card". And this should not be different in German. Karte is countable and is not a profession. Does there exist a rule why it should have no article or is it just a slang to say it faster in supermarkets?
Note, that German knows the so-called Nullartikel to be applicable:
Das Nomen erhält keinen Artikel vor Abstrakta, die allgemeine Eigenschaften und Gefühle ohne nähere Bestimmung bezeichnen. Die Abstrakta stehen im Akkusativ oder direkt nach einer Präposition.
It also applies to amount specifications (see Christian's last examples). In your example the card is completely unspecified and only used to distinguish from cash, so abstract perfectly matches.
When addressing tools (claiming card to be a payment tool), there are also quite close examples like mit Messer und Gabel, mit Schaufel und Besen where any article would seem strange.
Here is an (improvised) attempt to explain this.
The general rule is:
1) With a noun in singular, use always an article - either definite (Ich sehe das Pferd) or indefinite (Ich sehe ein Pferd), depending on what fits your communicative situation. You use definite article when you suppose that the object of your conversation is already known to your interlocutors because the object of conversation had already been mentioned. You use indefinite if you suppose that the object of conversation is new to this communication.
2) In plural, you use the definite article when you suppose that the objects of your conversation are known to your interlocutors (i.e. have been part of previous communciation): Ich sehe die Pferde. If not, you do not use any article: Ich sehe Pferde.
Now, in the case you presented:
Zahlen Sie mit Karte oder bar?
something different happens.
My first thought was that situation you present might be so common, and the sentence uttered so frequent that users of German have tacitly agreed that they can go without any article, just for brevity.
However, look at these cases:
Hier ist Ihr Tee. Möchten Sie Zucker dazu?
Hier ist Ihr Brotzeitteller. Möchten Sie Rettich dazu?
At least the last example should prove my first theory was wrong. It is not a frequent situation that one person asks another if she or he does want radish on top of his or her dinner plate.
(Note however that you also could say: "Möchten Sie einen Rettich dazu?", "Möchten Sie Rettiche dazu?", where just your way of referring to the vegetable is different. You are speaking of one individual root in the first case, and of multiple (but countable) roots in the second. - In "Möchten Sie Rettich dazu" the quantity is completely irrelevant. It could be just a piece cut off a whole radish, or any other practically possible quantity.)
Back to "Möchten Sie Rettich/Zucker dazu", I would suppose, both Zucker and Rettich are seen as general representations of their class of things that do not need to be separated explicitely into "things we already spoke about" vs. "things we have not yet mentioned". In Guidot's answer we find for this the notion of "abstract" use of the word. Which seems to fit well.
One of the properties of German nouns is 'Zählbarkeit' (countability). Traditionally, German grammars (and some dictionaries) make a difference between countable and uncountable nouns and take it as a fixed feature, e.g. 'Obst' being uncountable and 'Frucht' countable.
Christian's 'Rettich'-Example is a good demonstration of the fact that a noun's countability can be influenced and even 'abrogated'. Normally, 'Rettiche' can be counted, but you can also use it as an uncountable mass noun like in
ich mag Rettich nicht
You could also say
ich mag Banane nicht
As a result I think that (un)countablity is more an aspect than an unalterable property of a noun. Guidot calls it 'regarding a noun as abstract'.
Another example:
Ich finde, Brille steht dir nicht.
The sentence doesn't mean the specific glasses the person is wearing now but any glasses at any time, glasses at all. It works like an adjective, like 'Rot steht dir nicht.'
This example may be interpreted as a generalisation, which is also the case for 'mit Karte zahlen'. One could say 'mit meiner Karte zahlen', with one specific individual of a cash card, but that is not what he wants to say, he just doesn't want to pay cash.
Another point ist that there are numerous examples for the integration of a noun into the predicate by omitting the article. In those cases a noun is not regarded as a noun in the proper sense but as a part of the verb like a separable verb particle:
'ich fahre nicht mehr oft Auto, das ist mir zu stressig'
'sie fährt jedes Wochenende weg'
Before the spelling reform some of those predicates had to be written in one word like 'radfahren' and 'maschineschreiben'.
In the same way, there are many 'Funktionsvergefüge' without an article like 'in Frage stellen' or 'Kritik üben'. So 'mit Karte zahlen' could be a 'big verb'.
Finally, the prepositions 'mit' and 'ohne' might tend to be used without an article like in
versuch's mal mit Schraubenzieher,
which is not standard German but everyday spoken German. In the same way you can ask
fährst du nach Karte oder mit Navi?
In addition to that, you can find more or less idiomatic uses of the 'Nullartikel' like 'vor Arbeitsantritt', 'wegen Trauerfall geschlossen', 'bei Stau fahren Sie am besten ...', but they appear to be generalisations.
So, 'mit ø Karte zahlen' may be explained
As an historic example of this point, when John Kennedy made his famous visit to Berlin and said "Ich bin ein Berliner", his remark caused some amusement as it was humorously translated as "I am a doughnut", "Berliner" with the article "ein" meaning a local pastry. The more correct way to state what he meant would have been "Ich bin Berliner", i.e. without the article "ein".
