 With respect to its “external” dimension, an idea can have several (may be even infinitely many) objects, exactly one object, or no object at all. An idea that has no object at all is an empty idea; Bolzano calls it ‘gegenstandlos’ (‘objectless’). Bolzano puts forward the thesis with particular emphasis that there are empty ideas; his standard examples are ideas such as [nothing], [golden mountain] (WL I, 304 f., WL II, 329) or [winged horse] (WL III, 24). A special kind of empty ideas, viz. contradictory ideas (or, as Bolzano usually prefers to call them, imaginary ideas) cannot even have an object (WL I, 315 ff., WL III, 405 f.), examples being [a round polygon], [a round square], [a triangle that is quadrangular], [a regular pentagon], [a wooden iron tool], [an equilateral rectangular triangle] (WL I, 305, 315, 317, 321, 324, WL II, 329). Non-empty ideas are called ‘gegenständlich’ (‘objectual’) by Bolzano. They can be singular as, e.g., [the philosopher Socrates], [the city of Athens], [the fixed star Sirius] (WL I, 306), [an even integer between 4 and 8] (WL III, 407), [God] (WL III, 408), or general; if general, they can have a finite number of objects, such as [a heir of Genghis-Khan’s Empire] (WL I, 299) or [an integer between 1 and 10] (WL I, 308), or an infinite number of objects, such as [a line] or [an angle] (WL I, 298). For non-empty ideas (and only for them) Bolzano defines their extension (Umfang) (WL I, 297 f.); by using again the modern concept of a set (as we already did with Bolzano’s definition of the content of an idea), we can extend his definition to all ideas including empty ones; the extension of an arbitrary idea \(i\) (or Ext\((i)\), as an abbreviation) is then nothing but the set of all objects of \(i\).