application / proposal
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application
A formal, usually written request to an authority, company, or institution in order to receive something or have something approved.
Sie müssen einen Antrag auf Baugenehmigung bei der Stadt einreichen.
You have to submit an application for a building permit to the city.
In a formal or bureaucratic context, the most common verb combination is 'einen Antrag stellen' (to submit/file an application). You don't 'machen' (make) or 'geben' (give) an application; you 'stellt' (submit/file) it.
Für das Visum musste ich einen Antrag bei der Botschaft stellen.
For the visa, I had to submit an application to the embassy.
When saying what the application is for, you use the preposition 'auf' followed by the accusative. For example, you submit an application for child benefit or for funding.
Die Firma hat einen Antrag auf finanzielle Unterstützung gestellt.
The company submitted an application for financial support.
An important exception is the marriage proposal. Here the verb 'machen' (to make) is used: 'jemandem einen Antrag machen' (to propose to someone). The context is always romantic, not bureaucratic.
An ihrem Geburtstag hat er ihr einen Heiratsantrag gemacht.
On her birthday, he made her a marriage proposal.
In parliaments or committees, motions are often 'eingebracht' (introduced) or 'gestellt' (submitted). After that, the motion is 'beraten' (discussed) or 'abgestimmt' (voted on).
Die Opposition hat einen Antrag zur Steuersenkung eingebracht.
The opposition introduced a motion to lower taxes.
wish
'der Wunsch' and 'der Antrag' can both have to do with a request. But a 'Wunsch' is more personal and often informal: you say what you would like. An 'Antrag' is formal and usually written. You submit one, for example, to an authority, a company, or an insurance company so that something is approved. Learners may confuse the words because both express that you want something. When it is about a personal concern, 'Wunsch' fits. When it is about an official procedure, 'Antrag' fits.
Nach dem Umzug hat Maria einen Antrag auf Wohngeld gestellt.
After moving, Maria submitted an application for housing benefit.
Beim Frühstück äußert Tom den Wunsch, im Sommer ans Meer zu fahren.
At breakfast, Tom expresses the wish to go to the seaside in the summer.