Given this sentence:
I wish I had shoes like that
Is it possible to say:
I wish I had shoes like those
Why is the first correct?
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Both shoes like that and shoes like those are acceptable, because the referent may not be the shoes themselves but some (singular) property which they possess. For instance, you may admire a particular colour; or an ingenious fastening device; or a peculiarity of design suited to a particular activity.
HE: I got these football shoes, you don't change the individual cleats, you snap a whole set on and off, like this (demonstrates) ...
YOU: Wow, I wish I had shoes like that.
So like that has such a broad range of possible referents that nobody really stops to worry about whether the referent in a particular instance is properly singular or plural.