So i read the crusade and I understood What the Diminished 5th was.. And all the other intervals.

But in my book there's also an Augmented 4th and Augmented 5th

So Augmented 4th = Diminished 5th?

Augmented 5th = Minor 6th?
Yes, they are the same thing. But depending on the context you use one or the other.

Lets look at two scales that have that Augmented 4th/Diminished 5th interval, Lydian and Locrian.

Lydian is spelled 1 2 3 #4 5 6 7

Locrian is spelled 1 2 b3 4 b5 b6 b7

The reason different numbers are used is because it is easier to write down. It wouldn't make sense to spell a seven note scale with a number repeated.
^^The above is a Cryptic Metaphor^^

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