C°7 actually includes a B doubleflat .
A major seventh above C is B and a minor seventh above C
is B♭.
This means that a diminished seventh above C is actually B♭♭, which is enharmonic to A.
But the seventh is not A, because A is just a sixth above C. A is enharmonic to B♭♭, but since we want the seventh above C, it must be B♭♭, not A.
But it gets more interesting:
since fully diminished sevenths are just stacked minor thirds, they can be spelled four different ways, with each pitch as root; we say that these chords are enharmonic, just like we say that A is enharmonic to B♭♭.
This then means that C–E♭–G♭–A is a diminished seventh chord, but with A as the root: A–C–E♭
–G
♭.
If E♭ is the root, we'd rather spell it as D♯–F♯–A–C.
If G♭ is the root, we'd spell it as F♯–A–C–E♭.
These chords all sound the same as C°7, but they are spelled differently.
music.stackexchange.com/users/21766/Richard
Other answers have pointed out that it's a B♭♭, not an A.
To answer the question "
why not a B♭":
The chord in question is the chord of the diminished 7th.
As David Bowling and Richard have stated, if the root is C, the diminished 7th is B♭♭.
The chord is also called the "diminished 7th chord".
This name and its chord symbol Cdim7 are perhaps confusing.
I don't know if this is what you thought, but, just for the record, "Cdim7" does not
mean
the diminished triad C-E♭-G♭
as indicated by "Cdim" the pitch B♭ as indicated by "7", just as C7 means a C triad plus B♭
It's not a [ diminished triad ] with a seventh.
It's a chord of the [ diminished seventh ].
music.stackexchange.com/users/29126/Rosie F
A Cdim7 chord is actually spelled C-Eb-Gb-Bbb, where the interval from C to Bbb is called a diminished seventh.
The interval from C to A is a major sixth, but sometimes you do see people spell Cdim7 as C-Eb-Gb-A for convenience to avoid the double flat, it is just technically incorrect.
This chord is sometimes called a fully-diminished chord, in contrast to a half-diminished chord.
A half-diminished C chord ( Cmin7b5 , C-7b5 , or C ∅ ) is spelled: C-Eb-Gb-Bb.
These are often found in ii-V-i progressions in jazz.
music.stackexchange.com/users/39614/ex nihilo
It actually contains a Bbb (double flat).
Often (mis)spelt as A for convenience.
You'll also often see it notated as C, Eb, F#, A.
A dim7 chord is a pile of minor 3rds.
A minor 3rd up from Gb is Bbb.
music.stackexchange.com/users/17003/Laurence Payne
Adding a little to the others.
Cdim7 cannot contain a Bb note.
Bb only produces a minor 7th from the root.
To make it a diminished 7th, the space between root and m7 needs to be one semitone smaller.
That's the reason it gets called Bbb.
It sounds just like A, but since C>A is a M6, that wouldn't be correct if the chord was to be called dim7.
music.stackexchange.com/users/2808/Tim
CM7 - C, E, G, B  C7 - C, E, G, Bb  Cm7 - C, Eb, G, Bb  Caug7 - C, E, G#, Bb  But...  Cdim7 - C, Eb, Gb, A.  Why Does Cdim7 contain an A and not a Bb?
Why does Cdim7 contain an A instead of a Bb?
music.stackexchange.com
<theory><chords><chord-theory><harmony>
music.stackexchange.com/questions/75548
music.stackexchange.com/users/53114/Alex G-I
Since fully diminished sevenths are just stacked minor thirds, they can be spelled four different ways, with each pitch as root; we say that these chords are enharmonic, just like we say that A is enharmonic to B♭♭. A  Cdim7  chord is actually spelled C-Eb-Gb-Bbb, where the interval from C to Bbb is called a diminished seventh. The interval from C to A is a major sixth, but sometimes you do see people spell  Cdim7  as C-Eb-Gb-A for convenience to avoid the double flat, it is just technically incorrect. Sometimes people Cdim7  as C-Eb-Gb-A for convenience to avoid the double flat, it is just technically incorrect. 
A  Cdim7  chord is actually spelled C-Eb-Gb-Bbb, where the interval from C to Bbb is called a diminished seventh. Sometimes people Cdim7  as C-Eb-Gb-A for convenience to avoid the double flat, it is just technically incorrect. 