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Pressure == 0 at the center of a mass?!

Oh dear, I really messed that up didn't I? Maybe I should go
back to emacs... The joke is on me, but please wade through
this again.

Gene Roberts wrote:
[...]
> With gravity pulling away equally in all directions from the center of a
> planetoid there would be little or no weight and little or no pressure.

Seems like a physics problem!

I am having trouble with this idea of zero pressure at the center of an 
object. I have no problem with zero weight and unchanged mass.

Firstly, gravity attracts, or pulls towards, not pulling 'away equally 
in all directions'. I would say that all parts of the object are 
compressing and attracting equally at the center, so you fell pressure 
from all the weight in every direction
above you.

I have tried to analyze this with thought experiments and all come to 
the same conclusion, please enlighten me in my mistakes in the
following reasoning:

Suppose you start with a homogeneous sphere of material as the only mass 
in the universe.

Now carve out an element from this shaped like ice cream two cones 
formed by rotating a diameter about an angle. Like this gross ascii 
diagram:

     -------
     \     /
      \   /
       \ /
        *
       / \
      /   \
     /     \
     ------- 

I hope that is clear - there are two 3-D ice cream cones and a '*' at 
the center. Pray that my word wrap doesn't mess that up.

Now comes the experiment. Take one half of this cone. Nothing else 
exists in the universe. Now place yourself, and the lens of the 
Peekskill car, at the '*'. 

You will be standing on a sharp point feeling a gravitational force that 
would seem to radiate as if all the mass of the cone were positioned at 
the center of mass of that cone. 

No doubt the sharp point would pierce your skin and you would sink a bit 
under your weight. The lens might even smash a bit, surely it will be 
scratched. 

Now superimpose the other cone opposite you so you are at the '*' and 
the cones face each other as in the original diagram.

The two equal masses pull you equally (and each other) so you feel no 
net weight, but the two points are now starting to really hurt a lot 
more since the two cones are attracting each other as well. The lens is 
now shattered and two people just snatched them away.

Here comes a leap, applying superposition (all I remember from college 
physics):

I see no reason why the effect would change depending on the angle of 
the cone that we started with, so let's make the angle 180 degrees and 
now you are quite smashed, but weightless at the center of a sphere. The 
pressure isn't zero at all but increases as the angle increases and as 
the diameter (total mass) increases. There's nothing but small particles 
of the Peekskill lens, enough for us all to have a crumb.

Of course there's levity above, not to offend, but to enlighten. But 
still I can't see any flaw in the logic even if the lens is scratched.
References: