Snow Leopard 10.6 default boots into 32-bit kernel, but for developers working on 64-bit kernel extensions, Snow Leopard could boot a 64-bit kernel. This gave developers over 2 years to create 64-bit kernel extensions and drivers.
Lion 10.7 default boots a 64-bit kernel, but was able to also boot a 32-bit kernel for users that had 32-bit kernel extensions and drivers that had not been updated to 64-bits yet, giving developers another year to create those drivers or the users more time to find a replacement, if the developer was not interested in creating the 64-bit kernel extensions.
Mountain Lion 10.8 "Always" boots a 64-bit kernel, and as far as I know does not have a 32-bit version.
You profile says "Mountain Lion 10.8", but the "Categories" line says this is about Snow Leopard 10.6 and your quesiton says Snow Leopard. I just want to be sure which system you are checking.
Which system are you checking. Mountain Lion Applications -> Utilities -> System Profiler does not need to tell you 32 vs 64-bit kernel, since it is always a 64-bit kernel. Snow Leopard and Lion System Profiler should both tell you if the kernel is booting a 32 or 64-bit kernel.
You can also check via an Applications -> Utilities -> Terminal session:
system_profiler | grep -i kernel # this is from Lion, but Snow Leopard will be similar Kernel Version: Darwin 11.4.2 64-bit Kernel and Extensions: Yes
NOTE: Regardless of the flavor kernel booted, Mac OS X will run both 32 and 64 bit apps on any of these platforms. Actually Mac OS X has been running 64-bit apps on 32-bit kernels since at least Tiger 10.4.
Can you explain the why you wish to know the flavor kernel running on your system?
Message was edited by: BobHarris