Been tired of repeating this a thousand times.
AMD invented the 64-bit architecture which was backwards compatible with x86 or 32-bit. This architecture is known as 'amd64'. Intel because they are morons, on the other hand were pursuing a completely new 64-bit architecture named Itanium 64 or IA-64 which was NOT backwards compatible with x86 or 32-bit. This was to earn more money $$. IA-64 failed because of this reason. Then Intel licensed 'amd64' from AMD and calls it "Intel 64" now for obvious reasons.
amd64 is also called x86_64 or x64. Intel prefers calling it "Intel 64" because of obvious reasons.
So, every processor out there in the market now is an amd64 processor. This is the reason WHY you are able to use a 32-bit OS currently in your Core 2 Duo processor. And this is again the reason why you will be able to use 32-bit applications in a 64-bit OS if you install it.
Cleared?
Ignore what Windows 7 upgrade advisor says. Every 32-bit application works fine in Windows 7 64-bit.
There you go. See in the screenshot and see how many 32-bit applications I am running completely fine. example, Avast AntiVirus.
The only thing you need to take care of is, install 64-bit hardware drivers. That's it.