If a cpu is not hooked up to a fan and heatsink for
more than 5 seconds(with power on) it will overheat
and the core will break down.
No it's not true, but it could be.
You're trying to say something very specific when there are
several variables such as which CPU, and what "hooked up"
means.
In general, if you have a heatsink properly installed but no
fan, (including no duct with a fan at the exhaust like many
OEMs might use), it will always survive far longer than 5
seconds, would gradually overheat after a few minutes or so.
At that point, some would throttle to a lower speed, but
either way it would become instable and the operating system
or other software/bios menu/whatever would crash. If it
continued to heat up further and the system was newer, it
would shut off. If the system was older, continuing to run
in this state, after it had already crashed if it were left
like that overheating for a long time, it would tend to
cause damage many minutes later, but again it depends on
variables like how well the system case was cooled, what
peak temp it actually attained.
If there is no heatsink on it at all, yes it will probably
be damaged if turned on from cold-off state. "Maybe" if you
have a modern CPU with integral sensor and shutdown, you
might get lucky and it wouldn't be damaged before it could
shut off, but it is still quite possible because the
shutdown circuit is not designed well enough to cover
situations where the heat goes up as rapidly as it can
without any heatsink at all on it, that is just too fast a
rise in temp and the CPU can heat unevenly which will pop
off the heat spreader and crack the core.
In general if you didn't have a heatsink and fan on for 5
seconds and had tried to use the system, the CPU should be
set aside as an unknown variable until it can be tested in a
known working system (if the system it was in was not
working at this point).