Why has Earth's core not become solid? The Earth is billions of years old, yet its core has not yet cooled down and become solid. Will this happen in the foreseeable future?
 A: *


*Earth is big, so takes a long time to cool down.  Note that Mars, being smaller, has already cooled down to this point.


*The heat from radioisotope decay over something with the mass/area ratio of earth is significant.


A: I generally take "foreseeable future" to mean "in my lifetime" in which case the answer is No. However, if you are really asking, "What does science say about the solidification1 of earth's core?" then we can answer this.
In the core of the planet, we actually have this pretty picture:

The "solid inner core" is pretty much pure iron at a nice pretty temperature of about 5700 K (strangely close to the sun's surface temperature), which seems to be greater than iron's melting point of about 1800 K, but the Clausius-Clapeyron relation shows that melting points change with an increase in pressure, so all is well there.
The reason the heat has been retained in the core is that the only processes to move the heat are convective transport and thermal conduction, the latter being a fairly slow process (aside: it's the convective currents that cause the magnetic field). Degeun et al (2013) state,

...the dynamical time-scale of the thermal convection in the inner core [is] $\sim$1 My or more.

Which, though shorter than the age of the planet, is still pretty long. But the inner core is also growing at a rate of about 0.5 mm/year (possible paywall) and has been doing so for about a billion years. Though unreferenced, this Earth.StackExchange answer suggests that the core has cooled only 250 K since it was formed (a rate of 55 K/Gyr). At that rate, "...it would take something like 91 billion years to cool to 0 K." So no, not in our lifetime for sure!
Note also that the mantle is being partially heated by radioactive decays of Uranium-238, Uranium-235, Thorium-232, and Potassium-40, all of which have half-lives of greater than 700 million years (up to about 14 billion years for Thorium). This will protect us for some time from the complete cooling of the core.
So, fortunately for our tiny planet, the core will stay nice and toasty warm for a few more billion years (my estimation). Unfortunately for the planet, the sun will go red giant phase and eat it up before the core cools.

I'm being liberal with this word. As I state elsewhere, the core is solid. OP intends it to mean a cold lump of iron rather than the toasty lump of iron that we have
A: Hot stuff on the surface of the Earth cools fast because air can carry away the heat pretty easily by convection. This heat is eventually radiated back to space.
The outer core, on the other hand, touches the hot mantle. Due to the relatively small temperature difference, there's not much heat transfer to the mantle. The mantle itself does slowly cool, but this is at the subduction zones in the ocean (which are relatively few compared to the entire surface area of the mantle), as the rocky crust insulates the mantle. 
Besides that, radioactive decay is a significant contributor to keeping the outer core molten.
So, no, the core won't cool significantly in the forseeable future.
A: Earth's inner core is already solid. However not because it is cold, but because of high pressures. Now for whether Earth outer core will freeze, well radioactive decay heats up Earth's outer core. Pressure also prevents Earth outer core from losing heat. Of course a part of Earth's outer core is solidifying. Lighter elements are solidifying on the surface of the outer core as the outer core loses heat to the mantle. This causes dynamical patterns and is one reason for why the Earth has a magnetic field. As this happens molten iron moves through these solidified lighter elements. Amplifying the magnetic field coming from deeper in the outer core.
A: Don't forget the gravitational title forces of the moon and the sun combined, that stretch and cool the earth, also generate heat within the planet.
Example, take a plastic spoon or something like that and start to slowly bend it – bend it back and forth slowly so it doesn't break and it will slowly start to get soft and as it gets soft, bend it back faster and faster until it actually gets hot enough to cause a first real first-degree burn in the spot that's been bending the most.
If we didn't have the moon, we would've ended up like Mars. Mars does not have any real moons to speak of – it has two large asteroids. If Mars had a moon similar to ours, it would've kept the center of the planet liquid and hot, which would have its magnetic field and prevented the atmosphere from being blown away by solar winds....
Mars once had a huge ocean and an atmosphere but then lost it… And the only way to lose that is to lose your magnetic field… And the only way to lose that is for the core of the planet to cool.
The amount of radioactivity based on the materials in our planet earth are not enough to keep our planet as warm and hot as it has been underneath the crust... Certainly radioactive decay has played a role in keeping the center of our planet warm… But the two main factors are high pressure and density combined with the title forces of the sun and moon. When the sun and moon are in opposition, the planet is slightly stretched out and is more oval-shaped...when the sun and moon are at 90° angles of the planet that pulls on the planet in a different manner… And when the sun and the moon are on the same side of the planet, that pulls that side of that side of the planet even further towards the sun and the moon.
So our planet is constantly being kneaded like bread dough....and if you ever kneaded bread, you know that the act of doing it keeps the dough warm.
Plus don't forget all of this tugging and pulling creates a huge amount of friction… And as we all know when you rub your hands together and generate friction – they get warm. The same thing with our planet.
Just do an Internet search for "what if the earth had no moon" and you will find many scientific articles that say we would be just like Mars or close to it nowadays. I'm not saying a moon is required for a planet to stay warm, but in our case, it is the most contributing factor to prevent our core from being cold by now.
