Have you ever tried to explain/inflict forcing on your non-set theory friends? Let me tell you it is hard. In my ongoing effort to try to explain everything to everyone here is my attempt at explaining the idea of forcing, without explaining forcing.

First, I direct you to this post which explains the simulation argument.

tl; dr– Imagine that our civilization could simulate (artificially) intelligent civilizations. They wouldn’t know that they are in a simulation and could also run their own simulations.

Reading this post reminded me about forcing. Here is my reply to the OP (who happens to be my brother-in-law), and I would appreciate feedback on this so that I can refine my analogy:

Cool. This illustrates a key observation that underlies a lot of the mathematics (set theory) that I do. It goes like this for the interested parties:

We run a simulation as you’ve described, but we make sure that the simulation only has “a small number of things in it”. Perhaps we have some sort of minimal simulation, like we don’t include the letter Z in their languages or something (Call this SIM1). Now we check that the simulation can come up with its own simulations. It can? Great! So now we know that “having the letter Z in your language” is not a requirement for coming up with simulations. Or we could add a whole bunch of new crazy letters (in say SIM2) and see if they can still run simulations. Lets say they can still come up with simulations.

So what do *we* now know? The ability to come up with simulations is independent of the alphabet a civilization has. Moreover, a civilization cannot tell whether it has “the entire alphabet”. SIM1 doesn’t know about the letter Z, but can still run simulations. SIM2 does know about Z (and some other stuff), and can still run simulations. So the ability of a civilization to run simulations does NOT depend on its alphabet.

This is the idea of a ‘model’ of something. Here a simulation is a model of civilization. It is a structure that has most of the defining properties of a civilization (like intelligence and the ability to run simulations) but lacks some other properties (it might have a different alphabet). This is used to show that certain things cannot possibly be proved or established as fact by a civilization.

Another example that might help you digest this:

We can imagine a simulation where everyone in the simulation is (say) blind. This would tell us that sight is independent of the ability to make simulations. Something like that.

The math part of this is called “[Paul] Cohen’s method of forcing”. You ‘force’ some statement P to be true in a model (of mathematics), so that it is actually independent of mathematics. This is how we establish that there are things that are neither provable nor refutable. BAM!

Awesome analogy! See Joel Hamkins’ set-theoretic geology work for why it is simulations all the way down…

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It’s not only about forcing, but like you first said, model theory in general. When one model “can simulate” another model, it is said to interpret that model.

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Nice! (and nice tags! :))

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