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Let me give an alternative ending to Noah's road map. The splitting point is at symmetric models.

After you've understood the basics of forcing well, you can switch to Kanamori's The Higher Infinite. In the chapter about the real numbers and forcing he again reviews forcing (and if you're new to this - such review is always good) and constructs Solovay's model in a very clear approach.

He avoids [1] talking about symmetric models (which can be a rather complicated tool) by using the "external" construction: we add some sort of generic set to $V$ then we consider an inner model of $V[G]$ which is $HOD(\mathbb R)$ or $L(\mathbb R)$, the latter being thrown around a lot in discussions about models of set theory without choice.

In Kanamori you can find a good introduction to large cardinals (if you haven't run into them in previous steps) which also play a role in Solovay's construction, although that appears in another chapter of the book.

I want to add that studying the construction of symmetric extensions is a good idea. This is an extremely illuminating construction which sheds a lot of light on how set theory works, at least this is how I felt in the past year. However for this particular case I think that using the approach of relative constructibility is better.

Footnotes:

1. This is not entirely true that Kanamori avoids the symmetric models because as it turns out all symmetric models are $HOD(A)$ (whatever that means) of some generic set $A$. In the case of Solovay's model it is just much simpler to use this sort of construction rather going through the complication of symmetric forcing.
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Let me give an alternative ending to Noah's road map. The splitting point is at symmetric models.

After you've understood the basics of forcing well, you can switch to Kanamori's The Higher Infinite. In the chapter about the real numbers and forcing he again reviews forcing (and if you're new to this - such review is always good) and constructs Solovay's model in a very clear approach.

He avoids [1] talking about symmetric models (which can be a rather complicated tool) by using the "external" construction: we add some sort of generic set to $V$ then we consider an inner model of $V[G]$ which is $HOD(\mathbb R)$ or $L(\mathbb R)$, the latter being thrown around a lot in discussions about models of set theory without choice.

In Kanamori you can find a good introduction to large cardinals which also play a role in Solovay's construction, although that appears in another chapter of the book.

Footnotes:

1. This is not entirely true that Kanamori avoids the symmetric models because as it turns out all symmetric models are $HOD(A)$ (whatever that means) of some generic set $A$. In the case of Solovay's model it is just much simpler to use this sort of construction rather going through the complication of symmetric forcing.