Reddit reviews Constructing the World

David Chalmers's newest book Constructing the World takes up this challenge, particularly in the vein of Rudolf Carnap's Aufbau.

Here's the link to his book.

And here you can listen to the audio files of his 2010 John Locke lectures from which it came. In the first lecture he also talks about other contemporary philosophers who are working on projects to establish a universal logic to construct the world with a finite number of fundamental concepts, so that might give you some additional resources.

edit: added a few words to clarify

There's a lot of renewed interest in logical positivism these days. E.g., from David Chalmers, Michael Friedman...

It's a much more sophisticated philosophy than the caricatures suggest. There's been a polemic interest in caricaturing logical positivism since it was the movement we were supposed to be distancing ourselves from through the mid-to-late twentieth century. But as this polemic interest fades into history, there's naturally an increasing interest in more nuanced engagements with the movement.