The Maths necessary to understand Logic, Model theory and Set theory to a very high level

Solution 1:

Apart from actually learning logic, set theory and model theory you would probably benefit from some basic understanding in

• Abstract algebra (group theory, ring theory, etc.)
• General topology
• Some basic measure theory
• Computability and complexity

While these things are not necessary per se in order to gain understanding in logic, or set theory (although model theory deals a lot with actual mathematics, so you can't escape it there); in order to fully understand set theory I think that one has to see "usual" mathematical theories and understand them at a basic level. If not for anything else, then in order to understand what makes set theory special.

It seems, if so, that the better part of an undergrad degree in mathematics is needed. But then again, it is needed if you wish to understand any mathematical theory in depth.

Solution 2:

I would say that if you get hold of three books titled respectively "Mathematical Logic", "Model Theory" and "Axiomatic Set Theory" then that is pretty much all you need. Some or all of them should probably have the word "Introduction" in the title too.

I do not believe that calculus would be much help (but I understand that the elements of abstract algebra would be.)

If you want more precise recommendations for books, I am sure we can help there too.

Solution 3:

This is a bit tangential to your question, but if your ultimate interest is in the philosophy of mathematics, I would believe that some knowledge of category theory, as a possible alternative to set theory as a foundation for mathematics, would be important. The texts by Steve Awodey (Category Theory) or F. W. Lawvere and Stephen Schanuel (Conceptual Mathematics) may serve as useful introductions.