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My undergraduate University does not offer advanced courses on logic, I know truth tables, Boolean algebra, propositional calculus. However I want to pursue Mathematical Logic on the long term as a mathematician.

Can anyone suggest a study-map of Mathematical Logic.

such as

(1) Learn The following topics : a,b,c,etc..

(2) once you learned topics in (1), advance to these topics.

(3) ..

(4) etc..

Thank you

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closed as off-topic by Najib Idrissi, Claude Leibovici, Mike Miller, Michael Medvinsky, Justpassingby Dec 14 '15 at 12:54

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "Seeking personal advice. Questions about choosing a course, academic program, career path, etc. are off-topic. Such questions should be directed to those employed by the institution in question, or other qualified individuals who know your specific circumstances." – Najib Idrissi, Claude Leibovici, Mike Miller, Michael Medvinsky, Justpassingby
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up vote 5 down vote accepted

Your first step would be to learn about predicate calculus (first order calculus or propositional calculus with the addition of 'for all' and 'there exist' quantifiers). This will then lead into the major subcategories:

  • proof theory (syntactic manipulation of proofs, proof calculi, ordinals)
  • model theory (models and structures of other areas of mathematics)
  • set theory (axioms of sets, ZF, the axiom of choice, the general continuum hypothesis)
  • recursion theory (decidability, recursive functions, lambda calculus, with connections to theoretical computer science subjects like computational complexity)

Of course, as with any body of knowledge, these categories are not mutually exclusive.

These are specialty areas, meaning that research is done under one of these headings. To get into one of these areas, there are many 'elementary' concepts leading in that you probably would want familiarity with. An accessible topic like Goedel's theorems (the completeness theorem and two incompleteness theorems (these are not the same 'kind' of completeness)) will help get some of these necessary concepts.

Also a good area to be familiar with is philosophy of math and foundations; though not exactly mathematics or logic, it informs a lot of the motivations of study of the above categories.

Logic that is not mathematically based can be pursued in philosophy (human language arguments, deductive and inductive reasoning, vagueness), mostly in the area of epistemology.

Your mathematics department may not have advanced logic, but other departments may (though not necessarily by name. Consider looking in the philosophy department (some mathematical logicians get hired there) or the computer science department (the research may be more applied but they may give courses that are theoretical/basic math).

Executive summary: start with predicate calculus and Goedel's theorem.

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This(similar question on MO) would be helpful.

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I recommend Teach Yourself Logic by Peter Smith as an extremely helpful resource. In addition to detailed textbook suggestions, it has descriptions of the basic topics and the key points that you want to learn regardless what book you learn them from.

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