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I'm currently studying abstract algebra and topology. And I want to prove theorems and exercises more precisely in terms of logic. So I'm studying logic and set theory with [Introduction to Mathematical Logic: Elliott Mendelson] but it's cumbersome for me. How can I study logic and write precise proofs? (I prefer axiomatic logic systems with simple axioms like hilbert system rather than lengthy and complex axioms.

prime235711
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  • A textbook on mathematical logic won't help much in learning how to write mathematical proofs. For a math-oriented introduction to formal proof, may I humbly suggest the tutorial that comes with my proof-checking freeware downloadable from my homepage at http://www.dcproof.com – Dan Christensen Apr 15 '20 at 14:16
  • Enderton is a very challenging read because of its notation. Enderton is also very flippant about proofs, at times. You may like Kueker's notes on mathematical logic. He does a better job at explaining and giving formal definitions: https://www.math.umd.edu/~dkueker/712.pdf For set theory, check out Roitman's axiomatic set theory: http://www.people.vcu.edu/~clarson/roitman-set-theory.pdf –  Apr 15 '20 at 16:45
  • Thank you very much all for answering me though it's closed! I'm sorry for my horrible english but that's the answer what I wanted. Thank you. – prime235711 Apr 16 '20 at 11:46
  • @prime235711: Dan's system is useless for practical mathematics. What you want is a Fitch-style natural deduction system such as this one (which I have actually used for practical mathematical research). Hilbert systems are not practical either. If you need help with formalizing some mathematics, you're welcome to ask me in this chat-room. – user21820 May 02 '20 at 03:05
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    @ms._VerkhovtsevaKatya: While you may be technically right in that every true proof resides in some formal system, mathematicians often give what is accepted as "proofs" that are not formal but can be made formal by any expert in the field. You may be surprised how often they make unnoticed use of certain assumptions that are not actually trivially true in the formal system that they claim to be using. – user21820 May 02 '20 at 03:09
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    For example, most textbook proofs that sequential continuity implies continuity use the axiom of dependent choice, and many professors using those textbooks do not even realize it! For another example, consider the recursion theorem, which many take for granted without realizing that the proof of it is not really that short in ZFC. – user21820 May 02 '20 at 03:11

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