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I’m coming to realize that all of mathematics can be reduced to, and better understood by using formal logic.

I want to learn how to create a system of formal logic that can be used to explain any particular mathematic theory. I have a good book on axiomatic set theory, but it still feels as though if I had a stronger background in the formal logic used to create such systems, I’d feel more comfortable in my thinking.

I’d like some suggestions on what books to use to study formal logic for a mathematician.

Also, is there a general standard that mathematicians use for creating such systems? For example I noticed that ZFC uses the criterion of eliminability and the criterion of non-creativity in order to allow for such definitions to exist. Is such a standard used for all rigorous mathematical theory?

I don’t want to spend hundreds of hours learning topology or number theory to just ultimately realize that my proofs and justifications are incomplete due to a lack of understanding of the formal logic required to create a mathematic theory.

J. W. Tanner
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Lucas Willhelm
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  • This is an opinionated question, and so will probably end up being closed. I'm definitely no expert on logic, in fact I've only studied basic propositional logic, but that's definitely the most basic form of logic and is the best place to start – Riemann'sPointyNose Jun 01 '21 at 22:06
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    Principia Mathematica by Whitehead and Russell was an attempt to precisely express mathematical propositions in symbolic logic, most famous for managing on page 379 to show $1+1=2$. It is not now thought to be a good example to follow, though it did give Gödel a basis for his Incompleteness Theorem – Henry Jun 01 '21 at 22:34
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    What do you mean by "eliminability" and "the criterion of non-creativity"? – Rob Arthan Jun 01 '21 at 22:34
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    @Henry: Russell and Whitehead were 80 or 90 years ahead of the technology required to implement their programme. Logicism or formalism is now working very well indeed in implementing fully rigorous mathematics. E.g., read about https://en.wikipedia.org/wiki/HOL_(proof_assistant) or https://en.wikipedia.org/wiki/Coq – Rob Arthan Jun 01 '21 at 22:39
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    You might be putting the cart before the horse. That a formal theory succeeds in formalizing some mathematics can only be appreciated by a human being who understands both the mechanics of the formal theory and the mathematics it is supposed to represent. Your time is not wasted if you understand topology or number theory, and such understanding is critical to appreciating the formality of definitions and proofs. – hardmath Jun 02 '21 at 15:40
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    Your question is impossible to answer until you clarify the terms "eliminability" and "criterion of non-creativity" that you use. – Rob Arthan Jun 02 '21 at 21:56
  • You may find my educational freeware and accompanying self-study tutorial to be useful. Download it at my homepage http://www.dcproof.com – Dan Christensen Jun 04 '21 at 03:23
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    I agree with @RobArthan. Also, don't waste your time with dcproof. It's just another person marketing their pet project, and I've looked at it and found it to have little actual value. – user21820 Jun 27 '21 at 12:35

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