Fundamental axioms of logic used to interpret most of modern mathematics

Question

so i'm going through Terrence Tao's analysis 1 and he has a clear emphasis on rigor, however its kind of a contradiction when he didn't even go over the basics of mathematical logic.

A mathematical language needs symbols, and a structure of it needs an interpretation. Along with a mathematical structure, logical axioms and inference rules have to be accepted to do the mathematics and make deductions.

I also reading "A friendly introduction to mathematical logic", and it seems to me:

Logical axioms such as equality, tautology and quantifiers (in book im reading)
A rule(s) of inference like podus pones

And then a deduction is then a finite sequence of formulas that are either axioms, or inferred from already deducted formulas.

My question is that are these 4 axioms used to interpret all of the main of modern mathematics? Like set theory, algebra, analysis etc.

Thank you, I am a beginner and have no teachers, so please be kind to me :)

My recommendation, and it might seem controversial, is to forget about logic. I have taught intro-to-proof writing courses and I would occasionally have a few philosophy majors in my class who took an entire course dedicated to logic. Sadly, they all got 0's on the exams and were unable to write a single proof. If you want to get better at mathematics you should focus on doing mathematics. Once you get better at writing proofs then you can go back to the logic and rethink it again, but this time more carefully. Otherwise, I worry, you will prioritize less interesting stuff and get uninspired. — Nicolas Bourbaki, Dec 01 '23 at 06:18
@NicolasBourbaki I understand how to write basic proofs, I was at university for a short while and had a perfect gpa studying mathematics before rental become an issue. I have read book of proof blah blah, but none of these proofs in that book are actually formal, axiom isn't mentioned, and its all just boring to me. I am genuinely interested in learning this subject again though, with a proper foundation. — Fraser James, Dec 01 '23 at 06:25
Not exactly. Logic starts with a formal system made of 1) a syntax: rules that defines what is a term (a name) and a formula (a statement), 2) zero or more logical axioms (like $P \to P$ for propositional calculus and $x=x$ for predicate logic with equality), 3) one or more inference rules, like Modus Ponens. Then we define a semantics for the system, i.e. a way to interpret terms and formulas. And finally, we have results that link facts about the formal systems: e.g. derivable formulas, with the semantics: tautology (for prop logic). — Mauro ALLEGRANZA, Dec 01 '23 at 06:40
For details, you can see many many similar post on this site (like What is a Logic?) as well as the introductory chapters of every ML textbook. — Mauro ALLEGRANZA, Dec 01 '23 at 06:41
See Where to begin with foundations of mathematics AND Book on the Rigorous Foundations of Mathematics ... AND Does mathematics become circular at the bottom? What is at the bottom of mathematics? If you want to go deeper into this rabbit hole, this book is very nice, and see my answer to How do I convince someone that $1+1=2$ may not necessarily be true? — Dave L. Renfro, Dec 01 '23 at 06:47
@FraserJames Perhaps, you will find a book on advanced calculus more interesting than the "Book of Proof"? Take an interesting theorem and prove is carefully. For example, the intermediate value theorem. It requires a proper definition of "continuous function", and to prove it you need to carefully use the Nested Interval Theorem. Working through all those details take a lot of effort, but it is definitely a lot more interesting. — Nicolas Bourbaki, Dec 01 '23 at 06:47
@MauroALLEGRANZA so, at the end of the day, what are the zero or more logical axioms in place used for interpreting mathematical analysis? What is the inference rule(s)? — Fraser James, Dec 01 '23 at 06:56
It occurs to me (later) that Fundamentals of Abstract Analysis by Andrew M. Gleason might have what you're looking for. See my comments about this book here (FYI, since writing that I've managed to get a nearly unread copy of the 1966 edition -- libraries throwing out their treasures have been a boon to my book acquisitions in the last couple of decades!) and see this review. — Dave L. Renfro, Dec 01 '23 at 07:17
See also Taxonomy and structure of concepts of logic system as well as What is a formal definition of "predicate logic"? and Convincing yourself about choice of axioms for predicate calculus. — Mauro ALLEGRANZA, Dec 01 '23 at 09:25
You will find a useful appendix on mathematical logic in Toa's "Analyisis I." — Dan Christensen, Dec 01 '23 at 15:39

Peter Smith · Answer 1 · 2023-12-01T12:19:06.593

I really like the Leary/Kristiansen A Friendly Introduction to Mathematical Logic -- and indeed it is one of the one of the top recommendations in my Beginning Mathematical Logic: A Study Guide.

But note: they set up one way of presenting first-order logic, which has some nice virtues. However there are other ways of doing the job, which have other virtues. For example, there are systems of logic with zero axioms where all the work is done with rules of inference. And arguably such systems reflect the logical moves in ordinary mathematical reasoning much more closely than axiomatic systems (hence their label, "natural deduction systems"). For just one example, in ordinary mathematical reasoning, from something of the form $P\ and\ Q$ we just infer $P$ straight off, by a "conjunction elimination" step, and certainly don't invoke a logical axiom like $(A \land B) \to A$ and use modus ponens.

So don't get hung up on just one way of regimenting basic logic. Even for standard first-order logic there are some different ways of setting up the semantics (somewhat different ways of handling the quantifiers in particular). And there are multiple ways of capturing first-order logic consequence in formal deductive systems -- some linear, some using upward branching trees, some downward branching trees, some playing with formulas, some with sequents, and more options besides.

For some pointers, see the Study Guide which you can download here.

I'm currently short for time and will come back to mathematical logic when I have my life sorted out. But for now, Would I be ok in taking the presentation of first-order logic as presented in "A friendly introduction to mathematical logic" by Leary a fine framework for the basis of a rigorous mathematics system? I also heard all of the logic systems applicable for math are equivalent, in terms of the rules of inference and logical axioms taken, why is this so? — Fraser James, Dec 01 '23 at 10:49

Fundamental axioms of logic used to interpret most of modern mathematics

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