New math book rescues landmark topology proof

I think, with your explanation, there's still a big handicap for the halting machine code.

To illustrate.. If I were to say to you: Is the correct answer yes or no, considering that later I will reverse my answer, and enforcing that you can't later change your answer? It's like, okay, of course, with that model, you can never win. I don't think it reveals any limitation of your understanding, you just got pulled into a game that you can't win.

The limitation that is put on the code and in your example too is, although the machine can take in extra input along the way, at a certain point the mathematician gets the last word.

If you have G(H,M) which at each point gives you [0|1]:H', a more interesting question is if what happens if you define M = "if H(M)[0]: print H(M)[1] and then loop". Then, does the printed out H(M)[1](M) produce any contradiction?

Hopefully I'm making sense here! Thanks again for engaging.

In any event, all of our mathematical proofs are predicated upon the idea that the representation of a mathematical object can be treated in terms of proof as equivalent to the mathematical object. Is "2" the number 2 or is it just a glyph that non-representationally depicts the successor of the first counting number? And if you believe the latter, then how do you go about proving anything about the actual successor of the first counting number without resorting to glyphs?

One possible starting point is Book of Proof by Richard Hammach[0]. It's free. It will show you the rudiments of how math is actually done and thought of. Another good choice is Mathematical Proofs: A Transition to Advanced Mathematics by Chartrand/Polimeni/Zhang. These books have no prerequisites(good to go if you know how to do arithmetic with fractions/decimals; not that these are strictly necessary, rather that's about the level of comfort with math one should posses). Search on Amazon and you'll see there are a ridiculous number of books on this subject of all varieties of ways of presentation. Almost every week a new book pops up, it seems. Note, intro-to-proofs books are not only for novices, though. There are kinds written for advanced undergrads and beginning grad students as well. For example: Mathematical Concepts by Jugen Jost. This speaks to the wealth of the kinds of math books available.

Concurrently and/or after that you can study introduction to any of linear algebra, discrete math, group theory, analysis of reals, combinatorics, number theory, graph theory, algorithms, probability, category theory...whatever. Each of these are very deep subjects and there exist hundreds(if not 1000s of books from the past, current and future) of every imaginable level, depth and presentational quirk. Again check Amazon for titles. Check libge*n for actual files.

The following intro books are so easy anyone and they momma can read these with utmost ease:

Discrete Math by Susanna Epp

How to Think about Analysis by Lara Alcock

Linear Algebra: Step by Step by Kuldeep Singh

Real Analysis: A Long-Form Mathematics Textbook by Jay Cummings

A Tour Through Graph Theory by Karin Saoub

The above mentioned books are a few examples of elementary books that you can get started with right this minute.

Note and remember the wealth of possible books, though. For example, consider the subject of math analysis.

We can break it into three general parts: real analysis, complex analysis, functional analysis. Each can be further taken apart into measure theory, vector spaces, topology etc. Further, topology alone branches out into point-set topology, algebraic topology, differential topology etc. Each of these fly under many different names. Say, vector spaces is a huge part of linear algebra. For the beginners, it's usually the finite dimensional vector spaces (aka the title of a famous linear algebra book) version whereas the infinite version will take you into func analysis. Vector spaces are also usually included as a chapter in books on abstract algebra as they are an algebraic structure just like groups, rings, modules. But if there is linear algebra, there must be non-linear algebra too. Well, yes. Roughly, it corresponds to what's called algebraic geometry. It never ends.

The point is every topic in math has loads of books dedicated to it. In turn every topic has a (sub)^{n}topic and there are tons of books on those. Each of the books come in elementary, quirky, armchair enthusiast, average undegrad student, middling grad student, kickazz graduate math aficionado, researcher, undergrad researcher wanna-be etc levels. There are also olden books, classics, those just published (say, at most 5 years ago) and up-and-coming ones. Also, each term in math has bagillion synonyms. So multiply the number of books above by n. I am not even gonna mention books written in different languages and lecture notes that are available for free online. A lot of published book often spend years as a free set of lecture notes before ending up on Amazon as a newly published book. There are also Olympiad books, proof compilations, capstone compilations, intro-to-research books, those that prove a single theorem and in the process go through tons of disparate math etc Take advantage of all of this.

[0] https://www.people.vcu.edu/~rhammack/BookOfProof/

There is a wonderful book by Torkel Franzén about the incompleteness theorem that is 100+ pages, but it is not purely about the proof. It's a popularization that discusses bogus ways philosophers have interpreted the theorem, and stuff like that. It is called "Gödel's Theorem: An Incomplete Guide to its Use and Abuse".

Meanwhile Aschbacher and Smith's proof of the classification of quasithin groups really is a book of 1200+ pages: https://books.google.com/books?id=SbO2irSQHMEC

And of course the proof of the classification of finite simple groups has afaik never been published all in one place, though there are some book-length expositions of it referring back to the original papers. It is a set of papers spanning around 20 years and adding up to around 10,000 pages. It is sometimes called "the enormous theorem": https://en.wikipedia.org/wiki/Enormous_theorem

In the case of Wiles' proof of FLT, someone found a bug, Wiles worked unsuccessfully for something like a year trying to fix it, basically concluded that FLT was not really completely proved after all (i.e. he had "merely" made enormous progress, but not actually closed the ticket). Then right after more or less giving up, he and Richard Taylor figured out how to fix the bug, so it came out ok after all. Of course Taylor now gets a slice of the FLT credit.