Reddit reviews An Introduction to Gödel's Theorems (Cambridge Introductions to Philosophy)
We found 3 Reddit comments about An Introduction to Gödel's Theorems (Cambridge Introductions to Philosophy). Here are the top ones, ranked by their Reddit score.
Used Book in Good Condition
Peter Smith is a philosopher and his Godel's Theorems book seems to me like what you are looking for. It considers the implications of the theorems and comes with a careful bibliography. Most university libraries would have it.
A shorter book is Torkel Franzén's.
> Second and third semester calculus
Is this vector calc? If so I enjoyed this book as it's very geometric, not at all rigorous and has lots of worked examples and exercises. Sorry it seems to be so expensive -- it wasn't when I bought it, and hopefully you can find it a lot cheaper if it's what you're looking for.
In general Stewart's big fat calculus book is a nice thing to have for autodidacts.
Obviously what you describe might include analysis, which these books won't help with.
>Formal logic theory (Think Kurt Godel)
I've heard Peter Smith's book on Godel is good, but haven't read it. Logic is a huge field and it depends a lot on what your background is and what you want to get out of it. You may need a primer on basic logic first; I like this one but again it's quite personal.
Physicist here so don't pretend I don't know what science is. (Though like the ancient Pythagoreans I'm sure as soon as I discuss something that has been proven that goes against a purely scientific worldview out comes the pitchforks.) And though I love science, unlike some people here I am willing to admit to the limits of science. Science can lead to all truth in the same way that rational numbers define all numbers: it can't! and Godel proved it.
The real problem with science is that it has been mathematically proven by Godel that there are more things that are true then are provable and thus you can't ever have a scientific theory that can determine the truth or falsity of all things. As soon as you write down that theory, assuming it allows for arithmetic, Godel's incompleteness theorem immediately shows if the theory is true there will be true statements about reality that are beyond provability. Read Godel Esher Bach or Incompleteness or work through it yourself in this textbook as I have.
So like I said above, science is great in it's sphere (and in that sphere let me emphasize it is awesome!) but leads to all truth in the same way that rational numbers leads to all numbers. (And the analogy is precise since Godel used the famous diagonizational argument in his proof.) Russell and Whitehead set out to show in the early 1900s that if we could determine the axioms of reality then through logic work out everything that was true and Godel spoiled the party.
It it would be one thing if these truths were trivial things, but they are not. Some examples of true or false statements that may fall into this category of being unprovable are:
Now, at this point critics almost always tell me: but Joe, Godel's incompleteness theorem is only relative to your set of logic. (Ie... we can prove Goldbach by just adding axioms needed to do so.) Fine. But two things: (first) adding axioms to prove what you want willy nilly is not good science. (Two) You now have a new set of axioms and by Godel's theorem there is now a new uncountable set of things that are true (and non-trivial things like I listed) that are beyond proof.
Now usually comes the second critique: But Joe, this doesn't prove God exists. And this is true. But at least it has been proven God gives you a chance. It has been proven that an oracle machine is free from the problems that hold science and logic back from proving the truth of all things. At least something like God gives you a chance (whereas science falls short).
Or, like Elder Maxwell says so well: it may only be by the "lens of faith" that we can ever know the truth of all things. He maybe be right, and hence the importance to learn by study, and also by faith...