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What are some good books on First Order Logic, set theory, and then analysis

  1. Sep 16, 2008 #1
    "Mathematics is a study which, when we start from its most familiar portions, may be pursued in either of two opposite directions. The more familiar direction is constructive, towards gradually increasing complexity: from integers to fractions, real numbers, complex numbers; from addition and multiplication to differentiation and integration, and on to higher order mathematics. The other direction, which is less familiar, proceeds, by analysing, to greater and greater abstractness and logical simplicity; instead of asking what can be defined and deduced from what is assumed to begin with, we ask instead what more general ideas and principles can be found, in terms of which what was our starting-point can be defined or deduced." --Introduction to Mathematical philosophy, Russell.

    I'm currently on the more familiar path, but the books my school uses are mostly gearded towards problem solving with very little theory in calculus I - III. My school library has books by Apostol and Spivak, but both of them look like difficult books to try and start learning analysis. Aren't there books that can help ease my way into the subject of higher-order mathematics?

    Also, I've been wanting to study more set theory and logic theory. I've been trying to read some Bertrand Russell books on logic and mathematics, like "On Denoting" and "The Logic of Relations," but I have no idea about Peano notation and thus cannot even get past the third page in the latter essay. Even his Introduction to Mathematical Philosophy is a bit confusing, I'm simply not too familiar with using methods that look like something from the binomial theorem to prove things, especially in regards to classes. Russell would define things like "1" as the class of all unit classes.

    I have given up on these more complicated Russell books but would still like to learn some fundamentals of arithmetic and peano notation.

    Can you get off the ground in set theory without worrying about the philosophical mathematics stuff?

    Also, what is a good generalized overview of mathematics:

    Principles of Mathematics (russell) or An Introduction to Mathematics (whitehead)
  2. jcsd
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