Set theory formula

Question

I picked up a copy of Jech's Set Theory at my school library and I'm reading through it and taking notes. Right at the beginning, though, he mentions something called a 'formula'. Here's the quote:

"Concerning formulas with free variables, we adopt the notational convention that all free variables of a formula $\varphi(u_1, ..., u_n)$ are among $(u_1, ..., u_n)$ [...]"

What does this mean? What is $\varphi(u_1, ..., u_n)$?

Answer 1

See :

• Herbert Enderton, A Mathematical Introduction to Logic (2nd ed - 2001), Section 2.1. : First-Order Languages, page 69-on (and you can see also this post for a relevant quotation from Enderton's book).

Consider an example in the language of set theory :

$∀x \ (x∈y \to x∈z)$.

In it, we have three (individual) variables : $x$ (bound) and $y,z$ (free); thus, the formula is "like" $\varphi(x,y,z)$, because all free variables in it are among $(x,y,z)$.

Answer 2

While Jech's Set Theory is my favorite textbook, it isn't suitable as a starting point into mathematical logic. Thus, even though I strongly encourage anyone interested in set theory to read his book, you should study (at least) some basics of mathematical logic first. Over at mathoverflow, there is a post asking for textbooks on mathematical logic that provides some suggestions where to start.

I haven't read it, but this course on mathematical logic by Stephen Simpson is available for free and seems to cover the basics (judging from its index alone) up to a point, where Jech might be remotely readable afterwards... (At this point, it will still be a huge stretch... Jech's target audience is graduate students and researchers in set theory).

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To adress miracle173's comment:

You are looking for the definition of a formula in first-order logic. To rigorously define these and their meaning, one has to introduce languages, quantifiers, connective symbols, free/bounded variables, structures, assignments and the modeling relation. Any introductory textbook on mathematical logic will define these objects. Therefore, I consider the above a reasonable answer to your question.