My first question is on the relationship between ZFC and first order logic: "A reader who has been confused by the discussion in this chapter says, "Now I'm completely mixed up. How can ZFC be used as a basis for first order logic, while first order logic was actually needed in order to build up ZFC?" Help such a reader out of his dilemma." Well, I might be the confused student in this case. Is the confusion rises from the fact that first order logic isn't actually needed to build ZFC, but to model it and process of building it?

My second question is on a proof of the undecidability of the set of valid sentences. They prove it using reduction from the halting problem, but it seems like the reduction is not computable. While building the sentence from the program (which should be valid iff the program halts), they distinguish between two cases, where the program halts, and where it doesn't. But we can't know whether a program halts or not, because we can't solve the halting problem. Please ask if you need more details on the proof. It is on page 160 in the first edition. Thanks!

  • $\begingroup$ Regarding your first question: First-order logic can be viewed as a class of formal systems, of which ZFC is just one. In this sense, ZFC is built on first-order logic. However, first-order logic isn't just a class of formal systems. Its actually a class of formal systems with a specific purpose; namely, reasoning about truth and falsehood in the language of functions, predicates, variables and quantifiers. That is, first-order logic has an intended semantics... $\endgroup$ – goblin Nov 22 '13 at 17:11
  • $\begingroup$ ... okay, but how are we to know that axioms and inference rules of first-order logic actually capture this semantics? Well, we can formalize statements like "first-order logic is sound" and "first-order logic is complete" in ZFC. So, ZFC is not a basis for first-order logic. Its just a tool we can use to convince ourselves that the rules of first-order logic are correct and complete. $\endgroup$ – goblin Nov 22 '13 at 17:11

I can't answer your second question, because I don't have the book you're referring to. But here's what I have to offer for the first one:

It is very easy, when first reading about these things, to be lured into thinking that formal reasoning in ZFC is "what mathematics really is", and the usual sort of English prose arguments found in textbooks and papers is just a sloppy conventional shorthand for the actual underlying ZFC reasoning. This is a very appealing picture for someone who likes to take things apart to find out how they work -- it struck me as a tremendous epiphany the first time I learned it.

Unfortunately, as you have also noticed, the world can't be quite that neat. This picture leaves no place for reasoning about ZFC and formal logic to stand on. On one hand this seems to make all of mathematics an ultimately circular exercise; on the other it would feel wrong to say that reasoning about formal logic is a special kind of mathematical thought that doesn't need to be built upon ZFC -- because it clearly feels like the same kind of reasoning we use in other mathematical fields such as calculus or graph theory.

The most common modern resolution of this dilemma is to say that formal logic (and with it formalizations of set theory in formal logic) is not what mathematical reasoning really is. Rather, formal logic is (merely?) a mathematical model of everyday mathematical reasoning. It's a pretty good model in that it is remarkably successful in predicting which kinds of arguments you can get a skeptical mathematician to accept as mathematical, but that doesn't change the fact that the model is not the thing itself.

One point to note is that mathematical reasoning has been going on for centuries (millenia!) but formal logic was only developed during the first half of the 20th century. It would be ridiculous to say that Gauss, Euler, or Cauchy were not actually doing mathematics, and it would be untenable sophistry to claim that they were "really" following precise rules of symbolic manipulation that were only "discovered" (rather than invented) in the 1900s.

Mathematical reasoning exists, inside the heads of mathematicians, without any need for a particular formalization of it to prop it up. The followers "Hilbert's program" around 1900 imagined it could be improved by propping it up by a reduction to clearly defined rules, but as the work to do so was actually done, it turned out that it couldn't really work that way. The results are nevertheless interesting in their own right. The word "foundation" lingers on from those early days, and continues to confuse students because formal logic doesn't really provide the solid, indubitable, "foundation" for the rest of mathematics that the word promises.

  • $\begingroup$ Why can't both formal reasoning in ZFC be "what mathematics really is" (Not(2nd paragraph)) and have formal logic as a model for mathematics (3rd-last paragraph)? $\endgroup$ – nodwj Mar 16 '13 at 15:16
  • $\begingroup$ @tsho: Are you asking why these two hypotheses are mutually exclusive? I don't know what to say to that. It's obvious to me that if X is what Y really is, then X cannot at the same time be just a model of Y. It's implicit in the concept of a model that it's distinct from the thing it models. Now set X=formal logic, Y=mathematical thought. $\endgroup$ – Henning Makholm Mar 16 '13 at 15:30
  • $\begingroup$ I meant formal logic to not be what mathematics really is, only a model of it. So formal logic is a part of mathematics that models mathematics. $\endgroup$ – nodwj Mar 16 '13 at 16:04
  • $\begingroup$ This question seems really interesting. Would you say that mathematics is, after all, just some reasoning system about the world conjured up by the human beings, but nothing really "fundamental" nor exists like some "fundamental laws of nature"? $\endgroup$ – xji Nov 13 '17 at 15:52
  • $\begingroup$ I find some parallels in linguistics, in that previous linguists (spearheaded by Chomsky) tried to find some rigid formal rules and universal grammars of languages that are supposed to reside in people's heads, but eventually it becomes widely accepted that there are really no such rules whatsoever (or rather, those "rules" are just higher-level abstractions/formalizations of more low-level phenomena), and a neural network approach simply captures linguistic phenomena much better, and might well be much closer to the way the analog world actually operates. $\endgroup$ – xji Nov 13 '17 at 15:59

The answer by Henning Makholm gives a very clear explanation of one of the most common answers to question 1 among mathematical logicians today: mathematical logic does not provide a non-circular foundation to mathematics, but instead gives a model of mathematical reasoning which can be used to study that reasoning and understand how it works.

Like many deep issues, this one has many interrelated aspects. Another important aspect is that we don't need all of ZFC to formalize the syntax of ZFC, if we don't care about also formalizing the semantics. We can formalize just the the syntax and provability relation using very weak theories of arithmetic, much weaker than Peano Arithmetic.

These theories can't help us talk about models - they can't even begin to describe the semantics of first-order logic. But they are strong enough to derive various theorems about ZFC-proofs and to prove metatheorems about ZFC. For example,

  • Weak theories of arithmetic can prove that ZFC satisfies the deduction theorem: $\text{ZFC},\phi \vdash \psi$ if and only if $\text{ZFC} \vdash \phi \to \psi$.
  • Similarly, the theorem that the Axiom of Constructibility ($V=L$) is independent of ZFC can be proved in weak theories of arithmetic, beginning with the syntactic assumption that ZFC does not prove any contradictory sentences. The same goes for other set-theoretic independence results: they are provable in weak theories of arithmetic beginning with an additional consistency assumption.

So the amount of circularity can be reduced, in some sense, although the result is that we are unable to look at the very rich theory of models of ZFC. To fully capture both the model theory and set theory of ZFC, we need to move to very strong theories.

Question 2

Regarding question 2, on page 160 they define the structure $\mathfrak{A}_P$ by cases, depending on whether $P$ halts, but on page 161 they define the sentence $\psi_P$ to be the same sentence in either case. For example, the number $\alpha_k$ is determined by looking at $P$, not by knowing whether it halts. They could have rearranged the exposition into the other order, but because the construction of $\psi_P$ is so much longer, it seems they decided to construct $\mathfrak{A}_P$ first.


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