Some questions regarding Smullyan's proof of Compactness Theorem for propositional logic

According to Jeremy Avigad’s description of Gödel’s original argument ( the second step in the proof establish the following result :

If a set $S$ of propositional formulas is not refutable, it has a satisfying
truth assignment.

Let $S$ = {$\phi_0, \phi_1, \phi_2$, . . .}. Build a finitely branching tree
where the nodes at level one are all the truth assignments to variables of $\phi_0$
that make $\phi_0$ true; the nodes at level two are all the truth assignments to
variables of $\phi_0 \land \phi_1$ that make that formula true; and so on. (The descendants
of a node are all the truth assignments that extend it.) If, at some level $k$,
there is no satisfying assignment to $\phi_0 \land \phi_1 \land . . . \land \phi_{k−1}$, then $S$ is refutable. Otherwise, by Konig’s lemma, there is a path through the tree,
which corresponds to a satisfying truth assignment for $S$.

I think that this is the basic construction used in R.Smullyan, First-Order Logic (1968 – Dover reprint) for the proof of the Compactness Theorem for propositional logic (pag.30-on).

Question 1) Am I right ?

At pag.34 Smullyan gives the following argument :

“we wish to point out that although we used […] Konig’s lemma in our proof of the compactness theorem, [it is not] really essential.”

After the construction (pag.36), he discuss the result :

“The above proof nowhere uses Konig’s lemma […]. The above construction in effect generates the leftmost infinite branch of the complete tableau for $S$.”

Question 2) In what sense he can dispense with Konig’s lemma ?

My last concern is with tableaux method as a proof procedure (for propositional logic).

Assume that $A$ is a tautology (or, if $S$ is a finite set of formulae, let $A$ their conjunction, so that there is not a really limitation in considering a single formula).

Applying the method, we can build, in a finite number of steps, a closed tableau $\mathcal{T}$ ($A$ is a tautology).

Due to the fact that we are dealing with a single formula (or a finite set of them) we exploit the property that we are building a tree $\mathcal{T}$ in which every branch is finite.

Question 3) If I do not need Konig’s lemma for the proof of this fact, may I say that this is a constructive proof procedure for propositional logic ?

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As Benedict Eastaugh says in a comment, the compactness theorem cannot be proved without some form of Konig’s lemma or other “nonconstructive” principles.

In the context of countable theories, which can be studied with Reverse Mathematics, the compactness theorem is equivalent to weak Konig’s lemma over the base system $\mathsf{RCA}_0$. Weak Konig’s lemma says that every infinite tree of 0s and 1s has an infinite branch.

In the context of set theory, the compactness theorem for arbitrary theories is equivalent to the Boolean Prime Ideal Theorem, also called the Ultrafilter Lemma. See this answer on math.SE.

It is hard to tell from the brief quotes exactly what Smullyan is trying to say. However, I can say that the principle “every infinite tree of 0s and 1s has a leftmost infinite branch”, which he seems to appeal to, is in fact stronger than “every infinite tree of 0s and 1s has an infinite branch” (which is weak Konig’s lemma). Trivially, the former is at least as strong as the latter. But, for example, in Reverse Mathematics, the former is equivalent to $\mathsf{ACA}_0$ which is strictly stronger than the latter (weak Konig’s lemma) over the standard base system. The fact that the leftmost branch can be constructed “hands-on” is deceptive, in a sense, because the construction requires us to repeatedly answer questions that cannot be answered constructively (namely, “is this subtree infinite?”).

Perhaps what Smullyan means is that he is able to avoid invoking Konig’s lemma by incorporating its proof into the proof of his result. That is certainly possible.

And, for the third question, both the tableau method and the truth table method are constructive proof procedures for classical propositional logic.

The older logic textbook in this example illustrates how the use of the term “Konig’s Lemma” has changed with the arrival of Reverse Mathematics. There are really four forms of Konig’s Lemma in play here, which in decreasing strength are:

  1. ZFC Konig’s Lemma. Every infinite finitely branching tree has an infinite path. This is the original form of the Lemma from ZFC based Graph Theory and requires the axiom of choice (in a weak form, at least).
  2. ZF Konig’s Lemma. For a graph based on a well ordered set G, then every infinite finitely branching tree has an infinite path. Here the proof can use the “left-most node” made available by the well ordering. In particular if G is countable then G is well ordered via Nat.
  3. $\mathsf{ACA}_0$ Konig’s Lemma. This is Konig’s Lemma in Reverse Mathematics and allows construction of “left-most node”. The sets are necessarily countable and so this result is provable in ZF. However as Mummert says it is highly non-constructive.
  4. $\mathsf{WKL}_0$ Konig’s Lemma. This is the Weak Konig’s Lemma of Reverse Mathematics. It is not immediately obvious why this is weaker than the $\mathsf{ACA}_0$ Konig’s lemma, but it is defined via trees of 0s and 1s. This form of Konig’s Lemma is also non-constructive, but is strictly weaker than any of the others.

Now Smullyan makes a distinction of his own between:

  1. Konig’s Lemma for unordered trees; and
  2. Konig’s Lemma for ordered trees.

The unordered trees form makes “choices” and so appears to be a ZFC Konig’s Lemma (ie the Axiom of Choice). The ordered trees version uses “leftmost node” and so is equivalent to our ZF Konig’s Lemma. In effect Smullyan is pointing out that his set of sentences S is countable, and so all trees constructed from S inherit this countability. Consequently all apparent appeals to ZFC Konig’s Lemma can be replaced by ZF Konig’s Lemma and not use the Axiom of Choice.

Thus Smullyan finds that the Compactness Theorem (for a countable first order language) does not require Konig’s Lemma (by which he means ZFC Konig’s Lemma).

But in Reverse Mathematics we have:

Compactness Theorem == $\mathsf{WKL}_0$

With the conclusion that (the still non-constructive) Konig’s Lemma is unavoidable with the Compactness Theorem after all.

Thus Smullyan (as well as some modern authors) are basing their development of Logic tacitly on ZF Set theory, which enables them to establish non-constructive Logic theorems without using the Axiom of Choice.