Talk:Original proof of Gödel's completeness theorem

message: Given the confusion about the proof as presented here, I would just like to point out again that there are in fact several good expositions of the proof in secondary sources:
# Avigad's essay, which is apparently a slightly updated transcription of a lecture presented to the Association for Symbolic Logic in May, 2006.
# The chapter by van Attan and Kennedy in The History of Logic 
# The introduction to the collected works edition of Gödel's thesis (https://antilogicalism.com/wp-content/uploads/2021/12/Godel-1.pdf, p. 50 onwards)
# The explanatory power of a new proof: Henkin’s completeness proof  by John Baldwin (publisher version: https://link.springer.com/chapter/10.1007/978-3-319-93342-9_9 ) 
The general sourcing rules on Wikipedia mandate that we should follow the secondary sources in our interpretation and exposition of the proof, so the fact that the page currently does not even mention any source beyond Gödel's original works is a problem. Felix QW (talk) 08:53, 3 August 2023 (UTC)
replies: text: I hope I have finally resolved the dispute with Dirsaka on my talk page. Germanomosconi1 (talk) 11:03, 3 August 2023 (UTC)
thread_title:  Secondary sourcing 

message: Since I could not resolve the dispute with @Dirsaka on my talk page, I repost here the objection argument, and also I explain why it's incorrect, at least for me. I do this because I'm sure that anyone else who knows formal logic will agree with me.
In the proof linked here, it's used the "functional substitution" rule of inference to get a provable formula from the formula
:\neg\Phi = \neg[(\forall x')(\exists y') Q(x',y') \wedge (\forall x)(\forall y)( Q(x,y) \rightarrow (\forall u)(\exist v)(P)\psi )]
The rule is set out in Principles of Mathematical Logic, ch. III, par. 5 by David Hilbert (1950), exactly as follows:
"Under certain circumstances, a predicate variable with n argument places may be replaced by a formula which contains at least n free individual variables. Let F be the n-adic predicate variable, and \mathfrak A the formula in which F is to be replaced. From among the individual variables occurring in the formula which is to replace F, we select any n whatsoever ordered in any arbitrary way, say, x_1, x_2, \dots, x_n. The formula which is to be substituted will accordingly be designated by \mathfrak B(x_1, x_2, \dots, x_n). The substitution is now permissible only if the remaining free individual variables which may still occur in \mathfrak B(x_1, x_2, \dots, x_n) do not occur in the formula \mathfrak A as bound variables, and if, further, there is no occurrence of F in \mathfrak A such that a variable occupying an argument place in F occurs as a bound variable in \mathfrak B(x_1, x_2, \dots, x_n), provided always that the result of the substitution is a formula. The substitution is accomplished as follows: Considering any specific occurrence of the predicate variable F in \mathfrak A, we find the argument places of F occupied by certain individual variables, which we designate, for the moment, by \mathfrak a_1, \mathfrak a_2, \dots, \mathfrak a_n. These need not all be different. We now replace F(\mathfrak a_1, \mathfrak a_2, \dots, \mathfrak a_n), at this occurrence of F, by \mathfrak B(\mathfrak a_1, \mathfrak a_2, \dots, \mathfrak a_n), i.e. by the formula obtained from \mathfrak B(x_1, x_2, \dots, x_n) when the variables x_1, x_2, \dots, x_n are replaced, wherever they occur, by \mathfrak a_1, \mathfrak a_2, \dots, \mathfrak a_n respectively. The corresponding substitution is to be made at every occurrence of F."
In our case, we have \mathfrak A := \neg\Phi, F := Q, and \mathfrak B(x, y) := (\forall u)(\exists v)(P)\psi, where x and y are tuples of variables, which have been selected.
Now, @Dirsaka is sure that, in this case, the substitution is not permissible, because, as he says: "there are remaining free individual variables which still occur in \mathfrak B(x, y) := (\forall u)(\exists v)(P)\psi and also they occur in the formula \mathfrak A as bound variables". But this is clearly a misinterpretation of the rule, since it's clear that by "remaining free individual variables which may still occur in \mathfrak B(x_1, x_2, \dots, x_n)" the author intended not the all free variables, but instead the unselected free variables; otherwise, why Hilbert (or the translator) would use the word "remaining"? There would be no need, he could only say "free variables occurring in the formula". Furthermore, why the author wrote "which may still occur"? If he meant the all free variables, it's obvious that they still occur! So, it's clear that, at least in my opinion, the objection raised is completely meaningless. @Felix QW @Bbbbbbbbba Germanomosconi1 (talk) 22:37, 3 August 2023 (UTC)
thread_title:  Dirsaka's objection to proof of reduction to formulas of degree 1