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I know this question has been investigated in other threads, but I would like to pose yet another question on Gödel's theorem incompleteness, and truth in 'the standard model' compared with provability in the 'formal model'.

I will use the language of Gensler's little book "Gödel's Theorem Simplified" (see here for a summary).

For what Gensler calls System C, he gives an outline for how to construct a wff in system C. The system C the symbols are /, (, ), - , n (for a variable), nn (another variable),...etc.).

The reference numbers for system C formulas (like Godel numbers) are defined too.

For example the ref # for / is 1, the reference # for the variable n is 8, etc.

Then Gensler gives instructions for how to create a formula called F which states

(F) "There is no system C proof for the son of the system C formula with reference # n ".

Here n is a variable. So the system C formula with reference # n exists, call that system C formula XX, and F is the statement that the system C son of XX, SonXX, is not a theorem in system C).

F is a string of symbols of system C.The son of F is obtained by taking the reference number (an actual whole number like 112278 etc) for F, writing that whole number as the system C string //.../ (with reference # of F many /'s appearing) and replacing each occurrence the variable n in (F) with that many ///.../.

The result is a new system C formula, the son of F, called G.

The 'standard model' interpretation of G is then

(G) "There is no system C proof for the son of the system C formula with reference # equal to the reference # of F."

In other words, there is no system C proof of G itself.

Then Gensler and other sources indicate this standard interpretation of G is true, so it is a true 'ordinary arithmetic' statement, for which the corresponding system C formula has no proof.

It seems to me that what you can really say is this:

If you assume that the standard interpretation of G is a true statement in the ordinary standard model, then the corresponding system C formula is an unprovable system C formula. So you have a 'true' ordinary arith statement which cannot be proved in system C, BECAUSE you have ASSUMED that the standard model version of G is true.

If you assume that the standard interpretation of G is a false statement in the ordinary standard model, then the corresponding system C formula is an provable system C formula.So you have a 'false' ordinary arith statement which can be proved in system C, BECAUSE you have ASSUMED that the standard model version of G is false.

But how do you know that one of the two statements " G is a true statement in the ordinary standard model" or " G is a false statement in the ordinary standard model" must be valid?

Isn't the best you can say that IF you could determine that G were a true statement in the ordinary standard model" or " G is a false statement in the ordinary standard model" THEN you can draw conclusions about non provability or provability of G as a system C formula?

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    $\begingroup$ The assumption is a part of the theorem itself: Gödel's First Incompleteness Theorem states that, for a model with sufficient "power" (can represent arithmetic), that its consistency implies its incompleteness. It stands to reason, then, that one would assume G to be true. If you wish to prove the opposite (that completeness implies inconsistency), then you would assume that a proof of G exists, thus rendering it false, and your model inconsistent. Now, the theorem does not actually require consistent as an assumption- weaker assumptions work- but that is the one your mentioned book uses. $\endgroup$ – Ispil Jan 23 at 6:32
  • $\begingroup$ You can see the lenghty post Is PA consistent? do we know it? $\endgroup$ – Mauro ALLEGRANZA Jan 23 at 7:32
  • $\begingroup$ Gödel's Incompleteness Theorems is : "if [system] $C$ is consistent, then [formula] $G$ is unprovable". But also $\lnot G$ is unprovable. The two are sentences of $C$ and thus - for every model $\mathcal M$ of $C$ - exactly one of them (it does not matter what one) - must be TRUE in $\mathcal M$ . $\endgroup$ – Mauro ALLEGRANZA Jan 23 at 8:00
  • $\begingroup$ Thanks for these two replies, which I do find helpful. $\endgroup$ – grell6954 Jan 23 at 15:43
  • $\begingroup$ Here is the way they have helped me understand the theorem. THE KEY IS THAT GODEL’S IMCOMPLETENESS THM SAYS IF A MODEL OF ARITH (LIKE THE STANDARD ONE) IS CONSISTENT THEN IT MUST BE INCOMPLETE. IN THE GODEL/ GENSLER MODEL: ASSUME STANDARD ARITH IS CONSISTENT. IF IT IS ALSO COMPLETE THEN ONE OF THESE MUST BE TRUE: " G is a true statement in the ordinary standard model" or " G is a false statement in the ordinary standard model" (and consistency implies exactly 1 must be true). So consistency makes completeness impossible and completeness means 1 of these MUST be true. $\endgroup$ – grell6954 Jan 23 at 15:52

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