The problems he elucidates in the Tractatus cuts an entire category deeper than just "language problems - we have not defined our terms clearly", but rather that the very project itself of finding the ultimate sound and complete formal description for reality is incoherent: not something we need to clarify before investigating further, but utter non-sense we were only curious about because we were confused: like asking what is the last digit of pi, or what is the most positive prime number. This is deeply related to German concept of Transcendental Illusion.

Wittgenstein had no interest in Godel, he felt he had long since done away with the whole idea of a sound and complete formal model for reality in the Tractatus, and yet another paradox in yet another attempt at this nonsensical project held no interest for him (obviously it was a bigger deal for everyone else, proving that such a paradox could always be found. Wittgenstein already knew this was a pointless goal).

Wittgenstein only ever published two works: the Tractatus, and the Philosophical Investigations. The "Remarks on the Foundations of Mathematics" are lectures notes from his students, which isn't to say they should be disregarded, but we should also keep in mind that they are on-the-spot comments transcribed by others.

Paper [1] is a great example of something that should send off BIG warning flags when reading this kind of stuff. It goes to great lengths to talk about how it is quoting Wittgenstein out of context (extra especially bad for lecture notes) and that the rest of what he had to say _in_ _that_ _very_ _paragraph_ just isn't relevant for their paper ("the last three sentences are omitted and will not be discussed in this paper"). Let's take a look at those omitted sentences (would be good to read the block quote in [1] immediately before, for context):

> ...If you assume that the proposition is true in the Russell sense, the same thing follows. Further: if the proposition is supposed to be false in some other than the Russell sense, then it does not contradict this for it to be proved in Russell's system. (What is called "losing" in chess may constitute winning in another game.)

Wittgenstein has no interest whatsoever in logical systems being sound and complete. He refuses even discuss this as a problem, making _constant_ reference to "true/false in the Russell system" and being _very_ clear in the strategically omitted sentences that the "proposition [can be] to be false in some other than the Russell sense", even compares it to chess, a game with arbitrary rules! Latter Wittgenstein believed we "play" a variety of different overlapping "language games", _none_ of which hold any kind of primacy: mathematics is just a "real" (or whatever) as poetry, they are just different games with completely different rules and different standards for correctness. (Briefly addressing a misreading in [2], that's also relevant for understanding this, for Wittgenstein the highest level of certainty is not analytic proof, for they are vacuous, but rather his standard for the highest level certainty is "I am certain that I have never been to the moon", while completely admitting the faint possibility that he is instantaneously transported there and back each time he blinks his eyes. For Wittgenstein, certainly is not impossibly of the opposite, but a claim made by a knower-of-things: staking your reputation as a "knower" on something being true).

The author's "Godelian" rebuttal mistakes the claim that "‘True in Russell’s system’ means, as was said: proved in Russell’s system" as Wittgenstein's assumption. It isn't. It's Russell's, and just one of the rules that comes along with playing this particular language game. Wittgenstein is saying: So what if we can show though some series of transformations that some proposition can be constructed that violates some other rules of truth and falsity? We already started this game with our definitions (the provable being definitionally true and opposite being provable being definitionally false _in_ _the_ _Russell_ _system_ ). This is very much so part of Russell's system, which Wittgenstein describes in detail in the previous seven sections of these lectures notes, which also get at best passing mention in this paper. Such Godel sentences are simply out of scope for what Russell's system is good at (Russell's language game about logic/mathematics), by the _very_ rules of the game. Russel hoped to describe the totality of the world with it so this was a problem for him, Wittgenstein always thought that whole goal was utter nonsense. Wittgenstein is not directly addressing Godel's theorems, he's just calling them irrelevant to this particular game ("the Russell system") which many people play, and he is teaching his students how to.

A language game not being sound and complete just doesn't matter to Wittgenstein. He proved the very idea of that was nonsense way back when he was a kid writing the Tractatus. Instead, they would have to claim it to a _bad_ or useless language game (or, better yet, come up with a better one that fulfills similar needs in our lives) to strike against it.

Link [1] above misses his point so entirely, the entire debate sketched out there just has nothing do with what he said, and that author is tying himself into knots trying to represent Wittgenstein's arguments with his symbolism. I can understand why Wittgenstein would get so frustrated he would stand with his back to everyone and read poetry... That selective quoting in this case is particularly egregious and not typical of philosophical scholarship, I wouldn't get discouraged. It looks like this author's primary focus is "to spell out a Wittgensteinian alternative to mathematical logic", so he _very_ much so has a horse in this race.

Cora Diamond and James Conant both have significant amounts of work on Wittgenstein and are both beyond excellent.