> The linked article is about how the theorems destroyed Whitehead et al's aspirations to find One Algebra To Rule Them All.

Sort of. Whitehead's contributions to universal algebra are still relevant and universal algebra is still a thriving field of study for mathematical logic. Although perhaps you mean "algebra" in a more informal sense?

Again, the conclusion of the article is a bit strong.

> Gödel’s proof killed the search for a consistent, complete mathematical system.

The consistency half doesn't make sense. (EDIT: I get it now, see the last sentence of this paragraph) There was never a search for a consistent mathematical system in the sense that Godel destroyed because again a system that can prove its own consistency has no positive value in evaluating the consistency of that system (Godel's big contribution here is contributing a strong negative result, if it could prove its own consistency you're pretty screwed). EDIT: On reflection I see that the sentence probably means to tie consistency to completeness rather than as a stand-alone quality. That makes more sense.

As for mathematicians, their reactions to Godel's incompleteness theorems overall are probably similar to "Who cares about your incompleteness theorems?" (there's a reason Russell and Whitehead are known primarily for being philosophers first and mathematicians second and why often times there is distinction between logicians like Godel and other mathematicians). Most mathematicians don't think about the foundations of mathematics because it is (perhaps surprisingly) largely irrelevant to the day-to-day work of mathematicians. Indeed the vast majority of mathematics is very resilient to changes in its underlying foundations.

To interpret Godel's incompleteness theorems requires a healthy dose of at least mathematical logic that can start veering quite close to mathematical philosophy.

An example from the article:

> However, although G is undecidable, it’s clearly true. G says, “The formula with Gödel number sub(n, n, 17) cannot be proved,” and that’s exactly what we’ve found to be the case!

Well no, that's not true in certain senses. Indeed in a larger axiomatic system subsuming the current system that corresponds to G, it is completely consistent to state that G is provable and that its statement is provable (see my example of S'), i.e. there are Godel numbers that correspond to both a proof of G and to a proof of its content. To interpret that statement that way requires certain philosophical commitments to the correspondence between a Godel number and truth, which not everyone would accept (do you accept the truth of the new axiom introduced by S'? Why then do you accept the truth of the statement "S is consistent?" and vice versa).

"In striving for a complete mathematical system, you can never catch your own tail." This on the other hand I think is a very good informal description of what's on with Godel's incompleteness theorems. Focus on the incompleteness not on truth.

That's why I'm not a fan of using the word "truth" when talking about Godel's incompleteness theorems. I am in fact deeply sympathetic to your desire to separate philosophy from Godel's incompleteness theorems.

I prefer to clearly delineate between its logical properties in mathematical logic and its philosophical implications and using the word "truth" by necessity muddles the two.

FINAL EDIT: I am being perhaps a bit too harsh on the article. I think it does a fine job of describing the arithmetization of the incompleteness theorems. But if someone else reading this also decides to create an informal guide to Godel's incompleteness theorems, please please please don't use the word "truth" and "true" or at least separate it out into its own section on philosophy.