Beyond that, Gödel wanted to arrive to an alternate set theory that had a definite answer to the continuum hypothesis, because consistency and provability are not really goals in themselves, but simply prerequisites.

It supposedly killed off the Hilbert program, which sought a consistency and completeness proof for mathematics, and a mechanized procedure for deciding arbitrary mathematical propositions.

What I'm trying to get at is that mathematical truth people care about goes beyond provability, and that is the case in good part because of incompleteness.

He initially writes

>S(E) = “This sentence has no convincing heuristic explanation accepted by E.”

>If S(E) is *true*, then it’s an example of a *true* arithmetical statement without even a convincing heuristic explanation for its *truth* (!). If, on the other hand, S(E) is *false*, then there’s a convincing heuristic explanation of its *truth*

The confusion is because the "proof system" isn't supposed to be outputting true/false for E. Its supposed to be an "explanation system" outputting the convincingness of E. So let me try to recast it.

>S(E) = “This sentence has no explanation that is accepted by E.”

>If S(E) is accepted, then it’s an example of an accepted statement without a convincing explanation. If, on the other hand, S(E) is rejected, then it is a rejected statement that nonetheless has a convincing explanation.

Finding a single false statement which a heuristic system erroneously accepts does not invalidate the entirety of the heuristic system. But finding a single false statement which a proof system accepts does invalidate the whole proof system.

>S(P) = “This sentence has no argument for it plausible according to P.”

>If S(P) is true, then it’s an example of a true arithmetical statement without an argument for why its plausibly true (!). If, on the other hand, S(P) is false, then S(p) is a false statement for which there is nonetheless a plausible argument it is true.

Notice that there is no contradiction if this system of formal arguments tells me "A is plausibly true" and later it turns out A is absolutely false. The reductio ad absurdum step of the the incompleteness proof doesn't work when the underlying semantics never asserted anything to be true with certainty. In other words, soundness is not really a necessary or even expected property of these non logic formal reasoning systems. Thus there is no incompleteness problem here because there is no soundness property that contradicts having completeness.

> Systems that are guaranteed to be correct 100% of the time are subject to Gödel. Systems that are required to be correct much/most/almost all of the time are not necessarily subject to Gödel. Human intuition is of the latter type.

Gödel still technically applies in the sense that if I claim to have a reasoning system which is usually sound and complete, you can still use "this statement has no heuristic explanation" to force an instance of my system reaching unsound conclusions, But having just this one example doesn't disprove that the system is usually sound. It just proves that the system is sometimes not sound (which is already assumed in the meaning of heuristic).

'In kindergarten, we all learned Gödel’s First Incompleteness Theorem, which given a formal system F, constructs an arithmetical encoding of

G(F) = “This sentence is not provable in F.”'

> Aaronson grew up in the United States, though he spent a year in Asia when his father—a science writer turned public-relations executive—was posted to Hong Kong. He enrolled in a school there that permitted him to skip ahead several years in math, but upon returning to the US, he found his education restrictive, getting bad grades and having run-ins with teachers. He enrolled in The Clarkson School, a gifted education program run by Clarkson University, which enabled Aaronson to apply for colleges while only in his freshman year of high school. He was accepted into Cornell University, where he obtained his BSc in computer science in 2000, and where he resided at the Telluride House. He then attended the University of California, Berkeley, for his PhD, which he got in 2004 under the supervision of Umesh Vazirani.

> Aaronson had shown ability in mathematics from an early age, teaching himself calculus at the age of 11, provoked by symbols in a babysitter's textbook. He discovered computer programming at age 11, and felt he lagged behind peers, who had already been coding for years. In part due to Aaronson getting into advanced mathematics before getting into computer programming, he felt drawn to theoretical computing, particularly computational complexity theory. At Cornell, he became interested in quantum computing and devoted himself to computational complexity and quantum computing.

His kindergarten was clearly different than ours :D

Aaronson's academic journey was... fairly nonstandard. The circumstances under which he learnt about this are likely to be correspondingly nonstandard. "We all learned about Gödel in our Undergrad/Highschool/Grad School/whatever else" will easily come across as either condescending or arrogant. Kindergarten strikes me as an amusingly flippant way of saying something closer to "at some point in our academic journeys".

> S(E) = “This sentence has no convincing heuristic explanation accepted by E.”

We can formalise in F what a proof is, but what is a "heuristic explanation"? So this doesn't really make much sense. This guy sounds like a physicist :-)