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Comment by braythwayt | Hacker News Reader

Were Raymond Smullyan still with us, he would point out that you just answered your own question. And yet didn't. In which case you did. Provided you didn't.

Another example:

"The smallest integer that has a description too long to fit in a Hacker News comment."

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But you can definitely say "what is the meta information or algorithm to compute X". E.g. Pi is an irrational number, but we have a bunch of ways to specify pi (we just can't explicitly write down the full sequence of numbers). Another interesting thing, while the full sequence is not computable, any particular precision IS computable (ie for any number Y that you can write down exactly, you can compute the Y'th digit of pi).

These sorts of self-referential statements are actually forbidden in the axioms of Zermelo-Fraenkel set theory because they require unrestricted comprehension [1]. ZF set theory specifically restricted comprehension to avoid Russell’s paradox [2] as well as countless other statements, like these, which lead to absurdities in math.

Fun stuff anyway!

[1] https://en.wikipedia.org/wiki/Axiom_schema_of_specification#...

[2] https://en.wikipedia.org/wiki/Russell%27s_paradox

Absolutely!

But it is more than fun to explore them, as Smullyan points out. For example, it leads to discussions like, “What is a description?” Which is near and dear to my heart, as it leads to “What is a program?” and, “What is the specification of the machine that runs the program?”

Or some of my favourites:

“What is a number? Do numbers exist?”

I’ve been having a fantastic time in my philosophy of math course this term. It’s incredible how deep and how long these debates have been running. Cantor, Frege, Russell, Hilbert, Heyting, Gödel, Quine, and on and on!

I forget which book is the source of this, but I recall Smullyan writing about (I hope I have it roughly right) asking a child whether they could prove something they knew about mathematics or logic, and the child replied "What is a proof?"

Smullyan said that this was--if you took it literally--an incredibly deep question.

What is the limit? This <textarea> element does not specify one, nor do I see any limit mentioned in the text of this web page.

I want to post an extremely long comment and see what the limit is now…

It seems to be 10k chars.

Otherwise, it displays a "message too long" error.

10k characters might be a browser limit, but I tried pasting 9,000 characters from one of my essays into a comment, and it was also rejected as being too long.

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