4 ms·

A fun question I have been pondering recently: what is the smallest positive integer that has not yet been written or uttered by a human being?

braythwayt 7y ago
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."
- zzo38computer 7y ago
  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.
  - saagarjha 7y ago
    I want to post an extremely long comment and see what the limit is now…
    - feikname 7y ago
      It seems to be 10k chars. Otherwise, it displays a "message too long" error.
      - braythwayt 7y ago
        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.
- ivalm 7y ago
  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).
- chongli 7y ago
  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#Unrestricted_comprehension https://en.wikipedia.org/wiki/Axiom_schema_of_specification#... [2] https://en.wikipedia.org/wiki/Russell%27s_paradox https://en.wikipedia.org/wiki/Russell%27s_paradox
  - braythwayt 7y ago
    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?”
    - chongli 7y ago
      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!
      - braythwayt 7y ago
        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.
booleandilemma 7y ago
Reminds me of this: https://en.m.wikipedia.org/wiki/Interesting_number_paradox https://en.m.wikipedia.org/wiki/Interesting_number_paradox
missblit 7y ago
You can try to get a rough idea by doing web searches for various random integers of different lengths (courtesy of random.org, which graced me with lots of 7's). 7 - A lucky number 92 - How old Lady Gaga wants to be when she retires 420 - A class of dinghy 7260 - A wireless card 5108 - A section in the U.S. Code about supplemental claims 51664 - A hydraulic filter 794101 - A postal code in India 7578146 - A patent number of a knitting loom clip 74703567 - A clothing trademark serial number for "authentic bad lands" 760314276 - No useful search results. 2641309883 - No useful search results 24235991397 - One (useless) search result 415865230233 - Zero search results
nabla9 7y ago
Every time you answer the question the correct answer changes.
- rsrsrs86 7y ago
  Side effects in math!!!!!!!!!!
Someone 7y ago
If you limit it to mathematicians, it was, at some time, 8795 ;-) (https://oeis.org/wiki/Frequency_of_appearance_in_the_OEIS_database https://oeis.org/wiki/Frequency_of_appearance_in_the_OEIS_da...) More interestingly, the set of integers seems to be split into two sets: those that occur relatively often, and those that occur relatively rarely (same page)

braythwayt 7y ago

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."

zzo38computer 7y ago
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.
- saagarjha 7y ago
  I want to post an extremely long comment and see what the limit is now…
  - feikname 7y ago
    It seems to be 10k chars. Otherwise, it displays a "message too long" error.
    - braythwayt 7y ago
      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.
ivalm 7y ago
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).
chongli 7y ago
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#Unrestricted_comprehension https://en.wikipedia.org/wiki/Axiom_schema_of_specification#... [2] https://en.wikipedia.org/wiki/Russell%27s_paradox https://en.wikipedia.org/wiki/Russell%27s_paradox
- braythwayt 7y ago
  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?”
  - chongli 7y ago
    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!
    - braythwayt 7y ago
      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.

booleandilemma 7y ago

Reminds me of this: https://en.m.wikipedia.org/wiki/Interesting_number_paradox https://en.m.wikipedia.org/wiki/Interesting_number_paradox

missblit 7y ago

You can try to get a rough idea by doing web searches for various random integers of different lengths (courtesy of random.org, which graced me with lots of 7's). 7 - A lucky number 92 - How old Lady Gaga wants to be when she retires 420 - A class of dinghy 7260 - A wireless card 5108 - A section in the U.S. Code about supplemental claims 51664 - A hydraulic filter 794101 - A postal code in India 7578146 - A patent number of a knitting loom clip 74703567 - A clothing trademark serial number for "authentic bad lands" 760314276 - No useful search results. 2641309883 - No useful search results 24235991397 - One (useless) search result 415865230233 - Zero search results

nabla9 7y ago

Every time you answer the question the correct answer changes.

rsrsrs86 7y ago
Side effects in math!!!!!!!!!!

Someone 7y ago

If you limit it to mathematicians, it was, at some time, 8795 ;-) (https://oeis.org/wiki/Frequency_of_appearance_in_the_OEIS_database https://oeis.org/wiki/Frequency_of_appearance_in_the_OEIS_da...) More interestingly, the set of integers seems to be split into two sets: those that occur relatively often, and those that occur relatively rarely (same page)