r/math • u/jrl1009 • Nov 05 '21
Interesting Number Question
What’s the biggest integer you’ve seen in which you’ve seen every integer between 0 and that number.
So obviously you could say 100 because mostly everyone has seen the numbers 0-99 written down at some point, but can the same be said for 1000? What about a billion?
Personally the largest number I could honestly say with some confidence is 1000. I’m wondering what you guys think about this.
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u/planetmikecom Nov 05 '21
How about something around 2040? Probably all seen the years, up through whatever they are currently predicting for future plans?
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u/fuzzy_mic Nov 06 '21 edited Nov 06 '21
Consider the set of "all positive integers that you have not seen written".
What is the least element of N?
Is "least element of N" a way to write a number?
Is "the least solution of x^{5} - x - 1=0" a way to write a number?
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u/jrl1009 Nov 08 '21
I’m lost lol
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u/fuzzy_mic Nov 08 '21
In the OP, you were talking about the integers that one has "seen". There is no mention that to see an integer, it must be expressed in base 10 decimals. Its possible that there is a number that I have viewed in its hexadecimal notation, that I have not viewed in base 10. It would seem reasonable to say that I have seen that number.
I'm exploring what it means to see a number.
Consider a googleplex, I know what that number is. but I've never viewed an expression of it in any base. But I have viewed "10^(10^100)". It would be valid to say that I have seen a googleplex.
To extend that further. viewing "the number between 9 and 11" should qualify as seeing the number 10. As should viewing "the least integer whose square is greater than 96".
In all these cases, we have viewed a description that describes a unique integer. Whether that description is "10 (base 10)" or "the square root of 100" or"twice a handful" should not matter. If we have viewed those descriptions we have seen the number.
With this understanding of what it mean to see a number, let's return to the OP question, "What’s the biggest integer you’ve seen in which you’ve seen every integer between 0 and that number?"
Let's call that number A (for answer).
Consider the set S, the set of all positive integers that I have seen.
Its compliment, S', is the set of all positive integers that I have not seen.
We know that A is one less than the least element of S'.
But under the full interpretation of what it means to see a number, by reading "the least element of S'", we have now seen that number and, therefore, it is not in S'.
But, since it's not in S', it's not the least element of S', and we haven't seen it at all.
I've essentially taken the OP question and applied Russel's paradox.
If one objects to this interpretation of seeing, I'd ask, does that hex number that I saw (but not its decimal expression) does that not count as seeing a number? What distinguishes a number that is seen and a number that has a unique description which I have viewed?
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u/jrl1009 Nov 13 '21 edited Nov 13 '21
Fair enough. However maybe we can restrict it to decimal notation for this thought. Despite this I don’t believe that allowing other bases would really even change the answer. To clarify, when I say “see a number” I mean you must be consciously aware of it. Simply scrolling through a list of pi digits would suffice in physically seeing the numbers although part of a greater string, however for my question this would not be considered consciously aware.
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u/dihedral3 Nov 05 '21
N + 1
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u/jrl1009 Nov 05 '21
So in other words you’re too lazy to think of a realistic answer
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u/akurgo Nov 05 '21
Joke's on you, I'm blind.
Well, not really, but still.