The smallest integer no one has ever thought of no longer exists.
Posted Jul 10, 2018, 2:35 PM UTC
217
1 is the smallest integer.
1
Please explain this to me in layman's terms, I am legitimately curious but very confused.
1
The first thing to consider is what I mean by 'thought of'.
If I give you a description of a number, and only one number can match that description, then I have 'thought of' the number. For example, 1+1, 2, and the only prime and even number, are all sufficient descriptions that at most one number matches, namely 2 in this case. However, if I tried to describe 2 by only saying that it's prime, then that would not be enough info to identify 2. Then I would not have 'thought of' 2.
We can say that there exists a set of all numbers no one has thought of. If I now give you enough info to identify at most one member of that set (like that it's the smallest natural number in that set), then I have thought of that number. Because the number now has to have the properties of having been thought of and never having been thought of, its properties contradict each other (like a number being both even and odd), so whatever I described can't possibly exist.
But it could have, before anyone described it.
Important thing to note, though: there has never been a smallest integer we hadn't thought of, because you can always go smaller. The smallest natural number we hadn't thought of did exist at some point, though. Before we were even around, it was 1 (or 0 depending on how you define the naturals).
1
That's very interesting, thanks for explaining.
2
do you define smallest as closest to zero or most negative?
One is the closest non-zero positive integer to zero, the most negative integer will never be figured out because there's always one more integer below it waiting to be named.
It can't be defined if nobody has thought of it.
It being defined requires having thought of it.
Yes... that's pretty much what I said.
Yes.
I am terribly tired. Hence the post itself.
You can define something without having thought of it specifically. I can define that an integer consists of any number of digits in any order with either a + or a - before the series of digits and that would define every integer
In which case you would be said to have thought of the set of all integers, but not any particular integer. If you create a set of criteria by which a particular integer could be chosen, then it could be argued that you have thought of that integer, even if you aren't aware of its other properties.
EDIT: Nevermind, the implications of my line of thought are staggeringly impractical.
Post serves as proof of both cases.
it does not, just one of them
go to sleep
Any object satisfying the predicate that it has not been thought of has, by the virtue of having been defined (EDIT: in part), been thought of, and so is a contradictory construction, so it cannot be said to exist once defined. Smallness being irrelevant, the proof serves for both cases.
most people would define an integer being "thought of" as actually conceiving of the specific integer, specifying it explicitly in some way (e.g "negative one thousand and one" or "negative thirty eight times six thousand and four")