Why aren't all numbers in the set uninteresting? Did someone make a mistake when defining it?
Perhaps the minimal element should be removed from the set; there will be plenty of members that still remain.
Why aren't all numbers in the set uninteresting? Did someone make a mistake when defining it?
Perhaps the minimal element should be removed from the set; there will be plenty of members that still remain.
Serious response? In that case the set still has a smallest member which can then be removed, if we keep going eventually there will be no uninteresting numbers remaining.
The problem with that is the explanation of why each number is interesting becomes:
the smallest member of the original set of uninteresting numbers
the second smallest member of the original set of uninteresting numbers
the third ...
...
That version of "interesting" quickly becomes "not interesting". The concept simply defies mathematical logic.
It reminds me about the logic puzzle of the criminal sentenced to death, where the judge says "you will be executed on or before Sunday, and you won't know what day it will be until we come for you."
The criminal knows it can't be Sunday, because he would wake up on Sunday and know he was going to be executed that day. But if Sunday isn't possible, on Saturday he would know he was being executed that day; so Saturday wasn't possible either. The same reasoning can be repeatedly applied to every day between now and Sunday.
It's obviously flawed reasoning (Surprise! they execute you on Thursday), but the flaw is difficult to articulate.
This isn't how math works.
When you get to the point in a proof of the irrationality of root two where you've demonstrated that if it is expressible as a fraction p/q, then both p and q have to be even, you don't then need to proceed to prove that if they're both even, then they both have to be divisible by four, and then if they're both divisible by four, that means they're both divisible by eight...
I mean, you can, but you don't have to.
You can just say 'if it's a rational number then it has a reduced form where p and q have gcf of 1, so if p and q would both have to be even, that is a contradiction'.
Same with the 'set of uninteresting numbers'. If 'being uninteresting' is a property numbers can have, then the 'set of uninteresting numbers' exists, and it has a least member. Being the least member of the set of uninteresting numbers is interesting.
You don't have to infinitely regress from here and get tied up in knots saying that surely there is some 'first truly uninteresting number' to prove that the set is actually empty - you can just see that you must have gone wrong somewhere. Either:
1) Being the least member of the set of uninteresting numbers isn't as interesting as we assume.
or
2) 'Being uninteresting' is not a property numbers can have
I think actually of the two, 1) is more likely the case.
But that doesn't defy mathematical logic. It is a consequence of mathematical logic.
There's a third option. The definition of uninteresting we're using may be flawed. Here's a quick stab at a more rigorous approach:
We could start by defining a set of "all numbers that are uninteresting other than by membership or position in this set".
That describes the set the proof naively called "interesting numbers" without the contradiction.
Then we could create a second set with all members of the first set except those that are interesting because of where they are in that set (smallest, whatever). This is a new version of "interesting numbers" that approaches the version in the original proof but is, in human terms, less interesting. As you said, "Being the least member of the set of uninteresting numbers isn't as interesting as we assume."
We could repeat that, making a sequence of sets that approach the definition of interesting in the original proof, but the definition of each set is progressively less interesting in human terms.
Then if we really want to be rigorous, we could talk about "first degree interesting" (what most people mean), "nth degree interesting", or "asymptotically interesting", but the last one is an empty set.
If someone tasks me to create a set of even/prime/blue/rectangular/crunchy/uninteresting numbers I have two options:
1) I list each and every number that is part of the set. It is OK if the set is countably infinite, we can wait.
2a) I grab my special black box that receives a number and lights up a red or a green LED depending on whether the input is a member of my conjured up set or not;
2b) I grab the other special black box, this one has a single LED (to indicate it is switched on) and a push button which prints out the next member of the set on infinite 7-segment displays. The box is a bit wider than the 2a) unit.
These are mostly traversable, e.g. my 2b) generator could be built from a counter and a 2a) tester, or my 2a) tester could use a table lookup backed by a 1) list for all I know.
What they can/should not do is retroactively change their mind on the membership of a particular number:
- It is either in the 1) list or not, no erasers, no backsies;
- 2a) should always respond with the same LED for a given number, no moon phase lookups, no RNG, no checking of previous LED responses;
- 2b) can not even be rewound so it is impossible to tell if it would produce or skip the number, should we coerce it somehow to start again (we can't).
So using any of the two and a half mechanisms lead us to a set where the minimal element should have the same property as any other element: it is exactly as even/prime/blue/rectangular/crunchy or uninteresting as the rest of the set.
3) Numbers can be uninteresting, but the property is not binary.