i’ve already demonstrated that there are infinitely many uninteresting natural numbers. all you have to do is assume that no natural numbers are interesting!
the proof of the “no uninteresting numbers paradox” is sound. you need simply to accept that natural numbers are well-ordered (they are), and that the first uninteresting number is interesting. the proof by contradiction necessarily follows. of this, there can be no dispute.
if you have an intuitive problem with the conclusion, you ought to consider examining the premises.
The naturals are well-ordered, whereas the integers are not. There’s absolutely nothing surprising about the “proof” working differently for these two sets.
I think I see what you’re saying: the “firstness” can’t be taken out of the paradox as easily as I thought it could.
That’s fair enough. What I’d say though, is that the following conclusion is NOT licensed:
–There are no uninteresting numbers
but instead, the following conclusion IS licensed:
–If X is interested in firsts qua first, then no number would remain uninteresting to X given an arbitrarily long amount of time for X to consider the properties of that number.
In other words, no numbers are uninteresting to X.
It’s an empirical question whether there really is a person who, like X, is interested in firsts qua first.
All right, let’s try again:
Instead of leaving “interesting” as some nebulous concept, let’s define it.
*A number is interesting if it is the square of a natural number. *
OK? There’s nothing underhand or self-referential about this definition. We can list the interesting numbers: { 1, 4, 9, 16, 25, … }
But, the original puzzle also stipulates:
The first uninteresting number is interesting
Even ignoring the contradiction in this statement, we have a problem. The set of uninteresting numbers which began with { 2, 3, 5, 6, 7, … } can’t actually start with any number. It can’t have a first number.
By the logic of the original puzzle, the set of uninteresting numbers is necessarily empty, and we can conclude that every natural number is either a square, or the first non-square.
…even though this is demonstrably false.
you have a contradiction in your premises, so it shouldn’t come as a surprise that you can prove any premise you like. clearly the statement:
the first number that is not [a square of a natural number] is [a square of a natural number].
is false.
this is why this “paradox” is meant to be a joke. the idea that an entity can have a property by virtue of it not having that property isn’t meant to be taken seriously. it only appears serious when our interpretation is confounded by the nuances of natural language.
then again, the set of classes that do not contain themselves stirred bertrand russell a great deal.
I was implicitly taking “number” to mean “strictly positive integer”. Of course, the answer will be different if instead one takes it to mean “nonnegative integer”, or “integer”, or something else.
As ultrafilter says, it’s a joke. But that doesn’t necessarily exclude the possibility of further analysis, although it may indicate that that analysis cannot be terribly formal…
My inchoate thoughts, for whatever they are worth:
There cannot be an ordinal number with the property that everything less than it is clearly interesting, while it itself is clearly not-interesting. I think we can mostly agree to this; there cannot be a clear cut-off point from interesting numbers to the first non-interesting number.
Does it follow by induction that all ordinal numbers must be interesting?
Well, it’s like any other soritical paradox. There is no number of drops of water at which there is a clear cutoff between a non-ocean and an ocean. But even though a naive inductive analysis would therefore conclude that there therefore cannot be any transition from non-oceans to oceans, we’d all agree otherwise that some things are oceans and some things aren’t.
Given a function from the natural numbers N to the discrete two-element set {true, false} which hits false at some point, one can always enumerate through it to find the first value at which it hits false. A simple “for loop”. In this sense, it’s certainly true that any function from N to {true, false} which hits false at some point does so for the first time at some particular clear cutoff point.
But I think it is a mistake to suppose of every ordinary language property that it is fruitfully modelled as a function from its domain into {true, false}.
Even among just myself and what I, myself, would consider oceans, if you asked me, I would say there’s no sharp cut-off. If you forced me to go through each particular number of drops and give a “Yes” or “No” answer as to whether it could comprise an ocean (in the hopes that, upon doing so, I would be forced to observe a resulting sharp cut-off at some point), I would say you are misunderstanding how ascriptions of oceanhood work.
Then I would say the definition of an ocean is not based on the number of drops. So obviously there cannot be a transition from ocean to non-ocean based on the number of drops. But I was just pointing out the fallacy of the logic based on our individual definitions of the what constitutes an ocean or not. If you can define an ocean in formal logic, you can determine whether something is or is not an ocean.
I would say there can be a formal logic for undecidable or ambiguous predicates, able to treat oceanhood without the supposition that every particular instance either definitely is is or definitely isn’t an ocean; for example, though I wouldn’t consider it the relevant logic to use in this instance either, there is formal intuitionistic logic which denies the automatic truth-hood of “P or not P”, and in that sense allows one to define predicates without the supposition that there must be a clear YES or NO answer to their holding in any particular case.
this is not strictly true. consider, for example, any infinite, but not well-ordered set. there is no guarantee that you will ever hit a false. in fact, even for well-ordered sets, consider, maybe, the set of numbers which have been enumerated.
