Comment by btilly
1 month ago
Spoken like a true formalist.
It doesn't really have to mean anything when we say that the reals are a larger set than the natural numbers - that's just the conclusion of the game that we are playing.
What fraction of people who "know" that there are more reals than natural numbers, do you think really understand that this is not an eternal verity of mathematics, but only a conclusion that follows from a particular set of rules that we're playing the mathematics game with?
> What fraction of people who "know" that there are more reals than natural numbers, do you think really understand that this is not an eternal verity of mathematics, but only a conclusion that follows from a particular set of rules that we're playing the mathematics game with?
The claim that there are more reals than naturals holds given classical ZF(C) set theory. But there are alternative set theories in which the reals are countable, e.g. NFU+AxCount. These alternative set theories ensure all reals are countable by rendering Cantor’s diagonalisation argument invalid, since their axioms are too weak to validate it. But, they contain all the same reals as the high school mathematics concept of “reals”. So, there are many reals, and that some of them are countable and others are not are indeed “eternal truths” (it is an eternal truth that whatever axioms have the consequences they do), but the everyday (non-expert) concept of reals isn’t any of them in particular - and it is unclear if the dominance of classical notions in mainstream professional mathematics was historically inevitable or a historical accident - maybe, on the other side of the galaxy, there exists some alien civilisation, in which different foundations of mathematics are mainstream, because their mathematics took a different evolutionary course from ours - maybe for them, reals are classically countable, and uncountability is an exotic notion belonging to alternative foundations of mathematics
As I pointed out at https://news.ycombinator.com/item?id=44271589, there are systems that can accept Cantor's argument, without concluding that there are more reals than rational numbers.
As you point out, there are many mathematical systems that contain all of the numbers in the high school mathematics concept of "reals". Since those with a high school understanding of reals do not know which of those systems they would agree with, they should not be asked to accept as true, any results that hold in only some of those systems.
And that is why I don't like mathematicians telling lay audiences that there are more reals than rationals.
"Cantor's diagonalization argument" is best understood as a mere special case of Lawvere's fixed-point theorem. Lawvere's theorem is really the meat of the argument, and it's also the part that is very easy for exotic systems to "accept", since it's close to a purely logical argument. Whether these systems truly accept "Cantor's argument" is perhaps only a matter of perception and intuition, that people may perhaps disagree about.
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I agree, but I also want to clarify that cantors argument was about subsets of the naturals (N), or more precisely functions from N to Bool (the decidable subsets). This is where the diagonal argument makes sense.
So to conclude that there are more reals than naturals, the classical mathematical argument is:
a) There are more functions N to Bool than naturals.
b) There are as many reals as functions from N to Bool.
Now, we of course agree the mistake is in b) not in a).
11 replies →
> Spoken like a true formalist.
Doesn't seem to be a bad thing. There are some famous cranks who reject the concept of infinity, since I suppose they have problems wrapping their head around it.
> What fraction of people who "know" that there are more reals than natural numbers, do you think really understand that this is not an eternal verity of mathematics, but only a conclusion that follows from a particular set of rules that we're playing the mathematics game with?
People misunderstand mathematics all the time. It's ok, it's part of the journey.