Comment by hanche
1 year ago
I am not sure I understand what you mean by “literally” here. For sure, if you use Zermelo–Fraenkel set theory as the foundation of mathematics, as is commonly done, every mathematical object is a set. The first definition of 1 encountered in that setting is the singleton set {0}, where 0 is the empty set. (And 2={0,1}, 3={0,1,2} and so forth – you get the picture.) This is precisely the sort of thing this is all about: The natural numbers are uniquely described up to unique isomorphism by some variant of the Peano axioms after all.
That doesn’t make 1 a set. Its representation in ZFC is a set. But its representation in eg lambda calculus is a function.
Saying that 0 belongs to 1 is false no matter what one uses to represent those numbers in any ZFC formalisation of numbers.
It’s a map-territory distinction.
Ah, that depends what the meaning of “is” is, does it not?
On a more serious note, if you are of a certain philosophical bent you may believe that the natural numbers have an existence independent of and outside of the minds of humans. If so, 1 is presumably not a set, even if we don’t fully understand what it is. I certainly don’t think of it as a set on a day to day basis!
But others may deny that the territory even exists, that all we have are the maps. So in this one map, 1 is a set containing zero, but in that other map, it is something different. The fact that all the different maps correspond one-to-one is what counts in this worldview, and is what leads to the belief – whether an illusion or not – that the terrain does indeed exist. (And even the most hard nosed formalist will usually talk about the terrain as if it exists!)
But this is perhaps taking us a bit too far afield. It is fortunate that we can do mathematics without a clear understanding of what we talk about!
If there are many different ways to represent what something 'literally is', then how do we know for sure that ASCII '1' isn't a true representation of the literal number 1, just considered under different operations? We can say that 1 + 1 + 1 ≠ 1 (in Z), and we can also say that 1 + 1 + 1 = 1 (in Z/2Z): the discrepancy comes from two different "+" operations.
For that matter, how do we know what infinite sets like Z and Q 'literally are', without appealing to a system of axioms? The naive conception of sets runs headlong into Russell's paradox.