[LudwigNagasena]

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.

[hanche]

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!

[LegionMammal978]

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.