[LegionMammal978]

By any common set-theoretic construction of Q (e.g., equivalence classes of integer pairs under cross-multiplication), 1 as an element of Z is not literally an element of Q: 1 ∈ Q merely corresponds to 1 ∈ Z, and this correspondence is carefully designed to preserve the ring operations of Z within a certain subset of Q. In this case, the distinction is so negligible that eliding it is harmless, but the whole point of the article is that such elisions can become harmful in more complex constructions.

1 ∈ Z and ASCII '1' can similarly be seen as corresponding in terms of having the same glyph when displayed. But of course, there are far fewer meaningful operations common to the two.

[LudwigNagasena]

1 is not a set, any set-theoretic construction of 1 is not literally 1.

[hanche]

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.

[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.