What exactly is zero?

Asked May 15 '22 at 03:07

Active May 15 '22 at 03:11

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I'm learning set theory and the construction of set of natural numbers by Peano's axioms, which says "$0$ is a natural number", and everything starts at $0$.

Then what is $0$?

In set theory, every element in a set is still a set. So I think $0$ is just a set, for example, you can declare it to be $\emptyset$. But I'm not sure if this is right.

Axiom of choice is welcomed.

edited May 15 '22 at 03:11

asked May 15 '22 at 03:07

Evan_Bradley

1

Does this answer answer your question? We can get a model of Peano arithmetic with a nice representation as a set where zero is $\varnothing$ by using the finite Von Neumann ordinals or the Zermelo ordinals. There are other ways to encode the naturals, but these two are especially nice. Is your question mostly about $0$ itself or more about exactly what kind of thing Peano's axioms give you if we accept them? – Greg Nisbet May 15 '22 at 03:13
@GregoryNisbet I believe this proved my hypothesis, thanks. – Evan_Bradley May 15 '22 at 03:18
Many people define $\Bbb N={1,2,3,\ldots,}$, so everything starts with $1$. But I suppose it is still the same question. – Dietrich Burde May 15 '22 at 08:31

What exactly is zero?

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