is this room active?

@user709833 - Comes and goes. Though lately it's gone.

@MaliceVidrine Do you know how induction is presented logically/formally?

like what is the correct way to write induction

@user709833 - In what context? It would probably be written differently for the theorem in ZFC about the standard natural numbers than as an axiom schema in PA.

how so

Well, the first I could write as a single sentence in the language of set theory. The second you can only sort of metamathematically describe the axiom schema; there is no single formula of PA that entails the induction schema.

In the set theoretic case, it's $\forall U(U\subset N . 0\in U . \forall x\in N(x\in U \Rightarrow x+1\in U) . \Rightarrow N\subset U)$

metamathematically describe the schema? what do you mean by that?

i don't really understand the difference between axiom and axiom schema and single formula, they all look the same to me

i googled it but hard to understand, it still looks the same

They are similarly phrased, but the axiom schema is phrased like "For every formula of arithmetic $\phi(x)$, the formula $\phi(0) . \forall x(\phi(x)\Rightarrow \phi(x+1)) . \Rightarrow\forall n(\phi(n))$ is an axiom"

We can't say $\forall\phi(...)$ because $\phi$ isn't an object in the theory of PA, it's an object in the metatheory that we're using to talk about PA.

If we have an axiom schema do we need to literally describe it like that? as an axiom schema metamathematically? and say that "for every formula" etc?

like if we just listed that statement by itself would it be clearly a schema?

That's more of a rhetorical issue; if you're writing or speaking for an audience that is familiar with the device of axiom schemata, you can probably get away with not saying something quite that specific. But for a general audience, or for an audience newer to the area, you probably want to specify somehow that you're characterizing a set of axioms with a particular form.