a function to specify arity-- what do these sentences mean??

Question

I have encountered a function $\pi$ mapping the set of function and predicate symbols to the natural numbers so that for each $k\ge1$, each of the sets {$i \in N | \pi (F_i)=k$}, {$i \in N|\pi(P_i)=k$} is infinite.

Then it says that the purpose of the function $\pi$ is to specify the number of arguments or $arity$ of each function and predicate symbol.

I guess my question is: How do they succeed in specifying arity?

Thanks for your time!

Also, does k vary? What would one of these sets look like? Why are they worth defining? — MerryMC, Oct 02 '14 at 05:11

score 1 · Accepted Answer · answered Oct 02 '14 at 06:01

1

Each function and predicate symbol has an arity, which is the number of arguments to the function or predicate. We can pack all the arities in a function $\pi$. If we have infinitely many function and predicate symbols of arbitrary positive arity, then we are in the situation which you describe. (We could also have infinitely many function symbols of zero arity, i.e. constants, and infinitely many predicate symbols of zero arity, i.e. constant logical values.)

Here is one way to arrange this situation: for each $i,j\geq 1$, define $F_{ij}$ to be a function symbol of arity $i$ and $P_{ij}$ to be a predicate symbol of arity $i$.

answered Oct 02 '14 at 06:01

Yuval Filmus

57,157

Thanks~so could I call the arityof a function analagous to the cardinality of a set? – MerryMC Oct 02 '14 at 06:04
Do whatever helps you. – Yuval Filmus Oct 02 '14 at 06:05
And then pi is partitioning the functions according to their arity? – MerryMC Oct 02 '14 at 06:08
Yes, that's the idea. – Yuval Filmus Oct 02 '14 at 06:08
If a function has arity 5, then does it need 5 variables? – MerryMC Oct 02 '14 at 19:46
1

Exactly. A function of arity 5 has 5 inputs. By definition. – Yuval Filmus Oct 02 '14 at 20:07

a function to specify arity-- what do these sentences mean??

1 Answers1