Re: Review of Mueckenheims book.

"Jesse F. Hughes" <jesse@xxxxxxxxxxxxx> writes:

lrudolph@xxxxxxxxx (Lee Rudolph) writes:

"Jesse F. Hughes" <jesse@xxxxxxxxxxxxx> writes:

In category theory, a function is not just a set of ordered pairs. A
function requires that we specify the source and target. So, for
instance, we write id_N:N -> N. The function you're talking about is
formed by taking the inclusion N -> P(N), but this inclusion is a
different function than id.

Of course, in set theory, a function is just a set of ordered pairs
and so there is no need to pick out a domain and codomain. Both id_N
and the inclusion are, in this sense, the same function.

They certainly are not, on any of the several usual accounts of N
(and on the account of "same" which makes it mean "identical";
if "same" means "naturally isomorphic", I guess you're okay).

No, I meant identical.

If a function is defined by its graph (as is often the case for set
theorists),

and which is cool with me

then the identity on N is the set {<n,n> | n in N} and the
natural inclusion N -> P(N) is also the set {<n,n> | n in N} (which
uses the fact that each n is also a subset of N).

Perhaps you thought I was speaking of the map n |-> {n}?

Well, yes. And I also seem to be a bit confused.

When I wrote "the several usual accounts of N", I was thinking of two
in particular. The less popular (but surely not totally unknown?)
has 0 = {} and s(n) = {n} for all n in N. On that account, surely
only 0 is both an element and a subset of N?

The more popular is has 0 = {} and s(n)=n \cup {n}, right?
I have just taken aleph-0 moments to check, and sho'nuff, you're
right there.

Huh.

Lee Rudolph
.

Relevant Pages

  • Re: Review of Mueckenheims book.
    ... lrudolph@xxxxxxxxx (Lee Rudolph) writes: ... function requires that we specify the source and target. ... formed by taking the inclusion N -> P, ...
    (sci.math)
  • Re: HP-UX "mksysb" ?
    ... So the complete command would be: ... -A Based on the files that are specified for inclusion, ... with what you specify for inclusion, if it's in another volume group ... then you may also get all of it's brothers, sisters, aunts, uncles, ...
    (comp.sys.hp.hpux)
Quantcast