Re: Review of Mueckenheims book.

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), 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}?

--
"It's my belief that when religion and pseudoscience achieve an
official status within a culture [...], then genocide, war,
oppression, injustice, and economic stagnation are sure to follow."
-- David Petry, on why |X| < |P(X)| is bad, bad, bad.
.

Relevant Pages

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  • Re: Review of Mueckenheims book.
    ... function requires that we specify the source and target. ... formed by taking the inclusion N -> P, ... on any of the several usual accounts of N ...
    (sci.math)