Pat, (01)
JFS>> I would prefer to say that a function is a mapping
>> between two sets. (02)
PH> But then we have to say what a mapping is... (03)
In set theory, you specify the mapping extensionally by pairs
or intensionally by rules or axioms. (04)
But in category theory, you talk about maps without actually
specifying them. In fact, many authors replace the word 'map'
with 'arrow' in order to avoid the implication that they need to
specify the maps in order to talk about them or prove theorems
about them. (05)
JFS>> For infinite sets, all specifications must be intensional. (06)
PH> What you say here could be interpreted as a claim that
> infinite sets cannot be specified (since sets are extensional
> and extensional must be finite?) which is of course not true. (07)
A specification of an infinite set by finite rules: (08)
1. The terminal symbols of a language L: {a, b, c} (09)
2. The nonterminal symbols of L: {S} (010)
3. The start symbol of L: S (011)
4. The grammar rules of L: (012)
S -> a (013)
S -> b S c (014)
5. The set of sentences of L consists of all and only those
strings of terminal symbols that can be generated by one or
more applications of the grammar rules in #4. (015)
John (016)
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