Pat, (01)
I have nothing against using sets, finite or infinite. I was
just making the observation that you can't give an extensional
specification for infinite sets. (02)
JFS>> ... the integers themselves must be specified by some rules
>> or axioms. (03)
PH> Most mathematicians would disagree. The natural numbers cannot
> be fully specified by any finite number of axioms, yet we all feel
> that we know what they are, and are quite happy to refer to them. (04)
I agree. But you can state a finite set of metalevel rules for
generating a denoting expression for each natural number. (05)
For example, I can say that any string of terminal symbols generated
by the following grammar denotes a natural number: (06)
Terminal symbols: {S, 0} (07)
Nonterminal symbols: {N} (08)
Grammar rules: (09)
N -> 0
N -> S N (010)
This is a finite specification that generates a denoting expression
for all and only the natural numbers. You can add a few Peano-style
axioms to get the standard model without the weird number-like things
of the non-standard models. (011)
John (012)
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