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Re: [HM] definit, definitheit
Posted:
Aug 26, 2001 6:48 AM
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I would like to add a rather obvious remark concerning the [alleged]
"wrongness" of the first-order "shadow" of the axiomatic characterization
of
* the [set of] natural numbers (the Dedekind-Peano axioms)
and
* [the universe of] sets (the current ZF or ZFC).
There is one essential difference between the two cases.
The very formulation of the correct Dedekind-Peano axiomatization (let
alone the proof that it is categorical -- ie, characterizes the natural
numbers uniquely up to isomorphism) requires an ambient theory, *within*
which this characterization is performed. This ambient theory is [some
kind of] set theory. This is what Dedekind did in 1888, and what one
normally does nowadays.
Now, what would the ambient theory be in the case of [the universe of]
sets? ... Presumably, second-order variables (or predicate variables)
should vary over the class of all ... what, exactly?
So, while all competent logicians know very well, and warn their students,
that First-Order Peano Arithmetic is but a pale shadow of what was
intended, their adherence to first-order ZF is motivated by horror of
circularity. For example, Azriel Levy, in his text-book on set theory
(which unfortunately I do not have at hand as I am writing) makes the
latter point explicitly. He seems to regard higher-order set theory as
suspect if not incoherent.
Now, I am not claiming that this [apparent?] difficulty cannot be resolved;
perhaps it can. What I am claiming is that there appears to be a difficulty
here, in the case of sets, that is absent in the case of natural numbers.
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