[FOM] Re: Definition of Large Cardinal Axiom
Roger Bishop Jones
rbj01 at rbjones.com
Mon Jun 14 16:26:49 EDT 2004
I'd like to express my thanks to Neil Tennant for
his answer to my question about the definition of large
cardinal axioms.
However, for a number of reasons I don't think his
answer is as decisive as he hoped.
Before asking some specific questions and making
comments on his post I should like to clarify
my original question.
Firstly, I asked whether there is a generally
accepted definition, and clearly there isn't.
So we may as well consider the merits of any
candidates we can come up with which might be
deserving of general acceptance, and in this new
project of course a negative proof would be important.
However, I didn't ask for a *formal* definition,
and not one specifically formalisable in first
order set theory. More than one of the proffered
definitions included the requirement that the
axiom be true, and it seems perhaps unlikely
(though I have not seen definitive proof) that
under any reasonable semantics set theoretic
truth will be definable in set theory.
Nevertheless, definitions which include truth
in them may be useful.
Thirdly, I did not have in mind that the
definition of large cardinal need be complete.
I was thinking in terms of what might be considered
minimal requirements for an axiom to be considered
a large cardinal axiom, i.e. what might be
put forward as necessary if not sufficient conditions.
Such may be useful in demonstrating limitations on
what may be done with large cardinal axioms, and an
example is the demonstration that large cardinals
cannot settle CH, possibly on the basis of the
supposition that a large cardinal axiom must be
invariant under small forcing.
I would be particularly interested in sufficient
conditions to ensure that large cardinal
axioms are all mutually consistent.
The point of these observations is simply that
even if Prof. Tennant has proven that the concept
of large cardinal axiom cannot be completely
formalised in set theory, this does not mean there
is no point in seeking an informal definition
or in seeking formally definable necessary conditions.
As to the truth of his negative result, I am inclined
to believe it, but have not yet understood his
proof.
On Friday 11 June 2004 8:37 pm, Neil Tennant wrote:
...
> On the inexpressibility of the notion
> "... is a large cardinal axiom (for V)"
> in the language of set theory.
I don't understand the significance here of adding
"(for V)".
I pretty much followed the line of argument until
the very end:
> So (*) implies
>
> for some rank x (in V), "(*)" is true in V_x.
>
> This contradicts G"odel's Second Incompleteness Theorem.
but I'm afraid I can't see why this is the case.
Even if the proof is correct or some other can be supplied
(which seems to me plausible) I don't think that this result
would reduce my interest in my original question (or perhaps
rather the one with "generally accepted" replaced by some other
criterion of merit).
Roger Jones
- rbj01 at rbjones.com
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