[FOM] Re: Definition of Large Cardinal Axiom

[Neil Tennant]

Roger Bishop Jones, on Tue Apr 13 15:24:40 EDT 2004, asked

"Can anyone tell me whether there is a generally accepted
definition of what is a large cardinal axiom?"

The subsequent discussion on fom did not yield what might be regarded
as a definitive answer. I am therefore tempted to offer the following
impossibility proof, in order to answer Jones's question decisively,
and in the negative. 

*** There cannot be any generally accepted definition of
"large cardinal axiom", because the definiendum is inexpressible.***

I make no claim for priority in what follows; for all I know, it may
be widespread folk lore, especially among large cardinal theorists. 
But, if it is folk lore, then it strikes me as rather odd that no one
on the list has shared that lore with the wider membership. There is
no implied criticism here; for sometimes the technical experts take
matters such as these as read, without realizing that merer mortals
within the wider forum might not be privy to the lore that underpins a
lot of their thinking.



		On the inexpressibility of the notion 
		"... is a large cardinal axiom (for V)" 
		in the language of set theory.


Consider various formulae F(x) expressing "x is a large cardinal of kind
F (to be found in V)". Hence "something is F" would be a large
cardinal axiom intended to express a truth about V; and we could 
say that the predicate F(x) is apt for the expression of a large cardinal
axiom (about V).

The question addressed here is whether there is a definition of
the notion "... is a large cardinal axiom (for V)" in the language of
set theory. 

Suppose there is indeed such a definition, in the language of set
theory, expressible as a condition C( ) on formulae F with one free
variable [or their code numbers]. I shall reduce this supposition to
absurdity. 

Note that the reductio will be perfectly general.  We are not assuming
that C( ) is an effectively decidable predicate, or an effectively
enumerable predicate, or anything similarly restrictive. We are
assuming only that C( ) is itself a formula in the language of set
theory, with one variable free.  So our reductio will establish, a
fortiori, that there is no effectively decidable predicate C( ) meeting
the requirement; that there is no effectively enumerable predicate 
C( ) meeting the requirement; ... and so on, for any such envisaged
restriction of C( ).

On the supposition above, C('F') would in effect say that 

	"something is F" is a large cardinal axiom (for V). 

That is, we would have the universally quantified biconditional
	
	for all formulae F with one free variable:
	"something is F" is a large cardinal axiom (for V)
	  iff
	C('F').

Clearly there are at most denumerably many such formulae F available
in the first-order language of set theory.

Now let H be an arbitrary formula with one free variable.
Consider the following formula G(H) with one free variable, x:

(G(H))[x]:	x is a rank (in V) such that 
		for all formulae F with just one free variable
		if H('F'), then "something is F" is true in V_x

Then consider the existential claim "something is G(C)". It says something
like "By some rank (in V), all large cardinal axioms (in the sense of C) are
true."

Intuitively, this would appear to be a large cardinal axiom in the
very sense with which we are concerned---for wouldn't such a rank in V
have to be very large? Moreover, it doesn't seem so powerful as to be
(obviously) inconsistent. For, as already noted, there are at most
denumerably many distinct formulae F that can play a role here, hence
at most denumerably many distinct kinds of large cardinal to be
accommodated, in V, by the rank postulated. If each of those different
types of large cardinal is instantiated in V, one could reasonably
expect there to be a rank in V by which they have all been
instantiated. Note also that on the intended reading of C( ), the
countenanced large cardinal axioms must form a consistent set, since
each of them is about a particular kind of large cardinal in V
itself. (The need to stress this particular point was made clear by
helpful comments from Robert Solovay.)

Hence we should have C('G(C)'). 

So the following is a formula apt for the expression of a large cardinal
axiom (about V):

(G(C))[x]:      x is a rank (in V) such that
                for all formulae F with just one free variable
                if C('F'), then "something is F" is true in V_x

and the large cardinal axiom in question is "something is G(C)", i.e.

(*)	for some rank x (in V)
	for all formulae F with just one free variable
        if C('F'), then "something is F" is true in V_x

Since (G(C))(x) is a formula with just one free variable, this implies

	for some rank x (in V)
	if C('G(C)'), then "something is G(C)" is true in V_x

Since by assumption we have C('G(C)'), it now follows that

	for some rank x (in V), "something is G(C)" is true in V_x

So (*) implies 

	for some rank x (in V), "(*)" is true in V_x.

This contradicts G"odel's Second Incompleteness Theorem.

Conclusion: the concept "F(x) is a predicate apt for the formulation
of a large cardinal axiom about V" cannot find formal expression as a
formula C('F') in the language of set theory.


___________________________________________________________________
Neil W. Tennant
Professor of Philosophy and Adjunct Professor of Cognitive Science

http://people.cohums.ohio-state.edu/tennant9/

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