FOM: sterility?

Vladimir Sazonov sazonov at
Tue Mar 23 15:11:39 EST 1999

Randall Holmes wrote:

> I reiterate the point that second-order logic does not have any more
> set-theoretical prerequisites than first-order logic.  Second-order
> quantifiers may be understood to range over _properties_.  "Property"
> is not a specifically mathematical notion (it is inarguably
> topic-neutral and arguably "logical").  So objections to second-order
> logic thus presented cannot be based on claims that second-order logic
> is set theory in disguise.

Does anybody understand what is an _arbitrary_ property?  Say, 
what about the following property of natural numbers

P(n) <=> `n is a natural number which can be denoted by
	an English phrase consisting of < 1000 symbols'?

What about the (paradoxical) least n such that not P(n)?

We have too unsteady ground for discussing 2nd order logic in a 
"pure" form outside ZFC or the like 1st order formalism.  The 
only existing way to make any informal idea or notion (like 
`property') precise and mathematically rigorous consists in an 
explicit presenting some formal rules of reasoning on this 
subject.  We are doing this usually via suitable extension of 
1st order logic by non-logical axioms.

I completely agree with Simpson's characterizing second-order
logic as a sterile myth. This logic can meaningfully exist only
inside a suitable 1st order context. Otherwise it is extremely
unclear what we are talking about.

Vladimir Sazonov

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