FOM: Re: SOL confusion

[Harvey Friedman]

Reply to Roger Jones 9:19PM 9/7/00:

>> >The point has been made that it is possible to express in SOL
>propositions
>> >which cannot be expressed in FOL (under its usual semantics).
>>
>> When expressed entirely in terms of the usual semantics, this has been
>> realized, at least in some form, for over 100 years. The machinery to
>> demonstrably refute is more recent, perhaps 70 years ago.
>
>Could you clarify this response for me.
>You *seem* in your first sentence to be saying that the point is long
>established, but I am unable to place any interpretation on your second
>sentence.
>What machinery are you speaking of and what does it refute?

There is a sentence in second order logic with only equality whose models
are exactly the domains that are finite. There is no sentence in first
order logic with only equality whose models are exactly the domains that
are finite.

When I said

>>The machinery to
>> demonstrably refute is more recent, perhaps 70 years ago.

I meant

The machinery to establish this fact is more recent, perhaps 70 years old.