Re: FO logic without equality

David C. Ullrich <ullrich@xxxxxxxxxxxxxxxx> writes:

On Mon, 05 Jun 2006 17:09:23 +0100, Alan Smaill
<smaill@xxxxxxxxxxxxxxxx> wrote:

David C. Ullrich <ullrich@xxxxxxxxxxxxxxxx> writes:

On 3 Jun 2006 11:40:16 -0700, "Keith Ramsay" <kramsay@xxxxxxx> wrote:

As long as you're here I have a question, which I should probably
ask at the top before I start with the snippiness:

How do you do the original exercise via the compactness theorem?

Of course if we know that there are arbitrarily large finite
models then the compactness theorem shows there is an infinite
model. But that can't be it, since I don't see how to show there
are arbitrarily large finite models without using a construction
that would also give an infinite model...

How about adding an extra element to the domain?

Deal with constants and functions as you did;
for relations extend the interpretation appropriately;

Well of course that works. But my point is I don't see
how that's any easier than just adding infinitely
many elements to the domain.

Seems like at most the _notation_ might be simpler,
but I really don't see how it's going to make a big
difference even from that point of view...

This is all kind of subjective, but I think this is
easier to find; looking for a one element extension
is a big hint as to how to define things, whereas
the infinitely many disjoint sums is harder to pin down,
IMO.

Once you have the definitions right I agreee there isn't much of a
difference.


************************

David C. Ullrich

--
Alan Smaill
.

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