FOM: consistency equals existence

[Andrew Boucher]

Stephen G Simpson wrote:

> Neil Tennant writes:
>
>  > The G"odel Completeness Theorem guarantees only that if T is
>  > consistent, then some *countable* model of T exists.
>
> Why ``only''?  All that I needed for my consistency-equals-existence
> argument was that *some* model of T exists, provided T is consistent.

I guess I shouldn't try to butt in, because I'm sure Tennant has more
cogent reasons than I, and I guess I shouldn't confuse or potentially
damage his response by my own peculiar views.  But it really seems to me
"only".   What Godel's Theorem shows is that any consistent theory can be
modelled by any countable set.  If someone, for instance, were unsure
about the assumption of the existence of a countable set, you'd probably
trot out the natural numbers (or some such) as an example.  So Godel's
Theorem shows that any consistent theory can be modelled by...the natural
numbers.

So, Hibert's dictum that consistency ensures existence is, on Simpson's
viewpoint, simply the assertion that consistency implies that the natural
numbers exist.  I'm afraid I'd say big deal to this (to state the
obvious, the conclusion is true even without the premise).  In
particular, if ZF set theorists are just talking about the natural
numbers, and if measurable cardinals are just some natural numbers,
considered in a particularly funny way, well then there you go.

<snip>

>
>
> As Hilbert said in a famous
> pronouncement, the points can be tables and the lines can be beer
> mugs, for all we care.

This is so in only one direction.  Once we have a consistent axiom
system, then indeed we don't care what models it.  But we *do* care
whether it is beer mugs or points when we make the axioms.  Take the
example of ZFC.  Some people (I am not one of them) would justify these
axioms by an appeal to their intuition of what sets are like:  that is
why they think it's okay to state these axioms--because that is how they
think sets are.  If you turned around and said, no, you're actually
talking about the natural numbers in a peculiar way, then I don't think
they would feel justified to make the same assumptions.  In brief, the
axioms are justified and stated only because they are meant to describe
the intended model.    And so they would lose at least a large part of
their value if, in fact, there is no such intended model.

For one thing, the existence of an intended model tells us that the
system is consistent.   Why do we believe the Peano axioms are
consistent?  Well, because we think that they describe accurately the
natural numbers (and the successor function).  If there is no intended
model of ZFC (if sets *aren't* like that), well then maybe it's just
inconsistent--and so maybe it doesn't have a model (natural numbers or
otherwise) at all.