Re: Cantor's circular "proof" that evens = integers

WHY do you have to
retreat to semantics.

On May 28, 9:14 am,
stevendaryl3...@xxxxxxxxx (Daryl McCullough) wrote:
The reason is because something only counts as a "proof" in
first-order logic if it is a *finite* sequence of formulas
such that each formula is either an axiom or follows from
previous formulas by the rules of inference. But finiteness
cannot be pinned down by a first order axiomatization.

Exactly. Now compare that with this:
But what forces Negation(y,z) and And(y,z,w) to be about formulas?
What forces Proof(x,w) to be about proofs?
You can't determine *syntactically* that a formula
is about proofs or is about formulas.

That is irrelevant; as I said before, what is determined
is ALWAYS determined NOT to be about any particular
thing (unless you are lucky enough to get a first-order
theory that is simple enough to be categorical; and for
these purposes, theories THAT simple ARE TRIVIAL;
for THIS discussion, THAT is THE appropriate dividing-
line BETWEEN trivial and non-trivial): what is determined
(i.e. proven) is determined to be true in ALL models of
the theory and therefore SIMULTANEOUSLY "about"
a CORNUCOPIA of different things (and therefore about
NONE of them specifically or in particular). That which
might be ABOUT some ONE thing (because it is a
result available in and only in some ONE model instead
of all of them) is NOT proven and therefore NOT determined,
not by the axioms and first-order logic anyway.

You have to establish, in some metatheoretic way,

No, not really.

that for the intended domain,

Again, no, not really.
You don't even KNOW yet WHAT an intended domain
might be or WHY you might intend a domain.
It is NOT because of ABOUTness. It IS because of
FINITUDE. You have to be able to say, in some way,
that the statement or proof that you are coding as some term
of your first-order language IS FINITE.

Proof(x,w) is true if and only if x is a code
for a proof, blah, blah, blah.
So Proof(x,w) only means proof
relative to that intended domain.

THis is VERY OBVIOUSLY *NOT* true.
Once you instantiate the x and w, once you IN FACT
HAVE finite x and w terms, Proof(x,w) IS going to mean
that IN ALL domains. It's when you're QUANTIFYING over
things under "proof" (and therefore possibly having to
consider "all" x and w) that you risk including some that
might not be finite. Just because SOME aspects of proof
"can't be formalized" does NOT mean that NONE can be.

Thus the need for semantics.

No, you STILL don't need semantics.
Thus you need SOMEthing stronger than vanilla FOL
in order to restrict the range of your quantifiers
to the finite, BUT THAT SOMETHING NEED NOT
be A MODEL, ESPECIALLY NOT a model that you
CAN'T fully describe or articulate NOT EVEN in your
metalanguage! Other approaches to metatheory are
clearly possible and arguably DESIRABLE.

What is to be established is that the formula Proof(x,w)
is true in the intended domain

SOME of the time, Proof(x,w) is going to mean that
IN ALL domains. The intended domain IS NOT relevant.
Whatever it TAKES to focus attention on what's finite
IS relevant. You can try to do that by trying to claim an
intended domain, but the point is, IF YOU DON'T already
HAVE A LANGUAGE for describing this domain, your
claim to "intend" it WILL NOT BE PARSEABLE, but IF
YOU DO, then WHY DIDN'T YOU JUST CONDUCT
*THE*WHOLE*ENTERPRISE* in THAT language??

"True in the intended domain" IS A *QUEER* locution.
It is SO queer that EVEN people championing YOUR PARADIGM
almost NEVER SAY "true in the intended domain" or "true
in the standard model". When they mean THAT, they JUST SAY
"true"!

if and only if x is a code

In the intended domain

for a formula and w is a code

In the intended domain

for the proof of that formula

In the intended domain.
I mean, I see why they would get tired of saying it.
My POINT is that it does MATTER knowing WHICH
parts of this are domain-dependent and which aren't,
IF you are going to go this way.


This is *not* established by a proof in the object theory.
It *can't* be, since the object theory doesn't mention proofs
or formulas.

That is just plain ridiculous.
What any theory does or doesn't mention
is just plain not relevant.
ZFC doesn't mention infinity but there
are still plenty of results about it that
you can establish in ZFC, AFTER YOU DEFINE
infinity, ALSO IN ZFC.


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