Re: Learning Logic and Set Theory


Daryl McCullough wrote:
> >> To me, it seems that for a formula to be said to define something,

I interrupted,
> >Oh, please. This is not a matter of opinion.


> If there is no formal definition of what it means to
> define something, then yes, it is a matter of opinion.

I insisted,
> >The general linguistic behavior of the community does
> >factually settle the issue.

And got THIS bullshit as a response:
> According to the community, "countable" is perfectly
> definable in the language of set theory.

That is just utter bullshit.
"The community" does not use terms like "the language of "
this, that, or the other, in any consistent way anyhow.
The standard classical paradigm of first-order logic has a
PRIOR definition of a FIRST-ORDER LANGUAGE.
The question of whether something is or isn't a first-order language
is LOGICALLY PRIOR to the question of whether it is or isn't
"the language of set theory" or "the langauge of arithmetic" or
the language of whatever. Finally, "the community" has been taught
the Lowenheim-Skolem theorem, so the community knows damn well
that countability is not first-order definable, not in the language of
set
theory OR IN ANY first-order language.

> >> you have to rely on an *interpretation* of the language
> >
> >And this is just idiotic bullshit, Darryl. The WHOLE POINT is that
> >first-order theorems are INDEPENDENT of the interpretation.
>
> What relevance is that? Whether a formula defines something
> or not has nothing to do with theoremhood.

Again, I repeat, utter bullshit.
There is no DIRECT connection with theoremhood, but IN LIGHT OF
the completeness theorem, IN LIGHT OF the fact that (syntactic)
theoremhood
does not just HAPPEN to CORRELATE with (semantic) logical consequence,
with AGREEMENT ACROSS interpretations, whether a candidate definition
does or does not get elected has a LOT to do with theoremhood, IF the
definitions
are being phrased IN A FIRST-ORDER LANGUAGE and that paradigm is being
presumed as relevant.

> Perhaps you are thinking that a formula Phi(x) defines a set S if

Oh, shut up.
In order for that to be coherent, one would need a prior definition of
"set" --
GOOD LUCK!
ANd I told you in any case that "a formal definition of 'definition'"
was NOT relevant
here. What is relevant instead are some basic minimal standards that
anything
purporting to be a definition would need to meet; definability would
need to be
scientific rather than religious, i.e., to be falsifiable, NOT
verifiable.
And, moreover, NOT, necessarily, DEFINABLE.

> but by that criterion, only recursive sets are definable.

I repeat, shut up.
Sets simply have nothing to do with this.
That is a completely other can of worms.

> That is *not* the usual notion of "definable".

Dipshit, I repeat: if there were a standard, YOU WOULD KNOW IT.
You're an expert. Your disclaimers at the top to the contrary just
indefensible hypocrisy. I have a way of bringing out the worst in some
normally good people.

> The usual notion, as I've said,

Well, YOU'VE said YOUR opinion, but that IS NOT sufficient to make
your opinion "usual".

> is via a standard *interpretation* of a language.

This is so idiotically ass-backwards as to be insufferable.
In the first place, once there is more than one interpretation,
HOW EXACTLY is anyone supposed to INDICATE WHICH is "standard"?
If this indicability existed prior to the creation of the
thing-that-has-myriad-
interpretations (that thing is usually some sort of formal structure,
subsuming
some particular first-order language), then WHY WAS THERE EVER ANY NEED
for the creation of the formal approximation? My point plainly and
simply is
that THERE SIMPLY NEVER WAS any such need; it was ALWAYS the case that
IF the alleged "standard" could be specified via some other linguistic
means,
THEN THE WHOLE ENTERPRISE was conductable in THAT language.
The formal approximation is just irrelevant. One retreats to the
formal approximation
because theoretical results have been proven about it, because it
offers a modicum
of tractability as a result of that. But once that happens, the LIMITS
of that tractability
pose DIFFICULT PHILOSOPHICAL problems for ALL STRONGER treatments, VERY
MUCH INCLUDING the one via which the original standard was specified.

> >> So, in particular, a formula Phi(x) is said to
> >> define "x is countable" in case

Shut up.
YOU DON'T KNOW.

> >> Phi(x) is true in the interpretation
> >> <-> x is a countable set

This is incoherent. If "Phi(x) is true" has to be qualified
by "in the interpretation" then "x is a countable set" MUST ALSO
be similarly qualified. The fact that you didn't do this makes your
formulation UNGRAMMATICAL.
OBVIOUSLY, if Phi(x) can be true in some interpretations and false
in others, then you are going to collapse into the absurdity that
x is countable AND it isn't.

> >If Phi can be true in some interpretations and false in others,
> >then the question must arise about whether x can be countable
> >in some interpretations and not countable in others.
>
> You pick a *standard* interpretation, and then x is countable
> iff Phi(x) is true in *that* interpretation.

That is infinite regress, DUMBASS.
That MERELY removes the question of "definability"
FROM whether "countable" is "definable in the language"
TO whether "the standard interpretation" is definable in the
language. OBVIOUSLY, NO interpretation CAN be definable
IN the object language! Defining interpretations REQUIRES a meta-
language! If you are defining something in terms of the standard
interpretation then you are defining it in the language IN WHICH THAT
interpretation is defined/indicated AND NOT in the language you CLAIM
to be defining it in.

That point is moot in any case.
To the extent that any particular standard model occurs for any
first-order theory with multiple models (which, by Godel 1, is all of
them,
all of them with any decent amount of strength, anyway), everybody
advocating that standard suffers under the nearly unberable BURDEN of
proof
that THAT model DESERVES to be THE standard. There are A FEW first-
order theories where you can meet that burden, and PA is one of them.
But PA basically meets the burden by having only 1 model up to
isomorphism
AT SECOND order, and by further having only 1 first-order model that
agrees
with that one. FIRST-ORDER THEORIES IN GENERAL ARE NOT so lucky.
And in any case, even in this PARTICULAR case, to say "we've solved the
problem
of definability by appealing to the standard model" is just bullshit.
We were talking
about definability IN A FIRST-ORDER LANGUAGE and the "solution" is by
appeal
TO A SECOND-order construct.

> The equivalence
>
> Phi(x) is true
> <->
> x is countable
>
> only holds for the standard interpretation.

That was MY point, DUMBASS.

> For other interpretations,
> the equivalence may not hold.

DUH.
Ergo, ipso facto, phi DOES NOT define "countable".

> "Countable" may not even be definable
> for some interpretations.

"Countable" is not definable FOR ANY interpretation,
DUMBASS, because it is not a MATTER of interpretation!
It is a PRIOR concept TO interpretation! BEFORE you could know
WHICH of these various models was "standard", before you could
even DEFINE/CHOOSE the standard among all the models of PA,
you would HAVE to ALREADY *know* what COUNTABLE means!

> >This is ridiculous.
>
> >The question was whether the FORMULA ITSELF,
> >whether phi, by itself, defined countability.
>
> Without an interpretation, a formula doesn't define anything.

That's completely idiotic, Darryl.
Obviously, in the case of any and everything that IS provable,
at first order, the interpretation SIMPLY DOES NOT MATTER.
That's WHY we BOTHER proving things.

> As far as I understand (which doesn't mean a lot, since I'm
> not an expert), what I've been telling you is the usual
> notion of what it means for a formula to define something.

Your understanding is wrong.
Your acquaintance with what is "usual" is not as broad
as you think it is, either. Moreover, even if you were to prove that
69% of successful researchers believe that it is reasonable to talk
about interpretation-dependent definitions at least 69% of the time
(which would imply that your line is tolerated almost half the time,
which would, under normal circumstances be an adequate defense),
it would still be the case that a moment's reflection would reveal
that,
here as elsewhere, a practice that has become common/convenient,
among those who know, is, logically, just sloppy.

.

Relevant Pages

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