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

On May 24, 11:15 am, stevendaryl3...@xxxxxxxxx (Daryl McCullough)
wrote:
I don't see how this is a topic that requires so many capitals
and exclamation points.

You're about to contradict yourself.

The other side (yes, I know you would
prefer that people only say: "Yes, George. You're exactly right!

That's precisely the point.
Even as august a personage as yourself still
DARES to defend the OTHER side!
THAT'S why it requires capitals and exclamation points.
If you see how the other side is defensible, then you DO
see why the position that it is not requires emphasis.

"I've never heard it expressed better.
The people disagreeing
with you are just idiots."

You are still defending the potential relevance of a
a potential "other side". Until you stop, there is still A WAR on
and THAT IS WHY there are still capitals and exclamation points.


You can define what it means

No, really, I can't. YOU can, or think you can,
but *I* canNOT *stoop*.

to be an interpretation of a *language*
independently of any choice of axioms.

Your putting *'s around " a *language* " will NOT
suffice to identify one!
We are in a context where what the other
side CALLS "a language" is completely IRrelevant to
the point of UTTER arbitrariness! WHAT'S IN a name?!?!?
That which we CALL "arithmetic", by ANY
other name, would compute as effectively! And this is NOT
a natural-language-semantic question, like whether to call a rose
"a rose" xor "a thorn", or whether to call arithmetic "arithmetic"
xor "conjuring". It is RATHER a question about SYMBOLIC
labels for things, about SIGNS & SYMBOLS *as* proper nouns
in a FORMAL language. And precisely BECAUSE the languages
in question are purely formal, the symbolic visual/spatial/design
identity of the signs/symbols SIMPLY CANNOT matter: ANY
ISOMORPHIC SIGNATURE would be EQUIVALENT in the relevant
way, for ANY formal language defined via the "signature" paradigm.
WHAT'S IN a sign? That which we symbolize "+", by ANY
other sign would add AS accurately!

An interpretation is a structure

Shut up! I KNOW what a structure is!
I KNOW what an interpretation is!

consisting of a set

And in introductory classes, you just lie.
You over-simplify and hope you can correct in later years for
the people who actually care. Jesus.
ZFC may be the PREFERRED meta-theory in this
paradigm but it is certainly not the only possble one.
You are one level of design/specification too
detailed/implementation DEEP with this.
It DOES NOT MATTER whether you do this with
sets or not. The predicates have to be interpreted
as boolean functions. The terms don't HAVE to be mapped
TO ANYthing BECAUSE they CAN be mapped to THEMSELVES (although you may
need to define "interpretation" more broadly
to encompass things like non-standard models of PA, where
the domain must necessarily contain things that AREN'T named
by terms).

blah, blah.

Exactly.


2. Relative to a chosen interpretation,
one can say that a formula
is true or false.

Of course, but one CAN'T identify the formula until
AFTER one has identified THE LANGUAGE, and the way
YOU THINK you know how to identify a language IS NOT
legitimate, because it is [again] one [ at least one! ]
level of specification too deep! It specifies something
THAT DOES NOT MATTER.

First-order ZFC is, fundamentally,
a theory of a language with
ONE binary predicate AND THAT'S ALL.
It has AN AMAZINGLY poor signature.
THAT IS ITS WHOLE *VALUE*.
That is THE IMPORTANT thing about it!
The ACHIEVEMENT, the NOTEworthiness, is in proving
that you can prove this MUCH output with that LITTLE
input of signature!


3. Relative to a chosen interpretation, one can say then say that
this or that theory proves false statements.

Well, of course, that is what most people are usually
doing (which obviously proves it can be done).


You don't actually need to respond to this post.

Oh, yes I do.

I can generate
my own george-like response:

<george-like>
It is IDIOTIC to start with an interpretation of a LANGUAGE!!
How can someone possibly have a preferred interpretation of
a FUNCTION SYMBOL, if you don't ALREADY
have in mind the AXIOMS
governing that symbol.

You know this is over-simplified: the axioms may
be necessary, but at 1st-order, G1 proves they are
not SUFFICIENT. IF you in fact "know what you intend"
by +, then it is a priori certain that the axioms DON'T
capture it fully. My rebuttal of course is that since "+"'s
domain is infinite and human knowlede is finite, it goes
withOUT saying that any human understanding of the concept
is NOT going to capture it "completely" and that while your
first-order axiom-set may not complete THAT task, it
can and does complete the task of formalizing as much
about "+" as YOU CAN in fact understand (and a great
deal more), given a finite lifetime. Or, if not your personal
current particular axiom-set, some better one shortly later.

The symbol "+" ONLY MEANS addition if
it is accompanied by the AXIOMS OF ADDITION!!! To say that
the intended interpretation of the SYMBOL "+" is given by
a set-theoretic model makes NO SENSE.

Again, since a 1st-order axiom-set CAN'T get it right,
the other side will say that if we're getting it right, we MUST
have some independent understanding of the preferred
interpretation. "A set theoretic model" would be as good
as any, IF YOU COULD IN FACT SPECIFY one (which of
course you can't).

Someone can just as well use "*" or "&" or "@*$!!!"
as the symbol for addition!
The symbol means NOTHING without accompanying axioms.

As I will clarify, it means ALMOST nothing WITH them
either, the point being that any other symbol could've
been assigned THIS symbol's meaning by an ISOMORPHIC
axiom-set.

The choice of a symbols is COMPLETELY ARBITRARY!!!
So SHUT THE
*** UP about "interpretations of a language"!!!!
</george-like>

How did I do?

A-minus.

I should probably stop here and thank you for having
bothered to read what I said closely enough to understand it.

But I never was THAT prudent.

My next point would be that having understood the position,
you now owe something of a rebuttal, because if you actually
agreed with it, then you needn't've confused the issue by
presenting the alleged "other" side. Precisely as you have
already proven you know, I don't concede an ounce of worth
to "the other side". I think that what you just said constitutes
utter refutation, vanquishment, and banishment to the outer
darkness of "the other side". Since you just presented
both sides, you, despite understanding my side, do not
concede that it rebuts the other side. So why not?

Before you go there, though, I need to clarify the
difference between A-minus and A-plus regarding how you did.
EVEN IF you have the axioms, there is STILL arbitrariness
in the language, because you could've phrased completely
typographically ISOMORPHIC axioms with a different signature.
In other words, not only could you have chosen a different
symbol for "the sign you were going to interpret as addition",
you could ALSO have chosen a different symbol for "the
sign you were going to use to state the additive identity
axiom", i.e., you could've said Ax[x%0=x] instead of Ax[x+0=x].
The axioms are in some sense NECESSARY to "identifying"
the symbol but they are not SUFFICIENT.
"+" may be the only symbol playing-the-role-that-"+"-plays
in some axiomatization of arithmetic, so it may be properly
and uniquely identified by the axioms in that sense, but this
does NOT CHANGE the fact that ANY symbol (not already
in use in another role in the same axiom-set) COULD have
been cast in that role.

Distinguishing "types" of symbols in the first-order-language
paradigm involves caring about "result-type" (boolean for
predicates and term for term-functors) and -arity.
You have to know whether a name
names a term or a predicate and you have to know how many
arguments it takes. If there is more than one term or predicate
of the same -arity then you have to know how many there are
of each.

I am basically waxing overzealous with Occam's Razor here.
A signature of a first-order language arguably could be as lean
as a pair of multisets, one for predicates and the other for
term-functors, each telling how many predicates/functors of
each arity were needed by the language.

There are of course good reasons why mathematicians choose
more meaningful symbols than the minimal amount of information
that could be gotten away with. BUT from a LOGICAL perspective,
THEY CONFUSE the issue thereby. The issues thereby confused
are of course a lot more relevant to automatic provers than they
are to human mathematicians, but given the drastically lowered
level of the newsgroup, now more than ever it is important for
people NOT to be burdened with conventions, and that (a
cultural practice, not only not justified but in fact REBUTTED
by logical considerations) is all that talking about "the language of
arithmetic" is.


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