Re: Set Theory: Should you believe?


george wrote:
Obviously, I personally would NEVER concede that any of
THAT was Peano Arithmetic OR ANY OTHER kind of arithmetic
because IT NEVER MENTIONS MULTIPLICATION OR ADDITION.


abo wrote:
How is it relevant what you concede or not concede?

Like everything else it is relevant to whatever else is in
the literature. You might want to do a count of just who thinks
"the definition of Peano Arithmetic" does vs. doesn't include addition
and multiplication. Not that numerical might makes right here.
We are in fact arguing about what a term usually means, so there,
you do NEED to count.

I think you are
mistaken that this discussion is some kind of contest where George
might "win", as long as he doesn't concede.

Well, it is noble of you to realize that the facts are going to remain
the facts irrespective of who espoused (or contradicted) which of them.
But there is, despite that, a clear opportunity for winning and losing
here,
if only to the extent that you lose for not having even considered the
possible relevance of the MORE prevalent form of PA.

Because of this mistaken
idea, you try to bluff your way, pretending you know more than you do,
and this gets you into trouble and makes you look ridiculous.

This is ridiculous. I am not pretending to know anything
and I am certainly not invoking irrelevant psychology, WHICH YOU
are. So it is fairly clear who is making whom look ridiculous.



Back in 1998, Ali Enayat (then)of American University in
Washington, DC, was asked whether Wiles proof of FLT
could be carried out in PA. Wiles neither knew nor cared.
Enayat explained,
I suspect the question struck him (as it seems to strike many non-logicians)
as one which is of limited interest. Although most mathematicians know of the
so-called Peano's axioms, they know of it as what logicians refer to as a
"second order theory", i.e., one which is couched in a formal language
allowing not only quantification over elements of the domain of discourse,
but also subsets of the domain of discourse. On the other hand, what
logicians commonly refer to as Peano Arithmetic, is the "first order projection"
of the second order theory, i.e., one in which the second order axiom of
induction is replaced by the so-called induction scheme, which consists of
a set of first order axioms asserting that any first order definable subset
which includes 0 and is closed under successor is equal to the whole set of
natural numbers. The logicians move to the first order theory is motivated by
the fact that first order logic is "complete", in the sense that there is a set of
axioms and rules of inference for which the concepts "provable" and "true in
all models" coincide,

Well that quote is all very nice, but since I made clear I was talking
about second-order PA it is not relevant.

Dip***: "2nd-order PA" does NOT HAVE a UNIQUE referent in
this context! People from "my" side of that quote WILL MEAN,
BY "2nd-order PA", THE SAME axiom-set as 1st-order PA
WITH THE INDUCTION SCHEMA RAISED to 2nd order.
So your saying you are talking about 2nd-order PA does NOT
make it clear that ANYBODY CAN talk about ANY version of PA
WITHOUT addition and multiplication. Dip***.

My point is simply that it is reasonable and
NOT "confused" for you to intuit one thing and me to intuit another
when somebody says "Peano Arithmetic".

That's certainly a reasonable point. Good! You progress.
Unfortunately, that wasn't your "point" at the beginning.

It most certainly is consitent with it, however, which
your point has never been.

To recall,

abo wrote: "Consider
second-order PA without the successor axiom (that every number has a
successor) - call this F."

George replied: "There is no such thing.
PA has well-defined axioms and NOT ONE of them
says "every number has a successor".

Which is true.


And then when I gave an example of someone (Heck) using PA in my sense,
you said that was "UTTER BULL***".

Well, I wish Heck were here, but the point is, he was not
using 1st-order PA for anything, so until we establish a
context in which he was more comfortable, he is just
as irrelevant to any normal usage of Peano axioms as you are.
Obviously a lot MORE people can be quoted using the first-
order axioms, or even using them AT SECOND order, since
ONLY the induction axiom needs to change, and even it
doesn't need to change much. The separating issue between
these two treatments is NOT the order but the presence or
absence of function-symbols in the signature. IRrespective
of any order, s(.) IS TYPICALLY present in the signature, which
makes the possibility of its being partial IRrelevant.

The definitions of + and * are in some sense irrelevant precisely
BECAUSE they are definitions, but they are recursive and contexty;
they are not easily eliminated.

In second-order PA they are easily definable.

They are equally definable IN FIRST-order PA, IDIOT --
the axioms that mention them in 1st-order PA *constitute*
their definitions. Nothing materially different from that
happens in 2nd-order PA, except that the class of interpretations
that satisfy the definitions (the TYPOGRAPHICALLY IDENTICAL
definitions) shrinks.

I mean, why stop with s(.)? Why not make # partial as well?

Yes, exactly.

Why stop there? Why not +(.,.) and *(.,.) as well?

Yes, exactly.

Not "exactly". Even in the original Fregean treatment,
there were SOME functions. Making EVERYthing a predicate
and treating every existence/uniqueness assumption as a separate
axiom that could be relaxed IS UNUSUAL. The fact that there was
a long history of it in the thread (in the literature, not the
newsgroup)
to which you were contributing does NOT make it LESS unusual.
I am the only person who is bothering to engage but I am not doing
so from an Atypical standpoint: IN THIS newsgroup, people who
were more familiar with the 1st-order axioms and were more familiar
with functions-occurring-in-the-signature-being-lingustically-mandated-
to-be-total (as is standard for 1st-order languages GENERALLY, NOW,
REGARDLESS of who was using predicates in 1882) are who you would
GENERALLY be expected to encounter.


In PA, there is nothing in the universe but natural numbers.

Again you are confused.

Again you are full of *** as usual.

Maybe you should say "In PA as I intuit it" ?

None of these questions has anything to do with
anybody's intuition. EVEN IN YOUR version, it REMAINS
the case that all the 0th-order objects are natural numbers
and all the first-order ones are classes of them (if you admit
1st-order objects at all; Boolos, fearing "ontological commitment",
tries some lame dodge called "plural quantification", appending to
Quine's "To be is to be the value of a variable" the laughable
"or some values of some variables").
2nd-order PA is CATEGORICAL, remember? THERE IS ONLY ONE
model, so nobody's intuition is relevant. There are ONLY
natural numbers in that model. Indeed, if you were alleging
that I was mis-intuiting the 1st-order version, that version
DOES have models with MORE things in them than natural
numbers, so it would certainly NOT be PA as *I* intuit it that
has this problem.

And there I had hoped you were making progress...

Well, it would be nice if that were who and how you were,
but stop lying.


Read Boolos, or Heck, or Burgess, or pretty much anyone on the subject,

I have read Boolos on the subject, and have understood him better than
you have on the core point that is still confusing you, which IS
ACTUALLY
about whether the set/class distinction is analytic.

rather than just ranting on about something which you clearly do not
understand.

That is not a refutation; that is just a fallacious appeal to
authority.

It is not fallacious, and since the subject was what 'Peano Arithmetic'
means, then appeals to authority are certainly relevant.

Not to THOSE authorities, idiot.
Anybody could ask YOU to read up on the first-order model
theory, or to stop pretending that things that are functions might
not be (which was the real issue).

Me: "Peano Arithmetic..."
George: "That's not Peano Arithmetic! No one uses Peano Arithmetic
like that. That's UTTER BULL***! Dumbass. Go *** yourself and
die."

Me: you're a damn liar.
Exaggerated paraphrase IS NOT quoting and IS misleading.
If you quote people in context then you can have a rational
argument. If you just mix and match to suit your taste, then
you can go *** yourself as usual.


(If sci.logic were a soccer field, George would be head-butted
everyday.)
Me: "Well here is Heck and Burgess using 'Peano Arithmetic' in my
sense."

It doesn't matter and it is NOT "in your sense" in any case.
PA is what it is; nobody GETS to have " a sense ".

George: more ranting, expecting replies where none are warranted

You have NO EARTHLY fucking idea what *I* do or don't expect.
Jesus.


The only relevant thing that happened in those treatments WAS NOTHING,
i.e., they didn't think the addition & multiplication piece was worth
bothering with.

In the paper I cited Heck doesn't bother with addition and
multiplication because he knows they can be easily defined (again, in
the context of second-order logic).

Idiot: addition and multiplication ARE EQUALLY easily defined
IN FIRST-order PA and if ANYbody fails to bother, then the fact
that "they are easily definable" IS NOT THE REASON WHY
they failed to bother! The reason why they failed to bother is
that they are MORE INTERESTED IN INVESTIGATING PRIOR
DEEPER issues!

Burgess, in Fixing Frege, in fact
provides the definitions (at least of addition) explicitly.

Something that you could not be bothered to do, because, again,
you didn't think it was relevant. Which indeed it wasn't, to the issue
you were trying to discuss. But none of that changes the fact that you
need to import your own perspective instead of psychologizing about
people who don't share it.

.

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