Re: Set Theory: Should you believe?


abo wrote:
The different ways of writing down Peano Arithmetic are essentially the
same.

No, really, they're not.

Consider the two different ways which seem to be confusing you: (1)
with a successor function, and (2) with a successor relationship where
one assumes in the axioms that the successor relationship is a function
(including the Successor Axiom, which says that every natural number
has a successor, which is a natural number). These two different ways
of formulating the Peano Axioms are essentially the same.

Those are NOT the two different ways in question.
When you see what ARE the two different ways in question,
you will be greatly underwhelmed.

MY way involved something like this:
If you take 1st-order PA and replace the first-order induction
schema-variable
by a bound predicate-variable that is universally (2nd-order)
quantified
over all possible sublcasses of the naturals, THEN you get 2nd-order
PA.
If you do ANYthing else then you get
something that may OR MAY NOT be 2nd-order PA, but whatEVER
you get, YOU bear the burden of proof that EVERY difference on
EVERY issue that it has (from this) is unimportant as opposed to
important. Because IF it is important then what you have IS DEFINITELY
NOT 2nd-order PA.

There's a long answer to this, because you are obviously confused.

Between what and what?

The
short answer is: second-order PA without the Successor Axiom is indeed
not second-order PA.

No, THAT is NOT the short answer.
The SHORT answer is an exhibit of 2 different
kinds of axiom-sets.
Here is one that YOU thought was relevant:

Dedekind/Peano Axioms for Number Theory:
* 0 is a natural number.
* 0 is not the successor of any natural number.
* No two natural numbers have the same successor.
* If both (a) 0 falls under F, and (b) for any two natural
numbers n and m such that m is the successor of n,
the fact that n falls under F implies that m falls under F,
then every natural number falls under F. (Principle of
Induction)
* Every natural number has a successor.

This is fairly Fregean language (since I got it from a Zalta article
on How Frege Did It). Presumably these are theorems he actually
proved when he Did It. Here is another, of the SAME kind, from
Wolfram:

1. Zero is a number.
2. If a is a number, the successor of a is a number.
3. zero is not the successor of a number.
4. Two numbers of which the successors are equal are themselves equal.
5. (induction axiom.) If a set S of numbers contains zero and also the
successor of every number in S, then every number is in S.
Peano's axioms are the basis for the version of number theory known as
Peano arithmetic.

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.

One treatment I skimmed alleged that the right 3 of these 10
could be called "Weak Peano Arithmetic".

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,

And it is of course also notoriously INcomplete, and these two
nearly-contradictorily-named results are from the SAME person (Godel).
But that is not my point. 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". And the difference is not
MERELY that yours is 2nd-order and mine is 1st, because WE (the 1st-
order people) KNOW how to lift 1st-order PA to 2nd. But that doesn't
mean we automatically lift all the way back to a version from before
either 1st OR 2nd order logic as we now know them HAD EVEN FINISHED
BEING INVENTED yet, let alone a version that DOESN'T EVEN MENTION
addition OR multiplication, WHICH IS the version that YOU have
(wrongly)
been treating as though it was so standard that everybody else who
thought "Peano Arithmetic" meant something ELSE was "confused".
By whom and by what? would be the question. It was certainly not about
the meaning or order of PA. Or the current definition of first-order
language.


Here, by contrast, is the axiom-set that *I* thought was relevant:
Definition 5..47 (Axioms of Peano Arithmetic)

1. Ax[ ~s(x)=0 ]
2. Axy[ s(x)=s(y) -> x=y ]
3. phi(0) /\ Ay[ phi(y) -> phi(s(y)) ]
-> Ax[ phi(x) ]
4. Ax[ x + 0 = x ]
5. Axy[ x + s(y) = s(x + y) ]
6. Ax[ x*0 = 0 ]
7. Axy[ x*s(y)=(x*y)+x]

This is not even my preferred version of it because it is in
FOL *with* equality and you don't NEED equality to do this;
the actual signature of PA is {<,0,+,*}.

Lifting this to 2nd order does not require introducing
a successor axiom. Nobody is "confused" just for reminding
you of this, or even for insisting that it might be relevant.
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. These may not be what Frege
proved in the original, but who knows, YOU might get credit for
doing something original by BOTHERING to prove THIS version
from HP, instead of talking about what happens if we treat things
that obviously have to be functions, as though they didn't.
I mean, why stop with s(.)? Why not make # partial as well?
Why stop there? Why not +(.,.) and *(.,.) as well?
The existence of separate axioms for existence and uniqueness
in the original is artifactual to the point of irrelevance. When
you're
NEW, you don't KNOW what NEEDS to be a function and what doesn't.
WE KNOW *NOW*. There is just not any point to pretending that
any of these are not functions unless there is in fact no way to make
that workable. But there IS a way. Boolos did it already and it
therefore remains unclear what you are trying to achieve.


Look for instance at
Richard Heck, "Cardinality, Counting, and Equinumerosity," where he
writes,

"Stated in a form that is useful for comparison with Frege arithmetic,
the Peano axioms are:

... 6. (x)(Nx ==> (there exists y) xPy)."

But the whole point is, THAT is just UTTER BULL***.

It is certainly not just PA stated in 2nd order and a different form.
In PA, there is nothing in the universe but natural numbers.
It turns out that what rescuing the Fregean treatment ACTUALLY
requires is the limitation of size principle. That is NOT the same
thing as caring about what a natural number is. There is at MOST ONE
thing in the universe that is not a natural number. What you NEED is
therefore a name for IT, and NOT a predicate N.


2nd-order PA is categorical. It has ONE model.
In this one model, EVERY 0th-order object is a natural number.

If you just lift from 1st to 2nd, you don't introduce "#" as a 2nd-
order function. If you add definitions for it, you will eventually
get one more object (for #(X) when X is an infinite class).

"Frege Arithmetic" and "Peano Arithmetic" ARE JUST DIFFERENT
and if you try to move PA into the Fregean framework then you are
going to lose things that are sufficiently essential that you SHOULD
not still be CALLING the translation "Peano Arithmetic".

Having a cardinality operator and an infinity result for it certainly
are not part of the natural numbers per se, though it is arguably
an important distinction that some properties are satisfied by
infinitely many naturals and others are not. At 1st order you
could express that simply by saying that for any x having the
property, there is a larger x that has it as well. But the arithmetic
of the natural numbers certainly isn't going to care how many THAT is.


Read Boolos, or Heck, or Burgess, or pretty much anyone on the subject,
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.
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. Obviously a modern perspective is going to be
different;
that is ALL that is worth bothering with; it is the functionality of
the relations
that is not worth bothering (axiomatically) with because it is built
into
the language.

.

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