Re: Set Theory: Should you believe?


abo wrote:
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).

You cannot axiomatically assert that a predicate is a function.
These two things are GRAMMATICALLY DIFFERENT types of
things. You obviously CAN assert about a n+1-ary predicate
that 1 of its arguments is a function of the other n. But that is
saying that the predicate is the "graph" of the function, NOT that
it IS the function, NOT that it NORMALLY matters, BUT HERE IT DOES.


- e.g. whether one assumes whether there
is no predecessor of zero or whether there is
no natural number which is a predecessor of zero.

This is NOT a small difference.
This whole question of whether anything ever CAN have GAPS
is NOT a small question.

First, the question of how to formulate the non-existence of a
predecessor of zero has nothing to do with gaps.

You're being ridiculous. The fact that you personally use
"gaps" as a technical term does not stop the rest of us
from continuing to use the natural-language meaning in
talking about partial functions in general. This matters a
lot because the usual definition of a first-order language makes
ALL the functions total, purely as a matter of grammar.
If you want to have partial functions then you have to NAME
them as predicates or relations or sets INSTEAD OF functions.
By default, functions that actually occur in the signature of the
language MUST be total and that DOES mean they DON'T have gaps,
as "gaps" was understood before you decided to start talking about
some specialized application of the term.

I mentioned it to
show that there are indeed small differences in the way one can
formulate the Peano Axioms which are non-equivalent.

This is just ridiculous. IF they are non-equivalent then
BECAUSE they are non-equivalent they are NOT *both*
"The Peano Axioms".

Heck considers this in one of his papers.

Where he clarifies all these little points in ways that
are entirely UNobjectionable. Why is it so hard for you
to do likewise? It is simply unreasonable for you to assume
that readers are going to have the context of these papers.
Half of every paper's mission is to establish ITS OWN context.
You need to do a better job of establishing yours.
Which it appears you have, in the re-titled "Who Needs Hume's
Principle"
thread. I am glad that the something more constructive will ensue
there.

As to gaps: yes, gaps would be quite a difference.
But then arithmetic with gaps is not Peano Arithmetic, is it?

Your discussion of "arithmetic with gaps" in other threads
has nothing to do with whether any function you are talking
about is or isn't partial in PEANO arithmetic.


There's a long answer to this, because you are obviously confused. The
short answer is: second-order PA without the Successor Axiom is indeed
not second-order PA.

Since it's non-existent, of course it isn't.

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 UTTER BULL***.

Yes, it certainly 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.

***. Do you REALLY think I would be TEMPTED to retract
this after reading that?? ANYBody who is writing A WHOLE BOOK
is OF COURSE empowered to establish his own local context!
None of this changes the fact that NO version of Peano Arithmetic
has a predicate N(.)! In the context of Peano Arithmetic, Natural
Numbers
ARE THE ONLY THINGS THERE ARE, at 2nd-order! And even at
first-order, where there can be things in the domain of which N(.)
would
be false, N is STILL not formulatable to assert that about them!
Trying to cram N(.) into the universe of Peano Arithmetic is, I REPEAT,
UTTER BULL***. This does NOT mean that the book is bad.
It just means it is going somewhere else.

If smarter
people than I have decided to call it PA anyway then I am NOT
impressed.

Fortunately, whether or not you are impressed doesn't matter.

Of course it does.
Everybody's writings matter to the extent that the target audience
is or isn't swayed or influenced by them.

I made it very clear I was talking about second-order logic.

Oh, shut up.

Why should I shut up?

Because you'd be embarrassing yourself less frequently.

You went blathering on about first-order logic
when I made clear that I was talking about 2nd-order logic.

Obviously it was not clear to me. The fact that I missed some
earlier message does not imply that I am blathering.

Rather than apologize,

I don't have to apologize for asking you to clarify the differences
between your treatment and the SECOND-order treatment that
is most natural. There is nothing inherently first-order about
keeping a successor function.

you say "Shut up."
Do you realize how ridiculous that
makes you look?

Do you REALLY think anybody is judging BETWEEN US
on this? Even if it makes me look ridiculous to EVERYbody who
is reading/judging this, which is more ridiculous? Me looking
ridiculous to all ZERO of those people, or you being ridiculous enough
to think that that number is higher than zero?


Any discussion of Hume's Principle is in the context of second-order logic.

Well, usually, but not necessarily.
You are actually trying to talk here about the concept of "number".
This is sort of fundamentally hopeless. In both 1st and 2nd-order
PA, EVERYthing is a number (unless it is a class of numbers, in
the 2nd-order case). THAT'S ALL THERE ARE, in the universe.
Bringing in these other (e.g.Fregean) treatments where "number"
MIGHT be ANYthing other than "one member of THE ONLY class of
things that this whole discussion is about" is just doing something
fundamentally different from PA.

Read. Something.

Go. ***. Yourself.
It would do you a lot more good than what you have
been reading. And I *have* read some of this stuff.
The fact that you presume otherwise just proves you're an ***.

In the case of PA
all the numbers already existed anyway, and in the case of the Fregean
treatments, THEY DIDN'T. These treatments are just fundamentally
incommensurate.

That's funny, because there is something called Frege's Theorem which
relates the two.

I *know* what Frege's theorem is, dumbass.
And it DOESN'T relate the two, DUMBASS.
Frege's theorem was proving the Peano axioms from
#F=#G <-> F~G .
The thing "on the left" getting related TO PA
DEPENDS on a COHERENT treatment of #.
Frege didn't have one. ANY old coherent treatment of #
WILL DO. Frege's theorem is what follows AFTER that.
But "Fregean treatments" are ones trying, AS FREGE tried,
to ACTUALLY DEFINE #. And the more nearly they do it
to the way HE did it, the more Fregean they are, the closer they
are to WRONG the way his was.

Boolos and Heck are talking in the context of second-order logic. I
would really strongly recommend that you stop embarrassing yourself by
describing things as UTTER BULL*** when you don't even seem to
understand the groundrules of the discussion.

Oh, shut up. YOu say "I made it clear I was in 2nd-order logic"
But you did not even make it clear that you were Andrew Boucher.

You go ranting on about 1st-order logic when I made it clear that I was
talking about 2nd-order logic. You call something "UTTER BULL***"
when you don't have a clue what you are talking about. I point out
that you are embarrassing yourself. Your reply is that I didn't make
it clear who I was? How on earth is that relevant to whether I made it
clear I was talking about 2nd-order logic?

Because you presumably made both of these things clear
in the same place and/or time. OR not. The point being that
it is relevant to usual correlations within YOUR behavior that failing
to clarify one might change expectations about clarifying the other.
THAT'S how on earth that's relevant.


Pick a side. It MAKES a DIFFERENCE, you know.

"For the reader comfortable with first-order logic, including its
many-sorted variant version just described, higher-order logic may be
introduced as simply one special many-sorted first-order language, with
certain distinctive relation symbols as primitives." - Burgess, Fixing
Frege, p. 12.

Well, Burgess, here, is correctly saying "introduced".
WE HERE are NOT being INTRODUCED.
AS I SAID, IT MAKES A DIFFERENCE which of these
treatments you choose. It may not be an INTRODUCTORY-
level difference but BY THE TIME YOU GET UP HERE, it matters.

Personally I prefer FOL to be 1-sorted.

It is then possible to define N using the ancestral of Frege.

But doing *that*
IS NOT
doing PEANO ARITHMETIC,
dumbass.

"Dumbass"? Hardy har har!! When you don't have a clue about what you
are talking about, perhaps you should go just a teeny weeny bit easy on
the foul-mouthed insults?

Well, IF I didn't have a clue, that would be one thing.
The fact that you think I don't really reflects worse on you.

Peano arithmetic usually does all of s(.), +(.,.) and *(.,.)
AS FUNCTIONS.

Usually? It's more like always.

You need to pay more attention to the verbs.
I said "does .. AS" functions. That is an idiosyncratic
if not idiolectic (NOT idiotic) locution. I did that precisely
to avoid having to deal with idiots trying to tell me what
MY words mean.

Again, in the formulations of Peano
Arithmetic with a successor relationship, one states functionality in
the axioms.

But doing that does NOT change a binary PREDICATE, or
a RELATION, INTO a unary function. One is still in a very
DIFFERENT linguistic AND PROOF-theoretic universe DEPENDING
on whether the function IS IN THE LANGUAGE *as* a function,
AS OPPOSED to whether it is expressed predicatively.

If one weren't to state functionality, then one wouldn't
have Peano Arithmetic;

That is not the point; the point IS that if one doesn't actually
make it linguistically functional then one has a very different beast
as well. The linguistic functions are not as easily treated in the
Fregean framework and they are basically replaced by their graphs
IN ORDER to get the INcommensurate treatments INTO the same arena.
The original context of Frege IS NOT like modern standard notation
for 2nd-order logic.

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