Re: Set Theory: Should you believe?


abo wrote:
Or look at John Burgess, "Fixing Frege" (p. 25), where he writes, "The
so-called Peano postulates ... consist of ...", and he then lists the
successor axiom as axiom no. 1.

You need to quote that.

Why do I need to quote that?

Because PA * does not have * a successor axiom,
THAT'S why. You need to quote it so that people
can see, in context, whether this variation on the theme
is small enough to be reasonable.

Get the book - it's probably one of the
most important books on the foundations of mathematics written in the
past few years or so. If you don't want to buy it, get it from the
library. Look it up yourself.

Oh, you meant, "Why do I need to quote that" since
"you can just read it for yourself". OK, Here's Why:
THIS IS A NEWSGROUP. You NEED to know the ground
rules. You may have noticed that the people who DO own the
book are NOT BOTHERING with your bull*** here. The quality
of your audience here is low. You quote what's relevant for
two reasons: 1) it proves YOU KNOW what's relevant; that is
guaranteed to get you a better sample of replies; 2) it spares
would-be interlocutors the hassle. Anything I have to get up for
(like to go buy a book) is just not likely to get done. By the
time I read the book and get back, this ephemeral discussion
may just be old.

If it is 2nd-order then it is irrelevant.

Any discussion of Hume's Principle is always in the context of
second-order logic. It's your remarks which are irrelevant.

Then why are you bothering to engage them?!
And irrelevant TO WHAT?
What is the actual GOAL?
You presume that just because somebody once asked,
"Is Hume's Principle Analytic", your derivations involving M
"are relevant". One must repeat, relevant TO WHAT?
"Number" does not (yet) have any official pre-defined meaning,
so unless HP itself is going to be the meaning, it is obviously
NOT possible for anything introducing "number" to be analytic.

Instead of assuming a functional term s(t), you assume a relationship Sx,y.

You can do that if you want. Peano didn't do that.
What happens when you change
to doing it that way MATTERS. If you saw that in the context
of "Fixing Frege" then it also matters that that was about fixing
Frege AS OPPOSED to fixing PEANO. The whole importance of
Frege's theorem is that it is the PEANO axioms that become theorems.
Given that all the content of all these 2nd-order treatments of the
naturals
is supposed to be the same, it is very odd that you find any particular
dialect of phrasing them to even be important at all. If this is
really
sound then it ought to -- JUST like all the truths of arithmetic --
come up
THE SAME under all translations. If you are trying to do something
that
depends on deleting a successor axiom then THAT is irrelevant if the
standard treatment NEVER HAD ONE to begin with.

You are just making an uninteresting, boring semantical point, which in
any case probably incorrect as to current usage.

NONE of this has EVER been about CURRENT usage!
It is about FREGE'S Begriffschrifft usage from 120 years ago!
The first-order functional treatment is certainly MORE current
than ANYthing involving ANY 2nd-order treatment explicitly DESIGNED
for exposition of the original Frege! That is ALL VERY much THE
OPPOSITE
of CURRENT!

If for you it is a defining feature of PA, then call what I, and Heck,
and Boolos, and Burgess, and practically everyone else, are using
something called PA*. There, are you satisfied now?

No. I will be satisfied when you shave.
As in when you actually USE Occam's razor.
You are introducing M. Is Hume's Principle normally
stated with M? I am having some trouble even coming
up with an on-line statement. From plato.stanford.edu
I am coming up with something that has the same problem
that you have with M: it has a problem with #.
To quote,
" Using our notation '#F' to abbreviate 'the number of Fs',
we may formalize Hume's Principle as follows:
Hume's Principle:
#F = #G = F ? G "

This is just plain silly to begin with
because nobody has said what KIND of thing
"#" might be. The Stanford article includes a long
exposition of how Frege's attempts to give an EXplicit
definition of # all failed.
This treatment stresses that this is an IMplicit
(or "contextual") defintion of # because # can be
eliminated only in the context of statements like
#F=#G.

I don't see why you personally expect any respect
for choosing to write Mn,F as opposed to #F=n.
Do you intend to allow for the possiblity that MORE than
1 n could number F? If not, why are you creating a new
letter?

The Fregean treatment that inspired all this thought
that Hume's Principle was analytic because it actually
proved it, AFTER having given an explicit definition of #.
The proof was marred by (essentially) an appeal to naive
set-theoretic comprehension, which was in turn thwarted
by Russell's paradox. So the project you are now embarked
upon could have any of several goals, none of which you have
yet clarified, including 1) define numbers coherently, 2) rescue
Fregean comprehension&extensions, and thereby rescue
the original Fregean PROOF of the analyticity of HP, or
3) clarify why 1&2 are impossible and salvage the analyticity
of some other stuff. But if you are introducing any NEW
OPERATORS then obviously analyticity is NOT going to
be relevant to your work (nothing involving M in any meaningful
way is going to be analytic UNLESS you can DEFINE M!)

So, no, you have NOT laid out any ground rules, you have NOT
clarified WHY you are bothering, and you are NOT doing anything
except talking about the same general topic as Boolos and expecting
(unreasonably) to get respected for bloviating in the celebrities'
arena.

.