Re: Some basic set theory questions

On Jul 7, 11:10 pm, Transfer Principle <lwal...@xxxxxxxxx> wrote:
On Jul 7, 11:05 am, MoeBlee <jazzm...@xxxxxxxxxxx> wrote:

On Jul 7, 8:55 am, David C. Ullrich <dullr...@xxxxxxxxxxx> wrote:
Precisely. Although, curiously, I'm not entirely certain which
side you're wagging your finger at here. My interpretation
of the relevance of that paragraph is that you're agreeing
with me that getting the existence of the empty set from
the convention that something is a technicality of no
intrinsic interest, but it seems _possible_ that you're
instead referring to my insistence that the "right" way
to do it is to use the axiom of infinity, because that
shows it "really does follow from the axioms"...
But I am on no "side" that claims that using separation and universal
instantiation is "the right way" or a "better way" or "preferable" to
using infinity.

Speaking of "side," I never thought the day would come when
I'd actually be on David Ullrich's side of a debate. And
yet this is precisely where I find myself.

This debate concerns whether the Axiom of Infinity is
required to prove that the empty set exists.

NO! That is NOT the "debate" I've had with Ullrich. You're not even
READING The posts!

There appears
to be three schools of thought here:

1. An explicit Empty Set Axiom is required to prove that
the set 0 exists in ZFC.
2. The existence of 0 is provable from the axioms of
Infinity and Separation Schema.
3. The existence of 0 is provable from the axioms of
FOL= and Separation Schema.

These aren't even mutually exclusive.

Ullrich obviously adheres to 2.

So do I!!!

(1) The "existence of 0" is from logic alone ANYWAY. Ex x=0, no matter
WHAT 0 is.

(2) The existence of an object that has no members is provable from
the separation schema.

(3) That there is a unique such object is then provable from
extensionality.

(4) That object is given the name '0'.

(5) That that object is a SET (upon the definition 'x is a set <-> Ey
xey') then follows from pairing or from power set or from infinity.

Another set theorist who
adheres to 2 is Randall Holmes, who also uses an Axiom of
Infinity to prove that the empty set exists in his theory
PST, Pocket Set Theory.

So what! The the question of PROVABILITY was not about pocket theory,
but about ZF.

Then Ullrich said explicitly that he doesn't dispute that in a given
system we may prove the existence of an empty set without the axiom of
infinity. Rather, his point is that the axiom of infinity is BETTER
(NOT the ONLY one) to use for the purpose of a derivation that is
GENERAL across various systems. And I didn't even dispute THAT point
with Ullrich. I even AGREE that the axiom of infinity provides greater
generality in that sense. The only thing left is that I declined to
state a view as to whether that generality makes one approach better
than other approaches.

Notice that PST proves the
existence of proper classes, and the cornerstone of
Ullrich's argument is that FOL= and Separation Schema are
not sufficient to prove in a _proper class theory_ that 0
exists and is a _set_.

And I AGREE with Ullrich about that! Sheesh! THAT was not the
"debate".

MoeBlee obviously adheres to 3.

There's hardly any "adhering" that needs to be done. It is a plain
finite fact that in a certain common formulation of Z set theory
(perforce in ZF, which was the STATED context) and with the mentioned
definition of 'is a set', one can derive the existence of a unique
empty set without having to use the axiom of infinity. Period. That's
not even a matter of dispute with Ullrich or with anyone who has
simply read the proof.

Suppes, the textbook
which MoeBlee often cites, does something completely
different -- Suppes lets 0 be a _primitive_ and defines a
"set" to be either 0 or an object with elements.

Yes, and I mentioned that when I very first recommended the book to
you.

Then
Suppes uses Separation Schema to prove that 0 is actually
an _empty_ set. Although this isn't how MoeBlee presents
his argument, Suppes does support his side since Infinity
is used nowhere in the proof.

It doesn't even MATTER! There's no rational other "side" as to the
question that one may derive the existence of an empty set without
using the axiom of infinity.

Notice that Ullrich mentions a theory T' which consists
of the axioms of ZFC relatived to a new primitive called
"set," and a theory T'' which adds to the theory T' an
axiom guaranteeing that every object is a set. I actually
mentioned these exact theories to MoeBlee in a previous
thread, where I was talking about the so-called "crank"
Srinivasan and the ex-"crank" zuhair where discussing
particular models of what turned out to be Ullrich's
theory T'. And now the same theory appears in this thread
as well.

So what?! Such things are obvious anyway.

And so we see that once we leave MoeBlee's comfort zone
of ZF(C), we enter a world in which an axiom such as the
Axiom of Infinity is required to prove that 0 exists and
is a set.

You jerk, this is not outside my "comfort zone". If you had asked bout
this YEARS ago I could have explained it to YOU. All of this is not a
matter of debate. You didn't even bother to READ the exchanges you're
commenting on! (Or your reading comprehension and retention is
seriously impaired.)
The theories in which Infinity is required
include class theories such as NBG and PST,

I don't recall about NBG, but NB in one of its system forms does NOT
require the axiom of infinity to prove there is a unique empty set,
since the class version of separation plus extensionality prove that
there exists a unique class having no members, then "O is a set" comes
from EITHER the axiom of infinty OR from a redundant AXIOM '0 is a
set'.

Ullrich's
theory T', as well as the theories created or suggested
by Srinivasan and zuhair. And this is why finitists such
as HdB include an explicit Empty Set Axiom, for without
it they can't prove that 0 exists.

No, IF HdB's theory really is Z-axiom-infinity or stronger, then the
existence of a unique empty set (given the mentioned definition of
'set') is provable without an empty set axiom.

You are COMPLETELY confused about all of this, while you're acting a
jerk telling me what is in my "comfort zone".

And so we see that the bulk of the evidence is on
Ullrich's side of the debate. Only Suppes supports
MoeBlee's argument,

(1) As I've said, what you THINK was at difference between Ullrich and
me is not even what turns out to be at difference, indeed even if it
turns out there WAS a substantive difference, i.e., other than our
talking past one another in an unfortunate way.

(2) You are an UTTER clown when you sway "only Suppes". There are
OTHER textbooks in set theory that derive "ExAy ~yex" from separation.
It is a WELL KNOWN fact about set theory that the existence of an
empty set is derivable from separation (that it is a SET, upon the
mentioned definition, then would come from pairing, power or
infinity).

and even Suppes does it differently
by letting 0 be a primitive and defining "set" such that
0 is by definition a set.

So what?! This is not even the kind of thing that is at issue.

Prior to this morning, I merely pointed out that the
axiom of infinity is not REQUIRED. Sheesh! And then this morning I've
also pointed out a drawback to using the axiom of infinity for this,
though, still, I've not made any claim about what is "the right way"
or a "better way" or "preferable".

Bull! MoeBlee's made this exact same denial when I told
MoeBlee how he finds it "preferable" for theories other
than ZFC to provide for an axiomatization for an
application to the sciences.

YOU LIAR! Cut it out!

(1) I didn't say one approach is preferable to another here. I
recognized Ullrich's sense and I mentioned another sense that
disagrees, and I said I decline to state an overall preference.

(2) As to axiomatization of mathematics for the sciences, go back to
what I ACTUALLY said about such things, not to YOUR
MIScharacterizations.

Sure, MoeBlee might avoid
using the exact words "preferable" or "better way," but
as often as he mentions calculus for the sciences, it's
obvious to me that MoeBlee really does "prefer" that
proposed theories are so applicable,

You LYING fool. Your claim is IDIOTIC. First order group theory does
not provide an axiomatization for the mathematics for the science. PRA
does not provide an axiomatization for mathematics for the sciences.
PA does not provide an axiomatization for mathematics for the
sciences. Etc. Etc. But I never claimed that I "prefer" that they do!
I never said that a proposed theory must axiomatize mathematics for
the sciences or that I prefer that it do. There are all kinds of
different theories with different roles to play. The matter of
axiomatization of mathematics for the sciences concerns a proposed
theory in CONTEXT OF that theory being proposed as an alternative to
ZFC. Read my EXACT remarks on the subject to see what I ACTUALLY said.

just as it was
obvious to Ullrich that MoeBlee's "preference" is for
the existence of 0 to be derivable without Infinity.

How could it be obvious? Please point out EXACTLY where I said such a
thing. Please ENOUGH with your continual reading INTO my remarks what
is NOT there.

So
applicability to the sciences and the derivability of
the empty set's existence without Infinity represent
MoeBlee's desiderata in the same way that the repeated
preferences of the "cranks" represent their desiderata.

I mentioned a certain sense in which resort to infinity for this is
not preferable. I made it EXPLICIT that I don't necessarily take that
as DETERMINATIVE for what I prefer.

To conclude this post, let's look at MoeBlee's alleged

"alleged". Oh please, grow up.

proof that 0 exists without Infinity:

AxEbAy(yeb <-> (yex & (yey & ~yey)) ... an instance of the axiom
schema of separation
EbAy(yeb <-> (yex & (yey & ~yey)) ... universal instantiation

and Ullrich's response:

"You can't erase the initial "Ax" in the first line except
in a context where we're assuming that something exists."

That is NOT an issue as to whether the proof is correct. Rather, it
concerns how one would characterize the underlying assumptions in the
method of such a proof.

And when the smoke clears on that, I don't find a disagreement between
Ullrich and me on that point. Yes, it is fully granted that the first
order logic comes with a standard semantics in which every domain of
discourse has at least one member, and the rule of universal
instantiation supports that and our quantifier rules must conform to
that semantics if we wish to have soundness of our logistic system
with that semantics.

So according to Ullrich, we can't use the universal
instantiation rule unless we know that something exists.

And according to MOEBLEE the rule of universal instantiation would not
be valid in a semantics in which there are empty domains of discourse.
And according to MOEBLEE the rule of universal instantiation would be
inconsistent if for every formula P we had ~Ex P.

But doesn't MoeBlee's use of UI here look familiar. Let's
go back to the Nam Nguyen debate. MoeBlee writes:

1 AxAy x=y ... axiom
2 Ay x=0 ... universal instantiation

But the only axiom of Marshall's theory is "AxAy x=y",
which begins with a universal quantifier. So we don't know
that something exists in Marshall's theory -- which means
that MoeBlee's use of UI in line 2 is invalid!

No, it's valid in plain, ordinary, first order logic, you ignoramus!

Read the damn formulation of the rule, then read the clauses in the
damn proof of the soundness theorem in which the rule is proven
VALID.

Therefore, MoeBlee's alleged proof of "Ax x+y=0" in the
theory with language {"+", "0"}, and axiom "AxAy x=y" is
actually invalid! And so yet another of MoeBlee's proofs
falls apart!

You are SUCH a fool.

ASK ANY LOGICIAN whether plain first order logic permits the rule of
universal instantiation to be applied as I did. And whether the
soundness theorem for first order logic includes a proof that the rule
of universal instantiation is valid. Better yet, just get a book on
the subject and read for YOURSELF. Or better yet, just start with
plain common sense:

If a property is true of ALL objects, then it is true of any
particular object. Sheesh! Even Aristotle could tell you that!

MoeBlee


.

Relevant Pages

  • Re: Axiom of Pairing, Scheme of Replacement from others
    ... separation follows from the axiom of replacement together with the axiom ... Your hypothesis has uniqueness but not existence. ... Proof of separation from replacement: ... empty set is required. ...
    (sci.math)
  • Re: Some basic set theory questions
    ... with me that getting the existence of the empty set from ... using infinity. ... This debate concerns whether the Axiom of Infinity is ... MoeBlee obviously adheres to 3. ...
    (sci.math)
  • Re: Axiom of Pairing, Scheme of Replacement from others
    ... to prove the existence of the empty set, then the axiom of infinity, as ...
    (sci.math)
  • Re: Some basic set theory questions
    ... not be able to prove the existence of 0. ... the Axiom of Infinity is formulated in such a way ... empty set clearly exists -- since there's an explicit ...
    (sci.math)
  • Re: Some basic set theory questions
    ... that 0 exists uses the Axiom of Infinity, ... "Lie": Ullrich believes that a proof in ZF that 0 ... MoeBlee believes that the existence of 0 ... derivable through FOL= with the axiom schema of separation alone? ...
    (sci.math)