Re: Some basic set theory questions

On Jul 10, 11:13 pm, Transfer Principle <lwal...@xxxxxxxxx> wrote:
On Jul 10, 12:24 pm, MoeBlee <jazzm...@xxxxxxxxxxx> wrote:

Your "correction" is ANOTHER LIE.

Obviously, my corrections aren't sufficient, and will need
further correction. But the further corrections I'll give
in this most might still contain some "lies."

What I called 'lies' are lies. I don't know whether scare quotes will
be needed around 'lies' for what else you're about to come up with.

The hope is
that after finitely many corrections, all the "lies" will
be eliminated -- without having to resort to verbatim
quotes every time I want to refer to another's opinion.

"But let's appropriately up the ante. Is Katz's theory sufficient as
foundational theory from which to derive mathematics for the
sciences?"
SO WHAT? Where did I say that Katz's theory is a waste of time if it
doesn't derive mathematics for the sciences? I didn't say that. I
wouldn't say that, and I wouldn't say that because I don't believe it.
You are reading into my posts what is not actually there.

I wonder how MoeBlee would have responded in that old
thread if I had simply given the three-word answer "I
don't know" to his question as to whether Katz's theory
is sufficient as a foundation, instead of tried to guess
why he had asked the question and have my guess declared
a "lie." I doubt that MoeBlee would've called _that_
response a "lie," since one can't get much more honest
(in a math situation) than saying "I don't know."

Whatever your point about that, at least you know that I never claimed
a theory has to provide an axiomatization for the mathematics of the
sciences just for the theory to be worthy of study.

"Lie": Hughes/MoeBlee prefer working in ZFC.
Correction: Hughes/MoeBlee use ZFC as a _foundational theory_.
For CERTAIN purposes. But I never said that I prefer ZFC as a
foundational theory for the mathematics for the SCIENCES. For that
PARTICULAR project I'm more inclined to Z-regularity+"countable
choice".

Isn't Regularity (i.e., Foundation) one of the axioms
that's not included in Zermelo's original theory (i.e.,
like Replacement Schema, Regularity is an axiom of ZF
but not of Z)?

Z set theory is not necessarily taken to be the exact set of axioms
Zermelo himself gave in his original paper. Indeed, Zermelo did not
present a formal theory in his original paper. One may debate whether
regularity is to be counted among the axioms of formal Z set theory,
but at least some sources do list regularity among the axioms of
formal Z set theory.

If so, then Z-regularity+"countable choice" is
redundant

(1) '+"countable choice"' is not reduant. (2) '-regularity' is not
redundant given that it is common to include regularity as an axiom of
Z set theory; thus, no matter whether one takes regularity as part of
Z set theory or not, it provides definiteness to say whether one is or
is not including regularity in some particular axiom set.

-- so the correction to my "lie" is that
MoeBlee's foundational theory is Z+"countable choice",

No, you're leaving out all the rest I said about foundational theories
that contradict your characterizations of my thinking.

Why don't you just quote exactly from where I said certain of your
postings (or at least parts of them) are a waste of time? THAT, and
NOT what YOU later characterize, is what I said is a waste of time.
Your extrapolations to whole classes of theories is not MINE.

The reason I extrapolate is that, once I post something
considered to be a "waste of time," I want to know how
exactly what I posted is a "waste of time," so that I
can extrapolate to what I might say in future posts and
know how to avoid "wasting time" in the future.

Oh, how very selfless of you. As if it doesn't have anything to do
with your program to depict as intellectually intolerant.

Correction: A post about an alternate theory, or that
seeks to prove an established theory inconsistent, is a
"waste of time" if it is based on misconceptions on how
models and universes work, or how logic works (the key
concern in most Cantor diagonal threads) -- or the post
lacks rigor due to such misconceptions. Time is wasted
especially if such misconceptions are repeated over and
over again. The poster repeating such misconceptions is
often called a "crank."

That's at least closer to the general idea. However, I don't even say
that all such postings are a waste of time for all purposes
whatsoever; nor, of course, just to be clear, do I hold that 'waste of
time' is not a subjective evaluation.

Correction: My posts are a "waste of time" if I fail to
eliminate these misconceptions in the posts I defend,
or repeat the same misconceptions, or introduce even
more misconceptions.

That might be one component that might make a particular post (not
having much else to redeem it) a waste of time. Another kind of time
wasting post (not having much else to redeem it) is the kind in which
you go through a whole bunch of deductions just to arrive at the
conclusion you've already been told, or to arrive at the conclusion,
which you've already been told, that a certain crank's desiderata or
whatever does not come automatically from the crank's assumptions or
arguments. But again, there is no general rule I can state about this
any more than I can state a general rule whether, say, a certain movie
is or is not a waste of my time.

Case in point here is the exchange between Hughes/HdB:

"Lie": HdB is a "crank" because he's a finitist.
Correction: HdB is a "crank" because his belief that
ZFC is inconsistent is based on the misconception that
the theory ZF-Infinity has a unique model, a model in
which ~Infinity is true.

That is ONE of a whole set of claims and behaviors of HdB that led me
to say that he's a crank.

It would be interesting if there could be a finitist
posting on sci.math who's a finitist simply because
he is philosophically (or Platonistically) opposed to
the existence of infinite sets, and not because of
misconceptions about models of ZF-Infinity. Indeed, a
hypothetical finitist might even post, or include in a
paper, something like:

"Axiom of the Empty Set:
ExAy ~yex

The empty set axiom is not included in most known
axiomatizations of ZFC because the existence of 0 is
derivable from the Axiom of Infinity.

One may choose to put it that way; but I do not.

But since we are
considering a finite set theory, we must explicitly
include an Empty Set Axiom."

It's not true that we MUST adopt an empty set axiom to prove

ExAy ~yex

from the axioms of Z set theory without the axiom of infinity.

Note that I don't claim that the above is _true_ --
only that it would be _relevant_ (and in particular,
something that a hypothetical finitist that's more
relevant than what HdB posted).

Okay, it would be relevant to something or other to mention such
things. Meanwhile, frankly, I'm lost as to what is supposed to be the
point of this hypothetical exercise.

[Y]ou even carry that over to form a characterization of people as
having some kind of intellectual intolerance that they don't actually
have, which then, of course, has become your gravamen.

"Lie": People have some sort of intellectual intolerance.
Correction: ???

The correction is just not to cut my thought off at one sentence.
Please refer to whatever fuller statement I made about that. Did I
even list this as a lie of yours? (I'd like to see exactly what I
posted in that regard.)

I don't know how to correct this "lie." I tend to
believe that the very use of the word "crank" entails a
sort of intellectual intolerance, but it appears that
this belief needs a correction. But I don't know what
the use of the word "crank" really is, if it's not
intellectual intolerance.

First, of course, different people are intellectually intolerant in
certain ways regarding certain subjects. So, just to be clear, I've
not claimed that in general all the mathematicians and logicians who
use the word 'crank' aren't possibly intellectually intolerant in some
respect and in some degree or another about certain subjects. But I do
disagree with the general (and in certain cases, particular)
characterizations you make about people you call 'standard theorists'.
Second, it is not intellectual intolerance merely to say that a
certain person is a crank or that cranks have a strong presence at
sci.math and sci.logic.

Ongoing now, would you please quote me instead of putting words in my
mouth, or, if you have questions about what I mean, then just ask me
rather than put words in my mouth?

MoeBlee

.

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