Re: Skolem's Paradox and why is math the way it is?
From: J.E. (troubled6man_at_yahoo.com)
Date: 11/11/04
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Date: 11 Nov 2004 10:38:28 -0800
raf@tiki-lounge.com (Ross A. Finlayson) wrote in message news:<3c6b9c1e.0411101834.4b4f06fc@posting.google.com>...
> troubled6man@yahoo.com (J.E.) wrote in message news:<39d6e584.0411100706.6fbf1e7a@posting.google.com>...
> > raf@tiki-lounge.com (Ross A. Finlayson) wrote in message news:<3c6b9c1e.0411092307.6bf5fda4@posting.google.com>...
> > > troubled6man@yahoo.com (J.E.) wrote in message news:<39d6e584.0411091553.70ac985a@posting.google.com>...
> > > > raf@tiki-lounge.com (Ross A Finlayson) wrote in message news:<200411090734.iA97YJ230806@proapp.mathforum.org>...
> > > > > On 07 Nov 2004, J.E. wrote:
[snip]
> Hi, J.E.,
>
> I'm probably ignorant in both ways. Also, I'm ignorant.
>
> I try to have a general understanding of mathematical terms that are
> in current usage, and how they are used, and to some extent their
> background and surrounding historical arguments and developers. I'm
> not a polymath. I've posted the titles of mathematical books that I
> have read to sci.math, and I haven't taken any college math classes
> since "Calc III." So, I have never taken a graduate level math,
> physics, or engineering class at university.
>
> While that is some level of ignorance, I've "heard of" Clifford
> algebra and the four-dimensional space-time.
I think I understand what you are saying so far.
> So, I theorize the quantum, subquantum, and superquantum as scalar
> unity, infinitesimal, and infinity, and turned to sci.math to share my
> opinion. Words ensue.
And now I'm lost again. I've never seen any evidence for anything
other than quantum, neither subquantum, nor superquantum mean anything
to me. Infinity seems vague enough to mean most anything, and for an
infinetesimal I could probably follow what you meant within the
confines of a specific example, but I'd have to see and example to
know (and I'm not asking for one, I'm just telling you that I don't
understand you). Even scalar unity is something I don't know what you
mean.
> You can plainly read my mathematical development on sci.math.
>
> I am of the opinion that non-Euclidean "geometry" is Euclidean
> geometry without the parallel postulate, but that it is a superset of
> Euclidean geometry and not exclusive, ie, the parallel postulate may
> or may not be so for a given situation, ie, super-Euclidean geometry.
> Perhaps I am wrong about that.
I've never met someone who believed that, most people I meet consider
Euclidean geometry a subset of Non-euclidean geometry, not the other
way around, and the rest think they are exclusive. If you look at the
Clifford algebra of a N+2 dimensional space, you can find a set of
simple 3-vectors that are closed under addition, and that when
multiplied are isomorphic (as an algebra with multiplication and
addition) to the Clifford algebra of a Euclidean N dimensional space,
just as one example of Euclidean being a part of Non-euclidean). The
breakdown of multivectors into k-vectors in obviously different since
those original 3-vectors are 1-vectors in the other algebra. It's
really fun to see the example of taking space-time bivector and
interpreting them as 3-d spatial vectors though, especially if you
look at the way non-relativist QM is usually taught.
I don't know anything about super-anything, let along super-Euclidean
geometry.
> In the theory with ubiquitous ordinals, U contains only itself.
Why? Aren't orginals supposed to contain all their predecessors?
> I typed "dual set theory" into Google, and there were no results.
Well it's nice to know that Google doesn't follow me to the beach. I
thought I had made it clear that that was just based on a few moments
thoughts while on the beach.
> Paradoxes are not allowed because they are a sign of insufficient
> knowledge or prevarication. Consider Zeno, with a completed infinity
> and infinitesimals his situations are not paradoxes, thus a theory of
> reality, upon which those are modeled, with your completed infinity
> and infinitesimals is not paradoxical. Russell's set can not be
> constructed? It is not, although I have a line of argument about it
> confirming extensions to my logical theory.
Zeno got paradoxes by bad reasoning, I agree. "Russell's set" is
*not* a set in the sense that it is not the case for for any thing we
can consistently know whether or not something else is in it,
specifically we can't consistently know whether or not it is in
itself. If you have an excluded middle, that's a problem. If you
have a set of all sets U, AND you have separation that is allowed to
operate on U, THEN you can DEFINE R:={xeU : ~xex}, and "call" it the
Russel set. It's the question ReR that is hard to answer, but I don't
know what you mean by Russell's set "can not be constructed," if you
have U and R and by definition of U ReU is true, then you have ReR iff
~ReR. Are we really talking past each other here?
> About models in set theory, vis-a-vis models of set theory, you want
> your IF logic to say that with that logic-and-a-half you can operate
> in your first order logic, I just say to stay in first order logic in
> the first place.
"First order logic" is vague, most people mean Ordinary FOL, but
IF-logic is first order in that the only things you need to write it
down is the elements of the domain of discourse, it *can* be written
in terms of second order elements, but it is about the combinatorics
of the first order entities. I'm not sure why anyone would object to
IF-logic, the ordinary FOL is a subset of it, so it's not a
restrictive assumption.
Now regarding models, I think different people mean different things
by that word, so I'd have to know what your meaning is to know what
you are saying.
And as for "I just say to stay in first order logic in the first
place," what exactly *is* your objection to IF-logic?
> About the notion of "boring", do you mean that in the sense of dull or
> drilling? The simple interpretation is dull.
Dull as in repeticious, hence the name I choose of "dual." It doesn't
really provide anything above and beyond the original set theory,
unless you figure out how to discuss sets and dual sets together at
the same time.
> I'm pleased that you're back to a countable theory. It may well be
> that you can have both, a post-Cantorian set theory just like there is
> a super-Euclidean geometry.
I thought I was to "no model". And again I don't know anything about
a super anything.
> About infinity, in one of its many guises, shapes, forms, meanings,
> interpretations, uses, allusions, and mathematical constructs, it's
> forced to be in some cases zero, one, or negative one, and I believe
> i, an orthonormal vector to the unit vector, by its role within
> logical and mathematical constructs.
I will agree that i is linearly independant from 1 and from a vector,
and I heard you vague motions towards a big spiral being related to a
rotation by 90 degrees after having a bunch of undisclosed (by you)
meaning float in and do some undisclosed (by you) thing, but I haven't
a clue what you are talking about. P.S. I don't like calling a scalar
a vector, scalars commute with vectors, and only vectors don't commute
with each other unless they are colinear, which is really the case of
commuting scalars again anyway. If that's what you meant by "unit
vector" then I do take objection, and for the same reason, I don't
like to call "i" a vector, for the same commutation reasons. Yes in
2-d there is a linear mapping between the rotations and the vectors,
but that generalizes poorly in my opinion, so I find the mapping
rather useless conceptually. Whereas keeping the rotation 2-vectors
and scalars separate from vectors goes to n-d just fine.
> That's not always the case: it's infinity, there are lots of meanings,
> uses, and interpretations of the word that refers to the things. So,
> yes, please qualify what you say when you mean infinity.
I try to avoid the word because it doesn't seem to be playing fair to
*use* words whose meanings are always changing. If I used it, I
apologize, and frankly I was probably trying to talk about something
you brought up with the word that I didn't know what you meant by it.
> It's not my goal to be unclear, my goal is to find a logical system
> that supplants contemporary theory because it's better, besides being
> good. I've already reached the goal of having a logical theory
> adequate for my own purposes, and in a way portentious for the
> previous goal, for others. To some extent, the unclear points are
> that way for a reason, they're complicated or overly simple. That's
> one reason why I think it is key to examine these plain language
> statements about logic that are saying many things at once, or nothing
> at all.
I've had a lot of experience dealing with people being vague, and I'm
not sure that it's actually helpful to me, so if you are doing it on
purpose, then ...
> You pose no questions?
yes, I no questions for you.
- Next message: richard miller: "Re: how to subsample a vector field?"
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- In reply to: Ross A. Finlayson: "Re: Skolem's Paradox and why is math the way it is?"
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