Re: Skolem's Paradox and why is math the way it is?

From: Ross A. Finlayson (raf_at_tiki-lounge.com)
Date: 11/11/04 

Date: 10 Nov 2004 18:34:03 -0800

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.,
> >
> > No, I'm ignorant, I make it up myself. I barely read Jech's "Set
> > Theory", the most excellent reference, four years ago. I didn't even
> > view each page.
>
> There are two different ways to be ignorant. You can be ignorant of
> your own ideas, and you can be ignorant of the ideas of others. The
> second can be easily forgivable since there a millions of other ideas
> out there, and one can't reasonably be expected to know them all, but
> the first is usually taken as a very very bad sign.
>
> > Small set, large set: same problem as zero and infinity being limit
> > ordinals.
>
> I don't know what you mean, but feel free to try to explain ... if it
> matters.
>
> > That sounds similar to that NFU, a developing theory, which I may also
> > have influenced. The only reason I would ever hope to do that is
> > because I'm right.
>
> You lost me again.
>
> > You think sci.math has a large mathematically inclined readership? It
> > does!
>
> I thought I said it was unmoderated, but I'd expect it's readership to
> be finite, not large.
>
> > J.E., I'm trying to figure out how to get infinity equal to i, the
> > square root of negative one, and otherwise how infinity times e1
> > equals e2, a product of infinity and e1. Is not that ridiculous?
> > Think circles and big spirals, BIG ones.
>
> Mr. Finlayson, from earlier posts I thought you understood quite well
> that there were mulitple square roots of -1. You can look at the
> quaterions and see an infinite number of square roots. You can look
> at a Riemannian manifold and see two. If you want e2=(A)(e1), then
> the simpliest way to get that is to define A=(e2)(e1^{-1}), if e2 and
> e1 CAN be rotated into each other, then they have the same magnitude,
> which means A=(e2^{-1})(e1), now if (A)(A)=-1, then
> (e2)(e1^{-1})(e2^{-1})(e1)=-1, so
>
> (e1^{-1})(e2^{-1})=(e2^{-1})(-1)(e1^{-1}), so assuming -1 commutes
>
> ((e2)(e1))^{-1}=(-1)((e1)(e2))^{-1}, so again assuming -1 commutes
>
> ((e2)(e1))^{-1}=((-1)(e1)(e2))^{-1}, so again assumign -1 commutes
>
> (e2)(e1)=(-1)(e1)(e2), which is just fancy talk for saying that e1 and
> e2 where orthogonal to begin with. And clearly ANY two invertible
> orthogonal vectors that can be rotated into each other can create a
> "square root of -1", so the number of different such square roots you
> have depends on how many pairs of invertible rotatable vectors you
> have.
>
> Most mathematicians want to enforce a rule that sqrt(-1) is a scalar,
> specifically that it commutes with everything. I think that is silly,
> mostly because I've seen the theorems of multi-variable complex
> analysis, and compared them to the very generalized Stokes theorem and
> the generalized cauchy integral formula, and it's clear to me which
> kind of square root is more useful to physics. Mathematicians can
> have their complex scalar all they want, but I'll use a full clifford
> algebra any day for physics.
>
> I was a bit terse here, but that's because from previous posts I
> thought you were already aware of all this, I'd normally also infer
> that you were familar with the conformal embedding of euclidean space.
> You take a Minkowski space two dimensions larger, and fix one null
> subspace and call if "the point at infinity", then if you pick a fixed
> vector p representing it, then the set of all null vectors n such that
> np+pn=-2 forms a subset of the minkowski space that is homogeneous and
> has a euclidean group under the spacetime rotations that leave p
> fixed. Of course if p can move, then you get the conformal group,
> where p goes to the center of a circle, and the center of the circle
> goes to p.
>
> I get the feeling that this is similar to your big spirals, but I
> didn't know what you where talking about. I see you toss around
> "infinity" often, but if seems to either have no definition, or it
> changes all the time, so I'm never sure what you are saying. So
> basically I have no idea what your big spirals are, and I don't see
> how you are going to get a square root of negative one. Another
> geometruc square root of one with a bunch of really contrived helper
> operations, that I could see happening. But any old reflection is a
> square root of one. Isn't that neat how a square root of one
> multiplied by a differnt square root of one can be a square root of
> negative one?
>
> > I claim that is so.
>
> This is the biggest problem with one axiom that says "be logical",
> there is a race to define things, and everyone can make their own
> definitions (since there is no axiom against it), and they can
> mutually contradict (each other) when they use the same words (in
> different ways).
>
> > I must compliment you, you're remarkably astute and learned. There's
> > no question of actual obligation. I want to know more about you. All
> > I know of you is as you have posted to usenet as "troubled6man" and
> > the other e-mail name. That is some few months of regular usenet
> > posts, you appear to be well-known on sci.physics, or perhaps just
> > appreciated. I compliment Virgil regularly. So, tell us about
> > yourself.
>
> I think I'm likely to starve to death as a result of my own stupidity
> myself, but I'm sure I have more scientific breadth than the average
> liberal arts student, which might not be saying much. I don't post
> under other e-mail addresses, but sometime I use a colleage's computer
> so hopefully I haven't lost that useful privilage by accidentally
> posting under the wrong name, and I had no idea that I was well-known
> on sci.physics, maybe you have me confused with some other people?
> Anyone can post under the name J.E., it's not like I can reserve the
> name. I don't really think my background matters, I did take college
> courses in Geometry, Real analysis, Measure theory, Abstract Algebra,
> Topology, Set theory, Combinatorics, Differential equations, Vector
> calculus, Modern physics, GR, Quantum Theory, Electromagnetism,
> Classical mechanics, Lab courses, etc. If I come across as "not
> knowing" something basic, more often then not it's because I read more
> than one thing that contradicted each other, and so I chose to just
> "forget" both to avoid a contradiction, rather than that I never
> "really" heard of either basic fact.
>
> > One advantage you have is that you have an open mind that is not built
> > upon anything except itself. There are dozens of people among whom we
> > regularly argue on sci.math, and I'm almost always right.
>
> If people are using different defintions, then they can say different
> things and both be "right", so you have to know someone really really
> well to know when they are "really" not right. I wouldn't be
> surprised if most most people on sci.math. think that you are
> "unclear", which at least avoids "knowingly being wrong". As for my
> mind, I'm probably as closeminded as anyone else, but I've been on the
> recieving end of being called a crank just for being confused enough
> to have empathy for others.
>
> For instance their was this poor guy in Japan, he didn't understand
> what it meant for an axiom to be independant of another axioms system,
> and his English wasn't very good. So he wrote this entire thesis,
> that mmost people thought was a joke, but it basically said "(Euclidan
> geometry axioms+ Non-Euclidean axioms) => contradiction. Therefore
> Non-Euclidean geomtery wrong. Therefore GR wrong. Therefore 'photon
> has mass, please measure it for me physicists'" He spent many many
> pages saying this, and most physicists jump to the end after getting
> bored and see the "please measure photon mass" part and have
> hysterical fits of laughter because we already HAVE put upper bounds
> on the photon mass, very low upper bounds, plus there other effects of
> GR, like the double-the-beding-starlight during an eclipse that is
> strong evidence for a zero-mass photon. But unless I find a very good
> Japanese translation of John Allen Paulos' book on humor that explains
> axiom independance, or something similar, then that poor guy is going
> to remain confused about axiom indepedance forever, and as a result
> feel very misunderstood.
>
> I believe that honest people have points, but that sometimes what they
> think their point is is different than what it really is. For
> instance that Japanese guy is correct that a model of geometry (with
> points, and lines, etc.) cannot both satisfy the full set Euclidean
> and Non-euclidean axioms. The point he though he made, that the
> photon has mass, he simply did not demostrate that. As far as he got
> before making an error was right before proving that two axioms can be
> independant from each other. If he'd proved that instead of "proving"
> what he thought he'd proved then others would have agreed with him
> about what his point was.
>
> > If XeY then X=Y, that would be quite the set there. I don't know if
> > more than one set could fill that definition, or if any could. Then
> > again, that is any set that contains only itself. With ubiquitous
> > ordinals, that would also be U, Ord, etcetera, but not the confirmator
> > nor zero, except it goes right through there.
>
> You are right that "XeY => X=Y" is a set that contains only itself,
> and yes you can have two, A={A} and B={B} and then you have to define
> either A=B and ~A=B, if the former, then you have two different
> ur-elements. Those are the ONLY kinds of ur-elements to my
> understanding. U does not contain ONLY itself, it contains everything
> else too, so it's not an ur-element. So in dual set theory you have
> the dual of the axiom of pairing which says, for any dual set A and B
> there exists a dual set C such that ~AeC and ~BeC, and with acreation,
> you can take C to be the largest such dual set. So given U, U-{U},
> then there is another dual set "U-{U,U-{U}}", a set of "all sets
> except for two".
>
> > Ah, the Ross Finlayson paradise. It exists in my mind. You get your
> > four seasons, and a consistent and complete logical theory.
> >
> > Not wanting something is often enough to make it not so. I would not
> > be good as a springboard if I was not solid and flexible, no?
> >
> > J.E., when you say, "dual set theory", please clarify that.
>
> Take every axiom of set theory, replace XeY with ~XeY, take every
> definition of any term in set theory and replace XeY with ~XeY, the
> sets are now dual sets. The dual empty set is the universe. The
> dual-ordinals are the dual sets that contain all dual sets that are
> not dual-ordinals such that (~XeY and ~ZeX ) => (~ZeY). I know that's
> impredicative, I'm hoping and hoping that you see how BORING this is
> because ALL you have to do is write ALL things in terms of "XeY", then
> with one sweep of you hand change "XeY" to "~XeY" everywhere at once,
> in axioms, definitions, everywhere. You get "dual set theory", but
> it's theorems are just dual to the normal set theory theorems because
> a proof of one can be transformed into a proof of the other. Boring.
>
> > Paradoxes are not allowed!
> >
> > Warm regards,
> >
> > Ross F.
>
> Just saying something is not allowed is different than actually
> keeping the borders secure so that they can't get in. Separation to a
> dual set gives unrestricted comprehension, Accreation to a normal set
> gives unrestricted comprehension. You've get the Russel set or dual
> set, and if each belongs to the other type, then your two type theory
> might be just fine, I wasn't sure of the advantages, since frankly I'm
> still bothered by the excluded middle and the lack of models of set
> theory and this fixes neither of those.

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.

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.

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.

In the theory with ubiquitous ordinals, U contains only itself.

I typed "dual set theory" into Google, and there were no results.

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.

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.

About the notion of "boring", do you mean that in the sense of dull or
drilling? The simple interpretation is dull.

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.

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.

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.

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.

You pose no questions?

Warm regards,

Ross F.

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