Infiniti Q45

[mfgoode]

Further to my post 20 I give below some more axioms. Take it or leave it!

1) Infinity can only be represented as a point. I call it the Daleth point (3rd letter of the Hebrew alphabet) .

2) All else outside the Daleth is finite.

3) The infinity point Daleth contains all and is in all.

4) All lengths proceeding from it contract, so Zero cannot be reached no matter how far from the Daleth. Hence the universe has no boundary which is infinitely far and is expanding. No question of a Lorentz contraction here!

5) Matter can only react to consciousness at the speed of the limiting velocity of light and is proportional to its distance from the light source or Consciousness. .

6)Hence we say that Alpha Centaurus is 4.3 light years away.
That's the speed of a physical signal and the time it takes at the speed of 'c'. I think of Sirius and am transported instantly yet the signal from me will take 4.3 light years for it to react.

7) Consciousness is known by different names be it Prana, Akash, Tao etc.

These are some of the basics I formulated at 20 yrs and still am convinced of its truth. I still have to perfect my theory and am working on it.

Mally

[m_f_h]

boundary and boundedness ? Kindly clarify the difference if you care.

Basic definitions:
1) The boundary of a set in a topological space is its closure minus its interior: dA=[A] \ A°.
2) A subset of a topological vector space is bounded iff it is absorbed by any neighborhood of zero.
Alternatively,
2') A subset of a metric space is bounded if its diameter is finite.
Some more blabla:
ad 1):
The closure of A is the set of all points which are the limit of some sequence contained in A.
The interior of A, is the set A° of points in A that have a neighborhood which is contained in A. Removing the interior A° is usually(?) equivalent to taking the intersection with the closure of the complement of A.
Then the boundary is also the intersection of the closure of A with the closure of its complement. Going to complements, this is equivalent to say that the boundary of A is the complement of the union of A's interior and A's complement's interior.
Clearly, the boundary of a boundary is the boundary itself: dd=d. This is called idempotence.
EWMayer said the boundary of a boundary is empty (dd=0). This clearly is another concept of boundary. Not that I don't like cohomology, or exterior calculus... but I'd say this is much more abstract idea of a boundary, and probably not the first one Mr Everyday is thinking of when hearing "boundary".
It is left to the reader to decide if she considers a sphere (i.e. the boundary of a /solid/ ball), to be /without/ boundary, or to be it's own boundary. (Think of a, say, basket ball (or any inflatable toy of your preference, to leave more room for your fantasy...), and call its boundary the region where the material it is made of is in contact with air...)

ad 2):
"absorbed" means that by multiplying it with some big number, any (yet so small) neighborhood of zero can be made big enough to contain the set A. Or, since we're in a field, by multiplying
with a small enough number, the set A can be shrunk as to fit into that ("yet so small") neighborhood of zero.
Most topological vector spaces you can think of are in fact normed (or normable) spaces;
in that case you can take the neighborhood to be the (solid) unit ball. (Finally, it's radius does not matter.)
Well, there are less trivial topol.vector spaces, and, even more interesting, topological modules. There, it turns out that the good definition is : A is bounded if for any neighborhood V of the zero vector there's a neighborhood w of the number zero such that wA is contained in V...
ad 2'): The diameter of A is of course the supremum of all distance d(x,y) between points of A.
It turns out that for normed spaces, definitions 2) and 2') are equivalent. Fortunately...
Now (as if this was not enough...), let me end on a new thought:
According to basic common sense, a circle (set of points of given distance from its center, thus bounded) does not have a beginning or an end. Thus, it is (literally) endless, i.e., (literally!) infinite!
(regardless of the question if it has a boundary or not, and regardless of the cardinality of its points...) - I think this also replies adequately to

Originally Posted by ewmayer

The set of points is infinite, but the resulting curve is not.

Oh no, this made me think of another thing: you probably knew that, while balls are never empty (guys...we speak about math!), a sphere may well be empty... (I mean, an empty set: no points on it!)
But there are funny (ultra)metric spaces, in which any closed ball Bc(0,r) = {x | d(x,0) <= r } is an open set, and the collection of all open balls Bo(x,r) = { y | d(x,y) < r } of same radius, centered in any point x of the former, forms a partition of the former! Ain't that great?!

[m_f_h]

Originally Posted by mfgoode

Further to my post 20 I give below some more axioms. (...)

You seem to forget that there are different infinities.
There are already names for them : aleph[0], aleph[1],...
it is well known that there are much more real numbers than rational numbers, but as much rational numbers as positive integers.

Also, the boundary of the universe is not infinitely far away.
(In addition to the current folklore of saying that the radius of the universe cannot be bigger than ((age of the universe)/(speed of light))
= 10^10 light years, roughly, consider the following provocative reasoning:
The mass of the universe is finite, and since anything is quantized,
the diameter is finite.
no?

[mfgoode]

Originally Posted by m_f_h

You seem to forget that there are different infinities.
There are already names for them : aleph[0], aleph[1],...
it is well known that there are much more real numbers than rational numbers, but as much rational numbers as positive integers.

Good reasoning m_f_h ! But can these cardinal numbers Aleph_null .... be represented on the number line? If so then the Daleth point encompasses all.

[QUOTE = m_f_h] Also, the boundary of the universe is not infinitely far away.
(In addition to the current folklore of saying that the radius of the universe cannot be bigger than ((age of the universe)/(speed of light))
= 10^10 light years, roughly, consider the following provocative reasoning:
The mass of the universe is finite, and since anything is quantized,
the diameter is finite.
no?[/QUOTE]

Well I meant the geometrical hypothetical universe. Not the physical one.
The physical universe is expanding into the boundless geometrical one I would say as from infinity Zero cannot be reached. Putting the cart before the horse it would mean that infinity cannot be reached from zero.

I think you mean the diameter of the universe = age of universe x c ?

Please bear in mind in the physical world the Daleth Point has infinite energy!

Mally

[ewmayer]

Originally Posted by m_f_h

Clearly, the boundary of a boundary is the boundary itself: dd=d. This is called idempotence.
EWMayer said the boundary of a boundary is empty (dd=0). This clearly is another concept of boundary.

I wonder if you may be confusing "boundary" with "closure" here. The 2nd closure axiom states that the closure of a closure is the closure itself.

Now as you have pointed out, the boundary dS of a set S satisfies dS = closure(S) - interior(S).

Now the boundary dS of S is itself a set, so we ask, what is the closure of dS? Since dS is contained in the closure of S, closure(dS) can add nothing to closure(S), thus dS is identical to its own closure. But that implies that dS is "all interior," hence has the empty set as its boundary. QED.

[m_f_h]

Originally Posted by ewmayer

I wonder if you may be confusing "boundary" with "closure" here.

clearly not, since I use both in the same phrase

closure(dS) can add nothing to closure(S), thus dS is identical to its own closure. But that implies that dS is "all interior,"

wrong: e.g. a segment in usual space is closed, but with empty interior and identical with its boundary.
in fact, a boundary is "usually always" with empty interior (as common sense suggests), except for weird cases

Originally Posted by wikipedia

For any set S, ∂S ⊇ ∂∂S, with equality holding if and only if the boundary of S has no interior points. This is always true if S is either closed or open. Since the boundary of any set is closed, ∂∂S = ∂∂∂S for any set S. The boundary operator thus satisfies a weakened kind of idempotence. In particular, the boundary of the boundary of a set will usually be nonempty.

[ewmayer]

Well, then perhaps our definitions are different in some fundamental way. I am thinking e.g. of geometrical manifolds, for instance this simple classic case:

1) Start with S := open ball in R^3 or radius r, get its closure, which adds the set of points at distance = r from the origin, i.e. dS.

2) The boundary dS is just the surface of the sphere, which is a closed 2-manifold which is identical to its own closure, i.e. has no boundary.

To paint a cartoon: Imagine yourself being a 2-D being living on the surface of the sphere. Where is the boundary of your world?

[ewmayer]

I think I may have figured out where the disparity in my and m_f_h's takes on this lies. Again using the above example of the 3-sphere S, whose boundary dS is a 2-D closed surface.

Now, when we consider the boundary of the boundary of S, d(dS), in some sense the result depends on your point of view:

1) If you are still viewing things 3-dimensionally, then dS is (by definition) "all boundary and no interior" and so will be equivalent to its own boundary. (I believe this is m_f_h's point of view).

2) If you instead consider dS as a geometrical object in its own right when asking about d²S, then S (i.e. the thing dS bounds) is irrelevant to that argument, because one is now working in the 2-D world that is dS. Its boundary will be an object of geometric/topological dimension 1, which happens however to be null. In this approach, dS is "all interior," because every point in dS has only neighbors also in dS, where we now use the 2-D definition of neighborhood which is appropriate to the surface dS.

m_f_h, does that resolve the disparity for you?