I see that you are ill-prepared to deal with the topic in a rigorous manner. There seems to be a complete void in understanding on your part of the necessary complements -- something that you need to be aware of in order to proceed.
http://journals.impan.gov.pl/fm/Inf/171-2-1.html
I actually show that even if location is not introduced, still 1=3 according to pseudo-metric space reasoning.
According to pseudo-metric space the distinct x=(1,3) and y=(1,6) members are also indistinguishable.
Points are not any member, they are exactly the smallest existing members, and this property is significant, so your "(you may call its members points if that comforts you)" nonsense simply demonstrates your ignorance about members that have the property of the smallest existing things.
(1,3) and (1,6) are distinct members of A and your abs(1-1)=0 ignores the fact that these members are at least A={(1,3),(1,6)} such that |A|=2, or in other words, you can't break the members into pieces , because according to the notion of set theory, the identity of a given member is defined as a whole.
The application is not there and you never find an instance where a problem must be solved with pseudometrics. But I give you a clue. A sphere has two distinct points N and S (North and South). When you reduce the dimension of the sphere to 2-dim without junking N xor S, then there is a zero distance between N and S.
Another typical reply of jsfisher, which avoids detailed replies to:
The problem is that pseudo-metric space is inherently wrong, without any need to compare it to other axiomatic systems, simply because according to pseudo-metric space a pair of points are indistinguishable AND distinct, which is a contradiction.
In other words, if a given pair is similar, for example (1,1), then there is redundancy between them, which is essentially different than "nothing between 1 and 1" that is actually (1).
Start by the relevancy of the irreducibility of 1-dimensional element into a 0-dimensional element, and the non-extendability of 0-dimensional element into 1-dimensional element (there is no homeomorphism between them).
Really? Consider two distinct points S and M in natural 3-dim space. The distance between both points can never be equal to zero as seen below.
Wrong, the points in 2-dimensional space have two coordinates each, and there is 0 distance between 2 points only of there is actually a one pair of coordinates, or in other words, only a one point.
What do you mean?
You are definitely lost to Reason. What do you think that the word "pseudo" means? In the case of the eclipse, the 2-dim viewing space is a pseudo-space to the 3-dim space where the points were actually drawn into.
Since you can't extend 0-dim object into 1-dim object, you can't increase the magnitude of a 1-dim object.
Write that out again in English, and I'll have a go at understanding what you're on about.
“0” isn’t “approaching” anything (even 0) it is just, well 0. If you wanted to say that x is value approcing infinity and that “x<y, you could have just said it..
The actuality of nothing? What kind of kitchen talk is that? You're not peeling potatoes; you are in the process of submerging yourself into the deep philosophical waters where existence and non-existence dwells and that requires all the rigor available. Out of my undying kindness and love for OM, I help you this time to reformulate your thought, but then you are on your own.
Actually if you look at the formulae (note the educated plural I use
), both factors are reciprocal to the unit semi-circle, which is f(x) = Sqr(1 - x2). In those formulae (there is no language like the Latin language) there is a bunch of parameters substituting for x. Maybe the semi-circle is some kind of pseudo-sunset where the visible semicircle is "dry" and the one under the horizon line is "wet," when you watch it from your patio situated west of the Pacific Coast Highway.
Well, this is exactly your problem about this subject.
is simply avoided, because the used concepts (sameness;difference) are a generalization of concepts like distance and point.
All you did is to ignore b,c difference, which leads to partial understanding of what is really going on.epix said:
Please give an example of what you call "useful".
Nice epix, your "dry" "wet" sun fits to your pseudo metric flat universe.Actually if you look at the formulae (note the educated plural I use
I can develop the mathematical science, such that the term "mathematical branches" gets its actual meaning.
You are incapable of generating one near-rational thought given your natural resistance to do so. Our universe, especially the Milky Way, can be viewed locally as 2-dim space when it is desirable for a purpose and it has been done so. Gravitational lensing can be detected only through 2-dim viewing, for example. In the abstract domain, in order to investigate the nature of a 1-dim object, namely the length, 2-dim arrangement needs to be taken into a consideration, like in the case of the Pythagorean theorem. You can muse over the length of the hypotenuse ad infinitum your style, but unless you expand and include 2-dim objects, you won't figure the length.
In both cases you have at least two dimensional spaces, where the smaller dimension is local w.r.t the greater dimension and the greater dimension is non-local w.r.t to the smaller dimension.
There is no more then one element if there is no co-existence between at least two different dimensional spaces.
Yes I know, it is actually a pink snail.
What I actually say is that there is a domain starting with 0 and approaching infinity, where there are x and y variables in that domain, such that x<y.
In that case "dimension has nothing to do with points", which is a false proposition based on your "reasoning".
Only if one can't generalize what he\she reads.
Once again, you demonstrate your inability the understand the proposition "different version".
If something is considered as the negation of nothing, then without the actuality of nothing, something can't be defined.
You still do not get the actuality of nothing, and as a result you can't comprehend the difference of the following: