Strange

No. I am asking what this thread is about. I haven’t even followed your link yet (have you read the rules?)

It is not my website. It is not propaganda. It is an educational resource. People should be interested. As 2 of your first 3 posts appear to be spam... This is neither

poles

What?

Not really the point, though. Replace “proof” with whatever it is

You will need to present your proof here. https://www.scienceforums.net/guidelines/

Why?

Simpler but less useful. Your intuition, maybe. I have no problems with an intuitive understanding of the natural numbers. It wouldn't have been, if you hadn't been so unclear (verging on dishonest) about what you meant.

That's what I said. (Actually, what you mean is a+b is not in N - you can't even get your own notation right! ) So you have defined a set of numbers where the usual rules of arithmetic don't apply. I'm not sure what the point is. As has been pointed out several times, nothing that you prove about these numbers has any relevance to the natural numbers used in the real world (ie outside this thread).

Maybe you should ask the mods to change the title to "Cardinality and Bijection of FINITE sets" so it describes what you are actually talking about. p.s. I have suggested this to the mods

I think you can achieve this, even with Edgar's finite set of integers, using modulo arithmetic. However, he doesn't seem to want to do that, just saying that c is undefined which doesn't really make sense to me.

I have been trying to do this, for example when pointing out that it is not "gravity curves space" but "mass curves spacetime (which we perceive as gravity)". I don't know what level of (mathematical) detail you want to get into, but I think this is a really good overview of GR, and you can understand quite a bit even without following the math: http://math.ucr.edu/home/baez/einstein/einstein.html

Not a huge fan of Sinatra either. When he died someone said, "the greatest singer of all time". And I thought, maybe one of the best white, male singers of popular songs in the mid 20th century. But even then, he would only scrape into the top 10. (If someone had said the same when Aretha died, I probably would have just nodded respectfully.)

I don't have an opinion on whether pi or theorems are "real". It doesn't matter. (I don't even know what "real" means.)

These are not a cold or solid state fusion reactor, though. Just smaller plasma fusion reactors: https://en.wikipedia.org/wiki/Lockheed_Martin_Compact_Fusion_Reactor http://uk.businessinsider.com/boeing-just-patented-a-jet-engine-powered-by-lasers-and-nuclear-explosions-2015-7 I hope so. But progress has been incredibly slow over the last few decades. It always seems to be 30 years away.

Just that they are the ones we use all the time. It isn't clear that the rules of arithmetic work with your numbers (at least without some modification).

Just to expand on that, to avoid any confusion, "space" is a subset of spacetime: the 3 spatial dimensions. So spacetime is the measurements of distances and times.

True. But as your axioms define something different from the natural numbers, any conclusions you produce only apply to the objects you have defined, not the natural numbers. So maybe you can't define bijection on the sets you define with your axioms. But that says nothing about bijection of the natural numbers.

That could, in principle happen. So call the distance between things an "object" if you like. It may cause confusion with other people's concept of "object" but I don't really care. Mass warps spacetime. Spacetime is the set of measurements we make of space and time. (You keep focussing on space, but time is affected as well.) Field is a mathematical term (https://en.wikipedia.org/wiki/Field_(mathematics)). But that isn't the same meaning as in physics (https://en.wikipedia.org/wiki/Field_(physics)). A field is the set of values at different locations in space and time. So the electromagnetic field describes how electric charges and magnetism interact. The spacetime field describes how mass and energy interact. The electromagnetic field is described by Maxwell's equations. The spacetime field is described by the Einstein Field Equations. (Again, whether these fields are "objects" or even "real" is not something I have an opinion on. It isn't relevant.) OK. I will try and improve!

When you say things that are incorrect then it doesn't make any difference how many times people read it or how much they think about it. It is still wrong. Going back to your first post, you claim to be talking about the set of natural numbers. But now you are clearly talking about some finite subset with an arbitrary upper bound which you call ω-1. I don't think I can go back and reread the whole thread replacing the assumption that N is the natural numbers with N being some arbitrary subset.

So if you are talking about a finite set of numbers (not the set of natural numbers) then you some of your claims will be correct. But irrelevant to the set of natural numbers.

The analogy is supposed to show how changing the geometry of the lines that the people are moving on can cause an apparent force between them. Forget the "Earth" part of it and focus on the geometry of the lines of longitude. You can certainly say that magnets curve the electromagnetic field. Which is a reasonable analogy for how mass affects the spacetime field. Gravity doesn't curve space. Mass curves spacetime. One consequence of that is gravity. If you want to to stick to a totally Newtonian approximation, that's fine. I get the impression you don't really want to understand, so I won't put any more effort into this. Space IS the measurement(s) of distance. That is all it is. And those measurements are changed by the presence of mass-energy. So, not an object. And not "warped by gravity".

Sorry!

Space is just distances (in three dimensions). Would you call the distance between you and your coffee cup an "object"? It does require a new way of looking at things. The curvature of spacetime caused by mass describes the way that the relative paths of particles moving forwards in time change in the spatial direction(s). As an analogy, imagine two people walking forwards, side by side, on a flat plane. Their paths will remain parallel over time. We can consider the direction they are walking as the "time" dimension (they are moving steadily into the future) and the distance between them as one of the space dimensions. On the the flat plane, the distance between them doesn't change over time. Now put them on the surface of the Earth and have them both walk towards the North Pole (along lines of longitude). As they move forwards (in time) they get closer together. No force is acting on them, it is just a consequence of the curved geometry they are travelling in. You can consider them falling towards one another because of the gravity of the curved space-time they are in.

Dreams can be influenced by external factors, so I suppose they are "actual things" by this definition. In that case, our measurements of space and time (and the geometry relating them) are actual things because they are influenced by the presence of mass and energy, and by relative movement. Gravity is a force we feel. That force is created by the geometry of space-time. Space-time is a set of coordinates that we use for measurement. The equations relate space, time and the thongs that influence them. Whether any of those things are "real" is not something I can comment on. This conclusion seems to contradict everything that has been said. We have a model of how curving spacetime causes gravity. We have no model at all of how gravitons could cause gravity. Up to you. The best alternative then is to go back to Newtonian gravity where gravity is just a force proportional to mass.