I believe there is constructivism - a minority view in maths - which rejects actual infinity. — Devans99
??? — Devans99
The definition of Aleph-naught is contradictory:
1. Aleph-naught is the size of the set of naturals
2. Sets contain a positive number of whole items only
3. So Aleph-naught must be a natural number
4. But there is no largest natural number
5. So Aleph-naught cannot exist (or be larger than all the natural numbers) — Devans99
3. Who said? Maybe that's your personal problem. In any case it is defined as the first transfinite cardinal. That is, not an integer. And not a natural number. — tim wood
If there is a finite number of points in each interval, space must be discrete, so I will assume an infinite number of points in each interval: — Devans99
One of the following must hold true:
1. points(0,1) = points(0,2)
2. points(0,1) < points(0,2)
3. points(0,1) > points(0,2) — Devans99
The argument that you are making with the limits assumes that space is infinite but discrete, right? — Mephist
A piece of space is discrete if it allows only a finite number of possible positions (points). So I've assumed an infinite number of possible positions - if its not discrete, it must be continuous. From Wikipedia:
"Formally, a linear continuum is a linearly ordered set S of more than one element that is densely ordered, i.e., between any two distinct elements there is another (and hence infinitely many others), and which "lacks gaps" in the sense that every non-empty subset with an upper bound has a least upper bound." — Devans99
OKSo all continua are (in the above sense) alike in that they can be subdivided forever, so we can write:
points(0,1) = points(0,2) — Devans99
It comes back to Galileo's paradox - the above are equal in the sense of a one-to-one mapping but at the same time, one is clearly twice the other. I think that it is not valid to compare the size of two infinities (as Galileo believed) - they are fundamentally undefined so have no size and cannot be compared. If something never ends, then it can never have a size and never be fully defined. I do not believe Cantor has added anything our the understanding of infinity - he has detracted from it - Galileo was on the right lines. — Devans99
So I think that maths cannot model infinity or continua. Does that mean these things do not exist in the real world? I think that maybe the case. If continua exist, then that implies that the informational content of 1 light year of space is the same as the informational content of 1 centimetre of space - in the sense that both 'containers' record the position of a particle to an identical, infinite, precision. This flaunts 'the whole is greater than the parts'. I trust that axiom more than I trust Cantor's math. — Devans99
Discrete means that is made of parts that are distinct from each other: it can be finite, but not necessarily. The set of natural numbers is discrete but infinite. — Mephist
Galileo's paradox is about positive integers, not about continuous sets. In fact, I believe that the idea of a "continuous set" had not even been invented in XVII century. For what I know, Euclidean geometry never speaks about a line being a set of points: for Euclidean geometry, 1-dimensional objects (lines) are a completely different kind of things then discrete (countable) objects. And Galileo does not even consider the idea that a line can be made of a set of distinct objects. For what I know, the idea of continuous (uncountable) sets was invented after Cantor, 200 years after Galileo. — Mephist
So, if I understand correctly, you are trying to prove that the set of point of a line is finite (not countably infinite). Is it right?
If that's what you are arguing (that a line is made of a finite set of points), the obvious question is: how many points there are in a given segment? — Mephist
I think it is not necessary to fill space (as in a space-filling polyhedron like a cube). I am more imagining a grid of zero dimensional points in space. The particle, which has a non-zero dimension, would be centred on one of the grid points. If there are two neighbouring particles, they would not be in contact with each other, so space is not filled. Particles would move from point to point in the grid rather like the electron performs a quantum jump from one orbit to another - not passing between any intermediate space.
With QM, we have waves (and I suspect a particle is just a compressed wave) and so the waves would be centred on one of the grid points. — Devans99
Loop quantum gravity - the competitor of string theory - has space as discrete. — Devans99
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