A definition of infinity?

I’m interested in the idea of infinity, but like many others, struggle with some of the concepts involved. Further, if I look up the dictionary or encyclopedia, I see there are numerous definitions or ”understandings’ offered.As such, I’d like to explore one aspect only, using a simple hypothetical example.Imagine I’m walking along and come to a piece of string stretched across my path. At this point, if I choose to follow the path of this string, one definition of infinity might be that I could follow it forever, in either direction, and never reach the end, or return to the point I started at, unless I decide to stop and retrace my steps. However, if I choose to take a knife and cut the string, do I have the same definition of infinity? Sure, I can set of in one direction and carry on forever, but if I decide to retrace my steps at any point, I will eventually return to the point I started from, the ”end’ of the string, making it finite in one direction, if not the other.I don’t know if I’m making any sense, but can the second scenario be considered to be an example of infinity?

Cheers nemesis.I love this experiment, and I believe I once read that if you expand the ”wall with slits’ to encircle the particle gun, the spectrum will extend the full 360 around the gun, as determined by the probability distribution. Or perhaps my memory is playing tricks on me.I know we’ve wandered a bit OT here, but I wonder whether the outcome of the experiment is dependent on the status of the measuring device, i.e. does it need to be functioning for the waveform to collapse? What would happen to the results if the measuring device was randomly switched on and off during the experiment?I’ve often wondered also what would happen if instead of a measuring device, different coloured filters were placed in each slit when the single photons are fired.

I guess if our universe turned out to be infinite, and it was theoretically possible to realise such a line, how would it work? I mean, if you started at any point, you could presumably realise such a line in any direction, travelling forever. Presumably you could realise an infinite number of such lines radiating out in all directions, so what would cause them to be semi-infinite?I expect I’m not making myself very clear here, but something that’s always puzzled me is that infinity tends to be thought of as infinitely large as opposed to infinitely small. In some ways the idea of infinitely small intrigues me more than the idea of infinitely large. If we’re prepared to accept the possibility that the universe might be infinitely large (whatever that may be), I wonder if there’s also a possibility that there are things within this universe that might be infinitely small. In other words, in the previous paragraph, might it be that the lines that can be realised from infinity back to the point never converge because they keep on travelling forever into some infinite smallness ”within’ the point?

Don’t know. I’m just trying to get my head round some of these ideas.One observation I would make regarding the Planck length is that it uses the speed of light as part of the calculation. There has been some controversy recently about whether the speed of light is in fact a constant, Speed of light may have changed recently | New Scientist, leading to the possibility that the Planck length is not constant either.However, I accept that I am nit-picking here, and saying that the Planck length may be subject to small variations is not the same as saying that there may exist things that are infinitely smaller than it.

I don’t think I’m making a very good job of trying to explain what I’m trying to get at . maybe because I don’t really know what I’m trying to get at.I think what I’m trying to say is if you imagine a line coming from infinity and follow it back towards a single point in space (whatever that might be), what prevents the line carrying on forever ”into the point’?I expect the problem I have arises from my concept of what this single point might be. I tend to think of a point as something spherical. If I go to a physics textbook, I tend to see atomic and sub-atomic particles represented as spheres, so I tend to envision a point in space as a sphere also.Taking that as my starting point, if I imagine a sphere one meter in diameter, with a line extending infinitely from its surface, I will have a semi-infinite line. If I then imagine the diameter of the sphere to be halved, I still have a semi-finite line. I guess my question is, what prevents me halving the diameter of the sphere indefinitely? In other words, if I were to follow the line on its journey towards an ever-shrinking sphere, must I inevitably reach a point where the line can extend no further, and therefore be considered semi-finite?This is what I was trying to say when I clumsily used the word ”cause’. Perhaps I should have asked something like, does a line drawn from the theoretical largeness of infinity always have to be semi-infinite when followed in the direction of ”smallness’ in multi-dimensional space?Mmm . I’m still not sure that makes any sense.

Obviously I’m no expert in this area, which is one of the reasons I started the post, but I hope I’m starting to gain a better understanding than I previously had.It begins to appear that the way we view things goes some way to defining degrees of infinity. If we take, for example, the set of positive integers, it could be defined as infinitely large. If we then pick a positive integer at random, and proceed to increment it by one, we can do so forever, as the number of positive integers is presumably infinitely large. However, if we choose to deduct by one each time instead of increment, we will eventually arrive at the number one, a point beyond which we cannot go as there are no positive integers less than one.So it appears to me if we view the set of positive integers solely as that, a set, it may be defined as infinite. However, if we start to think of the set in terms of a sequence, it may be defined as semi-infinite, as there is a point beyond which it is impossible to go in one ”direction’.There’s another interesting point regarding positive integers and infinity. It concerns what happens when you pair each positive integer with its square, i.e. 1 with 1, 2 with 4, 3 with 9, etc. The original set will be a complete set of positive integers, but the set of squares is not, yet both sets are exactly the same size! I believe this apparent paradox has been resolved, but I cannot remember by whom, and am not aware as to what the explanation is. Maybe somebody on here can help us.

Is an ”infinite definition’ the same as a definition of the infinite though?

That sound you just heard was my head exploding!I can get my head around the Schrodinger’s cat thought experiment. As far as I understand it, there is no way of knowing whether the cat is alive or dead until the box is opened. Although I have seen this characterized in some places, as being the cat is simultaneously alive and dead, my standard worldview kicks in and dismisses this as purely hypothetical. A different analogy might be, a football match has taken place, but I am unaware of what the result is, so until I learn of the result, the outcome remains a matter of probability-based conjecture.I think the problem for those who hold the ”standard’ worldview as regards the double slit experiment is more complex. On the face of it, the interference pattern observed when no attempt is made to identify which slit the particle passed through, can be seen as the outcome of the event. Using the football analogy again, when I’m told the result, it is A wins 45% of the time, B 25%, the draw 30%. But if I want to know what actually happened, I need to have observed the game take place.However, I guess what you’re saying is that the interference pattern observed is not the outcome of the event, but the spectrum of possible outcomes, had we taken the trouble to observe it. Or am I still confused?

Cheers SG. Thanks for taking the time to try and explain these things. It is much appreciated.Hopefully I am starting to get a better grasp of the some of the issues involved, but am still hampered by my ”standard worldview’ in the way I see things.For example, with classical statistics, I can assume the chances of throwing a 6 when I roll a dice are 16.67%. To prove this hypothesis, all I need to do is roll the dice a sufficient number of times, and I ought to see an (approximately) even distribution of the six faces. Further, if that is not what I observe, I may start to infer some bias in the dice, with varying degrees of statistical confidence.However when I view the quantum example, would I be right in thinking that it is not possible to directly observe the negative probability of some of the outcomes? That is, I presume if all three paths were open in the quantum case, it would not be possible to observe the ball going down path C -66% of the time.Actually, the more I think about it with my ”classical’ head on, the less convinced I am of my presumption. I suppose we must be able to observe something, else how could we conclude that the probability of path C was not -50%, for example? And if we were to add further paths, which result in more negative probabilities within the scenario, how do we detect the contribution of the negative paths? Or am I still failing to comprehend?I guess some of the above might be explained by the answer to my next question, which is about the total probability of 0% in the quantum case. Is it the case that this probability could have been anywhere between 0% and 100%, and you just happened to choose 0% to help illustrate the weirdness of the maths?Incidentally, I’m still finding it really hard not to infer some undetected level of interaction in between the red laser and the atom in the Zeno experiment. But I guess I need to keep reminding myself that events at the quantum level are inherently different to what we are used to observing.Finally, I wonder what the quantum version of the Monty Hall problem would be? And it doesn’t bear thinking what the ensuing debate would be like.