I'm interested in your take on the nonexistent 'barrier' thing described at the lower half of my prior post in this topic. It also is a variation on something somebody else authored, but I cannot remember what it was originally called. — noAxioms
A man walks a mile from a point α. But there is an infinity of gods each of whom, unknown to the others, intends to obstruct him. One of them will raise a barrier to stop his further advance if he reaches the half-mile point, a second if he reaches the quarter-mile point, a third if he goes one-eighth of a mile, and so on ad infinitum. So he cannot even get started, because however short a distance he travels he will already have been stopped by a barrier. But in that case no barrier will rise, so that there is nothing to stop him setting off. He has been forced to stay where he is by the mere unfulfilled intentions of the gods.
The solution, similar to my proposed solution above, is that movement is not infinitely divisible (either because space is discrete or because movement within continuous space is discrete). — Michael
If movement is continuous then an object in motion passes through every 1nm marker in sequential order, but there is no first 1nm marker, so this is a contradiction. — Michael
This is not true. Perhaps you are reading a different account of the story than I did, which is the one on wiki, which says simply:
"In a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead". The 2nd bolded part is the non-sequitur, and the first bolded part follows from the 2nd. None of it makes the assertion you claim. The non-sequitur makes the argument invalid. There are ways (such as with the light switch) that make it seem more paradoxical. — noAxioms
Same non-sequitur. It is not true that Icarus always has more steps to take, only that he does while still on a step, but the time to complete all the remaining steps always fits in the time remaining in his minute. — noAxioms
OK, which premise then is false in the Zeno case? The statement is really short. One premise that I see: "the pursuer must first reach the point whence the pursued started", which seems pretty true to me. — noAxioms
Abstract space (as opposed to physical space) cannot be discrete because any minimum unit you propose can be halved. — keystone
The discreteness that ↪Metaphysician Undercover
↪Michael
are looking for is not in space but in measurement/observation. — keystone
Is minus one a natural number? And, is zero a natural number? Mathematicians' mathematics is not a strong suit for some. sad smile at one's own ignorance — kazan
Is minus one a natural number? — kazan
And, is zero a natural number? — kazan
Yes,very helpful.Thank you for taking the time.
Which begs the question, (smile) how, if it's possible, would "the lower case omega" concept of "upper (lower?) limit" be applied to all integers? Surely, if it's possible,this could be useful in some areas of mathematics ( besides arithmetic ). — kazan
This is a paradox I've come up with myself. But as Michael has mentioned it's very similar to Thomson's lamp. Where do you see problems with it? — keystone
You described it as endless, and yet claim he reached the end... The "paradox" is just you choosing to invent a story with contradictory concepts. — flannel jesus
Fair enough.
Nothing new of interest, comes to mind. Apart from adding negative l c omega with (to?) (positive) l c omega and getting the same answer as subtracting them,(still in the realms of arithmetic,) presumably zero? — kazan
Never in memory, has "pure" mathematics been of such interest as now. Feel like you've open a window and there's a gale blowing in, here. — kazan
Had more questions about l c omega, but will give them further thought first. And catch up with you elsewhere and later. Lounge perhaps in a few days? — kazan
As far as I understand about paradoxes, that's precisely what a paradox is about. It is a self-contradictory statement, but arrest our attention. — javi2541997
A paradox is a logically self-contradictory statement or a statement that runs contrary to one's expectation.[1][2] It is a statement that, despite apparently valid reasoning from true or apparently true premises, leads to a seemingly self-contradictory or a logically unacceptable conclusion.
Quantum Jump - Abstract space (as opposed to physical space) cannot be discrete because any minimum unit you propose can be halved. This is not an acceptable solution to Zeno's Paradox. I agree with you that Zeno's assumptions about motion are flawed, but you haven't offered an alternative premise that holds up. The whole point of his paradox was to highlight that the standard view of motion was flawed. Additionally, it's not definitively established that physical space is discrete. It's possible that only our measurement of space is discrete. This latter perspective is my belief which I'll expand on in a couple of paragraphs. — keystone
Similarly with the Thomson's Lamp case. When we ask "is the lamp on or off at one minute" we are asking for something that the set-up doesn't give us enough information to answer. The setup tells us whether the lamp is on or off at every instant in [0,60) and tells us nothing about whether it is on or off at 60 or later. We cannot infer whether it would be on or off at 60 because we know nothing about the physics of the world in question, which must be enormously different from that of our own, in order to allow complete switching of a finite-sized lamp in infinitesimally small time periods. I expect we could invent some physical rules to support either an on or an off assumption. — andrewk
`despite apparently valid reasoning from true or apparently true premises` - that's key! The premises and steps in reasoning have to make some kind of sense. — flannel jesus
The staircase problem is called an omega-sequence paradox, a paradox that involves counting 1, 2, 3, ... and doing something at each step, then expecting the behavior to be defined in the limit.
The answer to all those paradoxes is that you haven't defined what happens at the limit. — fishfry
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