matt grime

but it doesn't even prove that sqrt(2) is irrational. and the proof that any n'th root of any rational (except for perfect nth powers) is irrational is as short (rational integers = integers)

Somehow i thijnk that it ought to be {(a,b,c) in R^3:a+b+c=1} is not a vector space. The thing you're written down doesn't make any sense since 1 is not in R^3

None of the terms you use is standard. Indeed, I can't decide what you mean and I'm a group theorist. Are you asking for all subgroups of a given group? Well, obviously {0} is a subgroup of Z_4, as is {0,2}, and there can be no other subgroups for many reasons (simplest is because if the subgroup contained 1 or -1 then it is the whole of Z_4). What exactly do you mean be "realations"? what do the - symbols you've used signify?

no that is a completely spurious analogty. as i keep pointing out you logic also implies that the status will have to be off and on simultaneously if the notion of "did reach 2" were actually valid. which it isn't. it is also wrong and down right disingenuous to say that there is a last number in 10/3. "digit" would be better, not that it matters since there is not a last (nonzero) digit in the decimal expansion of 10/3.

no, it won't run for two minutes, that is the whole point of zeno's paradox. at no (finite) number of cycles have you ever reached time t=2, so your analysis of what happens there in terms of the finite pattern doesn't have any basis.

or if we're doing it in terms of numbers then yo'ure bracketing the sum as (1+1/2)+(1/4+1/8)+(1/16+1/32)+... where as why not bracket it 1+(1/2+1/4)+(1/8+1/16)+...

so, you''re stating that after 2 minutes we've added up an even number of cycles, despite the fact that this is an infinite number of cycles we're talking about and parity makes no sense in this context. let me do it for you in words: you are looking at (on off) (on off) (on off) with each on off cylce being half the length of the previous one and the off period being one half the length of the previous on period, right? however, that observation holds true for on (off on) (off on) (off on)... so what ever the conclusion was for the former the opposite holds for the latter. it is hard to make sense of what you're saying since it is written in some short hand that is meaningless to me (ony in equal /2 counts of this minute, for example) secondly, in your model, no wave peak or trough so described ever falls at the time t=2, only the previous times are defined. or to put it in less couched terms *you never get to time t=2 in your model* (this is merely zeno's paradox)

you can factorize the numerator. the point was you were supposed to work out the limit of the first definition as x tends to 6,. not simply bung in 6 and notice it was a divde by zero thing. as the first part isn't defined at x=6 this doesn't make any sense to do.

"if the clock is stopped at any even division of a minute from the starting position" nope, sorry, that makes no sense. going back to the original formulation you have not specified the wave from time 2 or greater, only for all stages before the time t=2. the coutn has not "stopped exactly" the definition of the wave you gave is not valid at t=2 or later. the self simliarity you use is dimensionless so it implies that the same logic is valid only looking from time t=1 to t=2, and thus concludes that the lamp must be off (if it were on in your orginal reasoning).

well, that is obviously nonsense: start the experiment at -2 seconds and have it in the oppposite state at the start of this period as at 0 seconds. now your logic applies perfectly there too so it is in the exact same state as when it was initiated at -2 seconds, only that is now the opposite one.

Yes, you've got me bang on. Just because I know something at every point *before* some fixed teim T deosn't necessarily mean that I can deduce what is happening at T. In the ideal situation I wuold have the assumption that things varied continuously (they don't here, it is very discontinuous) or that there was some known extrapolation, but in this case we don't know anything at all (nothing to do with never reaching 2 or some such Xeno type nonsense). There is a continuous variation problem version: sin{1/x} as x tends to zero. it is a pathological counter example to many things (especially in topology)

let em give another exmaple, suppose for the sake of argument that we know something, x(t) is a continuous function of t defined for all t, and that x(1- 1/2^n) = 7+2/n^2, then for n in the natural numbers, then what is x(1)? well, since x varies continuosly and 1-1/2^n tends to 1 as n tends ton infinity, uit must follow that x(t) tends to 7 since 7+2/n^2 tends to 7 as n tends to infinity. see, more information, and i can make a deduction. I can give a funtionf defined for all of t with range {0,1} that exactly models the bulb for the period of time youve defined it for and that is 1 (on) when t=2, and i could give one that is 0 when t=2, indeed i can give infintely many different models for either situation.

but the problem is that you *have* only specided what happens after a finite number of switches, you have not told us *how* one passes to the limit. passing to a limit is a delicate question that needs context; you've not give us any mathematical context and there is no physical context.

but there's the crux: you've told me how it behaves at every point in time prior the the elapse of two minutes but you've 1. not told me what happens at 2 minutes 2. not told me how i am to use the previous information to conclude what happens at the limit point. i need to know how things evolve through the "barrier" as it were. i have no absolutely canonical theoretical method of passing the behaviour to the limit, nor do i have any "fall back" physical intuition i can use. recall the eaxmples i gave about nested sets and so on. there is no reason to suppose that there is any meaningful way to pass to the limit point and have the "bahaviour carry on" that is we cannot infer from the previous information about all the preceding points in time what the behaviour at 2 minutes is. sometimes properties can pass to the limits, some times it doesn't make sense forget the elapsing of time which is a complete red herring anyway, suppose i told you that at time t=1 i have 1 banana, t=2 i have 2 bananas, t=3 i have 4 t=4 i have 8, at time t=5 i have 16 bananas. how many banans do i have at time t=6 or 7? you might suggest 32 and 64, assuming the pattern continues, or you might recognise that it is the infamous trick sequence (of the maximum number of segments you can partition a circle into by joining up dots on the perimter, and suggest that the answers are whatever that sequence is) but truthfully there is nothing compelling you to make either of these deductions. the ideas of limits and so on that you're talking abuot are actually totally irrelvant to the reason why mathematically this makes no sense. the problem is simply that given the information we cannot deduce what is going on at t=2.

and there goes xeno's (or zeno's) paradox.

As far as I can tell, the question in its purest mathematical terms is, reversing the direction of time as we may, defein f(t) to be n mod 2 (ie 1 for odd n 0 for even n) when t is in the interval (1/2^{n+1},1/2^n], ie start at 1 and count down to 0 for n in the natural numbers (including 0) is there any way to "naturally" extend this to a function valid at t=0? The answer is "no" not in the terms given. There is a reasonable, natural extension to negative integer n, that is we can extend to a function on the intervals (1,2], (2,4] and so on. OK, so what if i were to tell you now that we were modelling a theoretical object that can switch on and off? well, since it is theoretical there is no obvious way to say what happens when t approaches zero. we have no intuition to rely on since it is an inherently unintionistic (to invent a new word) situation. is time even infinitely divisble? might it not be discrete? (actually, all models for discrete time, i'm unreliably informed, have problems and would lead to contradictions with known observations at large scales; this doesn't mean that time is continuous jsut that no good discrete model exists). so it is practially impossible and theoretically not enough information is given. there is always danger in extrapolating beyond known data anyway.

Of course not. Nor should it.

any question that has a multitude of unstated assumptions (each of which could be one of many choices, most of which are inconsistent with each other) willl naturally lead in many directions. personally i don't think the question is at all interesting beyond showign that there is a lack of understanding of the word "function" in the physical sciences, yet somehow i keep trying to explain what maths does and doesn't say.

are you talking physically or purely from the mathematical vewi point of the 0,1 function interpretation I gave? It is *your* assumption that when the input is 2 the fucntion is fixed at 0. in your words, that is fair enough if that is how you wish to model the situation. there is no mathematical reason to suppose that is what happens, and no physical one either since the situation is unphysical. consider the following example. Suppose that a quantity x satisfies x(t)=1/t for all strictly positive t. what is x(0)? teh question makes no mathematical sense, obviously, but if i were to ask you to state some reasonable assumptions and then make a conclusion, what would you think the answer is? to me, as i've not defined x(0) i can declare it to be anything I wish and it is still a function of t for positive t (inckluding zero), and there is no way to make it continuous on the extended domain. that is all you can say about it. if you want to give some more physical meaning to it, do so, but i thought you weren't interested in the physics of it merely the mathematics of it.

nonsense; you've given no information at all that allows you to discuss its state at time t=2 or greater.

again that is a physical interpretation of the question and not a mathematical one. it is akin to xeno's paradox, and not a mathematical problem at all.

we're not talking about the physical possibility of this situation at all, simply what happens when we pass to the limit this is a very common mathematical sitaution. can we pass from some case defeind for all finite number of steps to say anything about an infinfite case? firstly we must deicide if the limit of the index exists, and then if the property passes through to the limit. example: let S_1, S_2,.. be the nested sets {1} {1,2} {1,2,3} etc then the limit of the sets is clealry the natural numbers in any reasonable sense, now consider some properties of the sets. eg S_n is a set of natural numbers. this property is passed on to the limit S_n is a finite set, this is not a property preseved in the limit in this case we have on or off (0 or 1) at each finite stage, but no canonical way of passing it thruogh to the limit (at time t=2) of course the situation so given is physically impossible; but apparenlty that isn't the point of the question (though this is purely my interpretation of "no practical considerations")

i thought practical considerations were to be ignored. the "theoretical" reason is perhaps this: replace on, off with the fucntion of time f(t)=0 when off, 1 when on. for 0<=t<2 as above. then removing all physical considerations, what is f(2)? or f(3)? well, we've not defined it. BUT there are ways of extending functions ot larger domains, but this usually depends on some structure of the function, and some natural "best" kind of extension. for instance, factorial thatm ost people think is defined oddly on 0 is simply a natural extension of it for the positives (useful and consistent and fits into the recusive definition of n!=n.(n-1)!. most naturally is the idea of continuous extension, or extension via continuity, that appears in say the arithmetic of the extended complex plane. or perhaps we want to take a continous function on the rationals and extend it to a continuous function on the reals. if possible. perhaps we want to extend to end points of an interval. note this isn't always possible. finally, there is analytic continuation where we try to extend a taylor series beyond its radius of convergence where possible. eg 1+z+z^2+z^3+... converges only for |z|<1, yet we all know that it is (1-z)^{-1} so it extends to a function on all of C except 1. however, youir fucntion has no nice natural properties that we may wish to extend in a natural way.