Angličtina pro matematiky II

COURSE MATERIALS WEEK XII.

Discussion (from Wikipedia)

The status of the lamp and the switch is known for all times strictly less than two minutes. However the question does not state how the sequence finishes, and so the status of the switch at exactly two minutes is indeterminate. Though acceptance of this indeterminacy is resolution enough for some, problems do continue to present themselves under the intuitive assumption that one should be able to determine the status of the lamp and the switch at any time given full knowledge of all previous statuses and actions taken.

One response is that one must consider how much time is spent moving the switch. Questions of lamp physics aside, one can simplify the problem to flipping a single bit of information to either a 0 or 1 state. If the flip takes any constant positive amount of time, then an infinite number of flips would take forever. So the only way this paradox will reach the 2 minute mark, under the assumption of constant flip time, is if the flip is not a delaying factor — essentially, if the flip takes zero amount of time. Yet if one can change the state of a bit instantly, then what does the question of a bit's state at a certain time mean? One could turn it off and on again without any time passing. One could even turn it off and on an infinite number of times. This response, however, does not deal with the case where successive flips take less and less time, so that the entire supertask can be performed in the given two minutes.

One possible solution to this problem, at least in the physical world, is provided by special relativity, i.e. the existence of a speed limit. That is, no one and nothing would be able to flick the switch infinitely fast, as would be required at the end of the sequence. There is a limit (the speed of light) to how quickly we can flip the switch.

 

1. What is known for times less than 2 min?

2. What is not stated in the question?

3. What is resolution enough for some?

4. In what time limit can this supertask be performed?

5. What is provided by special relativity?

 

 

Listening: Refuting Zeno´s Paradox – Series on Infinity Part 7
 
 
Listen and decide whether the Ss are true or false.
 
 
1)      The aim of the lecture is to explain Zeno´ s Paradox.
 
2)      Zeno wanted to demonstrate that a turtle can never pass Achilles.
 
3)      The lecturer wants to persuade the mathematicians that the argument is wrong.
 
4)      The turtle in the example is faster than Achilles.
 
5)      It takes Achilles 30 seconds to reach the place where the turtle was.
 
6)      Zeno argued that in each stage of the race the turtle is ahead of Achilles.
 
7)      The Greeks believed that you can not add infinitely many numbers because there is no time for it.
 
8)      In an attempt to explain the problem, you should consider time, not distance.
 
9)      We cannot calculate the time that has elapsed from the start of the race.
 
10) The total time is less than one minute regardless of the value of n.
 
11) After one minute Achilles is still behind the turtle.
 
12) The series is divergent.
 
 
 
 
 
 
Listening: Refuting Zeno´s Paradox – Series on Infinity Part 7
 
 
Listen and decide whether the Ss are true or false.
 
 
13) The aim of the lecture is to explain Zeno´s Paradox. T
 
14) Zeno wanted to demonstrate that a turtle can never pass Achilles. F
 
15) The lecturer wants to persuade mathematicians that the argument is wrong. F
 
16) The turtle in the example is faster than Achilles.F
 
17) It takes Achilles 30 seconds to reach the place where the turtle was. T
 
18) Zeno argued that in each stage of the race the turtle is ahead of Achilles. T
 
19) People believed that you can not add infinitely many numbers because there is no time for it. T
 
20) In an attempt to explain the problem, you should consider time, not distance.T
 
21) We cannot calculate the time that has elapsed from the start of the race. F
 
22) The total time is less than one minute regardless of the value of n. T
 
23) After one minute Achilles is still behind the turtle. F
 
24) The series is divergent. F
 
 

     

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