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Yet, if the segment of length 1 really consists of infinitely many subsegments of length  , as of ‘chopped-off’ wholes, then it is incompatible with the character of the infinite as the ‘incompletable’ that Achilles should have been able to traverse them all. If one admits this possibility, then there is no reason why a machine should not be capable of completing an infinite sequence of distinct acts of decision within a finite amount of time; say, by supplying the first result after  minute, the second after another  minute, the third  minute later than the second, etc. In this way it would be possible, provided the receptive power of the brain would function similarly, to achieve a traversal of all natural numbers and thereby a sure yes-or-no decision regarding any existential question about natural numbers!
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and 
as follows:
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The proper time measures the physical system time by clocks in an usual way.
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A discrete cycle time  characterizes an “intrinsic" time scale for a process running on the machine.
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For some unspecified reason we assume that the machine allows us to “squeeze" its intrinsic time with respect to the proper time by a geometric progression. For  we let any time cycle of , if measured in terms of , to be “squeezed" by a factor of  with respect to the foregoing time cycle. More precisely,
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approaches the infinity, the proper time 
approaches 
, so it remains finite. 
One might argue that such an “oracle" would require a geometric energy increase resulting in an infinite consumption of energy. Yet, no currently accepted classical physical principle excludes us from assuming that every geometric decrease in cycle time could be associated with a geometric progression in energy consumption, at least up to some limiting (e.g. Planck) scale. 
whose input is a binary string 
. Assume, for the sake of a contradiction, that there exists an effective halting algorithm HALT, implementable on a Zeno machine, which is able to decide whether 
eventually stops on or not. Using HALT
we shall construct another Zeno machine 
, which has as an input a program and which proceeds as follows: Upon reading the program as an input, makes a copy of it.
In the next step, our machine uses the code 
as an input string for itself, that is, forms 
, henceforth denoted by 
. The machine hands over to its subroutine HALT. Then, proceeds as follows:
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if HALT decides that eventually halts, then does not halt,
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if HALT decides that never halts, then halts.
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on its own code as input? Notice that is arbitrary, so there is no restriction to prevent us for doing this! Consequently, , which is representable by its code 
will be applied to itself. is restricted to classical bits of information. Then, whenever 
halts, HALT
forces not to halt, and conversely, whenever does not halt, then HALT steers into the halting state. In both cases one arrives at a contradiction, therefore, Zeno machines are logically inconsistent. is allowed a qubit _of information. Assume that 
and 
are the halting and nonhalting states, respectively. The computation can be performed if receives as an input a qubit corresponding to the fixed point state 
of the NOT operator
: 
proves the impossibility of to control the output as the probability to reach a halting (nonhalting) state is exactly one half. At the level of probability amplitudes, quantum theory permits Zeno machines, but at the level of observable probabilities, this super-power is nullified, as the result of the computation appears to be random. 
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Classical mechanics postulates space and time continua as a foundational principle.
This can be readily achieved, since the program 
is presented to 
in some encoded form 
, i.e. as a string of symbols.
This can be realized by an infinite loop.
Diagonalization operator.
