But mixing up actual stairs with models of stairs just produces a confusion, so the paradox is just an illusion - in my opinion. — Ludwig V
Despite the staircase being endless, he reached the bottom of it in just a minute. — keystone
This is not a paradox, but a confusion of concepts (like the number 1 or infinitely) with actual things (like a one step down one stair, and never reaching the bottom or doing so in a minute). — Fire Ologist
There are two cases in play at the moment - "0, 1, 0, 1, ..." and "1/2, 1/4, 1/8, 1/16.." Comments switch between them without always being clear. You are, however, quite right that the first sequence doesn't have a limit and the second one has what we could call a natural limit. — Ludwig V
Now if I can just get Michael to agree! — fishfry
@fishfry will speak for himself. But I think the point is that, even a convergent sequence, which does have a limit, does not have a end or last step defined - indeed, is defined as not having one. That means that any answer whatever is equally valid and invalid.I disagree with your claim that with respect to Thomson's lamp we can simply stipulate that the lamp is on after two minutes. — Michael
That means that any answer whatever is equally valid — Ludwig V
It is not just the case that whether the lamp is on or off after two minutes — Michael
I don’t understand. How do you ever arrive at the two minute mark? — Fire Ologist
That's how the world works. — Michael
The same is true if the lamp switches just the once — Michael
So we resolve the paradox by accepting the metaphysical impossibility of supertasks. — Michael
Is the lamp on or off after 2 minutes?
In all three scenarios the switch is "designed to function within two minutes." — Michael
Or more precisely, not designed to function at or after two minutes. — Fire Ologist
Because the switch is not designed to ever present the question. — Fire Ologist
Given that the lamp must be either on or off after two minutes — Michael
How is that? How is it on or off at or after two minutes? — Fire Ologist
It cannot be a function of a switch that operates by switching every half of the prior interval. — Fire Ologist
And that's precisely why supertasks are impossible. — Michael
And that’s precisely why the question of whether the lamp will be on or off at two minutes will never present itself. — Fire Ologist
at successively halved intervals of time within two minutes — Michael
the lamp cannot be either on or off after two minutes. — Michael
I don’t understand. How do you ever arrive at the two minute mark?
1 minute, half a minute later, quarter minute later than that, etc., infinitely…you never arrive at the two minute mark. — Fire Ologist
Two minutes just pass. That's how the world works. — Michael
Then stop talking about at two minutes or after two minutes. That’s some other scenario.
Don’t you see that? — Fire Ologist
Two months later Michael is still stuck in the same infinite loop. — Metaphysician Undercover
I have always agreed that the sequence "0, 1, 0, 1, ..." does not converge. — Michael
I disagree with your claim that with respect to Thomson's lamp we can simply stipulate that the lamp is on after two minutes. — Michael
See my previous post and my initial defence of Thomson on page 13. — Michael
A supertask is not simply an infinite sequence of numbers. — Michael
In our hypothetical scenario with hypothetical physical laws we are still dealing with the ordinary logic of cause and effect. — Michael
It is implicit in the thought experiment that it is only by pushing the button that the lamp is caused to turn on and off, but strictly speaking this premise isn't necessary as the logic applies regardless of the cause – even if it's magic. — Michael
If the lamp is on then something caused it to turn on, prior to which it was off. If it is turned on then it stays on until something causes it to turn off. — Michael
Given this, if the lamp is on at t1 then either:
a) it was turned and left on prior to t1, or
b) it was turned and left off prior to t1 and then turned on at t1 — Michael
But as Thomson says, "I did not ever turn it on without at once turning it off ... [and] I never turned it off without at once turning it on", and so both (a) and (b) are false. — Michael
Therefore the lamp is not on at t1. Similar reasoning shows that the lamp is not off at t1 either. — Michael
I'm afraid there was a typo in my last post. I posted "Infinity is certainly not a concept", which is rubbish. I meant to post "Infinity is certainly a concept". Apologies. — Ludwig V
Yes, I realized that and was hoping to produce a formulation that would allow a more constructive discussion. — Ludwig V
fishfry will speak for himself. But I think the point is that, even a convergent sequence, which does have a limit, does not have a end or last step defined - indeed, is defined as not having one. That means that any answer whatever is equally valid and invalid. — Ludwig V
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