It is clear that the impossibility of creating a CAD system that can correctly draw square circles places no limits on what computers can do. — PL Olcott
Not logically impossible. I suppose it depends on your definitions, and 'is perfectly round' is a poor definition of a circle. More like 'all points on a two dimensional surface that are equidistant from the center. Given that and 'four equal sides' (and maybe four equal angles to eliminate a rhombus) as a square, it is not logically impossible.It is logically impossible to draw a square circle because it must be perfectly round AND not round at all with four equal length sides. — PL Olcott
The inability to do an impossible thing isn't a limit? What is 'impossible' if not a limit?It is clear that the impossibility of creating a CAD system that can correctly draw square circles places no limits on what computers can do.
Godel showed that the program H cannot solve the task to which it is put. You seem to know this since you reference the halting problem. I was just discussing this very thing in another thread.It is less clear that requiring a program H to report on the behavior of another program D that does the opposite of whatever H says is a logical impossibility when we see that program H1 can correctly say what D will do.
Again, I don't find it logically impossible. All you need is to prevent access of H to the inputs of D. This again illustrates the fact that you've presuming conditions which have not explicitly been stated, same as the square circle.So the when the {halting problem} requires a program H to always say whatever program D will do includes programs that do the opposite of whatever H says this is requiring the logically impossible, thus the same as requiring a CAD system to correctly draw square circles.
Agree, squaring the circle is an exercise in what can and cannot be constructed with a compass and straight edge. But the thing you list as impossible can be done, even without drawing a rhombus, which fits your definition of a square.I am talking about creating a perfectly round thing that
cannot be round because it has four equal length sides,
thus a logically impossible square circle. — PL Olcott
Not logically impossible. — noAxioms
I am talking about drawing a circle that <is> a square thus not a circle. It must be in the same two dimensional plane — PL Olcott
On the surface of the Earth, imagine drawing squares centered on the North Pole with increasing length sides until the sides coincide with the circle of the equator. As ↪noAxioms said ,what can be done is all about unstated presumptions. — magritte
First you've mentioned 'straight sides'. Also first you've mentioned a 2D plane. This comes back to my point, which you predictably have totally missed besides it being emphasized multiple times: State your premises, because the assessment of 'impossible' or not depends on them.Please show how "all points on a two dimensional surface that are equidistant from the center"
and these exact same points form four straight sides of equal length in the same two dementional plane. — PL Olcott
Tip O' the hat to magritte, who actually paid attention to my point, and similarly did not see a restriction to Euclidean space, which is interesting because our universe is not Euclidean, so it's not an obvious assumption to make. One can draw a square circle on the 2D closed plane of the surface of (an idealized) Earth. It would have four equal length straight sides.On the surface of the Earth, — magritte
I think only a mod can change the subject of a topic, so it is safe.changing the subject to the not logically impossible — PL Olcott
I think only a mod can change the subject of a topic, so it is safe.
So yes, just be more careful when selecting your logically impossible thing, because so far none of them has qualified. Then, having found some suitable impossible requirement, in order to make your point in the subject line, you need to come up with a scenario that lists it as a requirement. That was also missing in the OP. So let's say for argument sake that a square circle (suitably defined) is impossible. What argument would plausibly list that as a requirement? — noAxioms
When such a contradiction is met, one ought go back and check one's assumptions. The assumption that must be rejected in your work is that there can be an algorithm that correctly predicts whether any Turing Machine will halt. — Banno
↪PL Olcott It's a reductio. The contradiction you point to is a direct consequence of assuming that the halting problem can be solved. It is what shows that the halting problem cannot be solved. — Banno
When an assumption leads to a contradiction, the assumption must be rejected. — Banno
↪PL Olcott Again, it just seems to me that you have misunderstood the structure of Turing's argument. — Banno
↪PL Olcott I, and most logicians, agree that
requiring the logically impossible is an invalid requirement
— PL Olcott
and yet see the argument as valid. — Banno
You keep saying that. Sure. Turing's argument is not an example of that. It is a reductio. — Banno
The Halting Problem is:
INPUT: A string P and a string I. We will think of P as a program.
OUTPUT: 1, if P halts on I, and 0 if P goes into an infinite loop on I.
Theorem (Turing circa 1940): There is no program to solve the Halting Problem.
Proof: Assume to reach a contradiction that there exists a program Halt(P, I) that solves the halting problem, Halt(P, I) returns True if and only P halts on I. The given this program for the Halting Problem, we could construct the following string/code Z:
Program (String x)
If Halt(x, x) then
Loop Forever
Else Halt.
End.
Consider what happens when the program Z is run with input Z
Case 1: Program Z halts on input Z. Hence, by the correctness of the Halt program, Halt returns true on input Z, Z. Hence, program Z loops forever on input Z. Contradiction.
Case 1: Program Z loops forever on input Z. Hence, by the correctness of the Halt program, Halt returns false on input Z, Z. Hence, program Z halts on input Z. Contradiction.
End Proof. — Prof Kirk Pruhs
Where's the flaw? — Banno
The flaw is that the whole notion of decision problem undecidability
is inherently flawed in that it requires the logically impossible. — PL Olcott
Does this apply generally? Are all supposed reductio arguments so flawed? They all contain a logical impossibility...
This by way of pointing out that your argument is not well-formed. — Banno
Yes this applies generally.
— PL Olcott
To all reductio arguments? — Banno
When it finds a contradiction is derived by a decision problem
then it is this decision problem that must be rejected. — PL Olcott
Yes, in the example argument above, Z is shown to imply a contradiction, and so is to be rejected.When it is contradicted that some H can correctly determine
the halt status of the direct execution of every D, then this
definition of the problem is rejected as incorrect. — PL Olcott
When it finds a contradiction is derived by a decision problem
then it is this decision problem that must be rejected.
— PL Olcott
Why? Isn't that just special pleading? — Banno
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