What you have done is to display the contradiction that we all agree on. It's what you conclude from that which is problematic.

Assume to reach a contradiction that there exists a program Halt(P, I) that solves the halting problem, — Prof Kirk Pruhs

:smile:

I wouldn't be here apart from trying to help articulate the point.

That's why I've asked you to show as explicitly as you can where Carol's question occurs.

In the other thread I suggested that the analogue would be "Will Program Z loop forever if fed itself as input?" — Banno

Repetition and appeal to (supposed) authority.

:ok:

General relativity is about gravity and acceleration. Special relativity starts with a thought experiment that shows that in a void, with one object stationary and one object moving at a constant speed, there's no fact of the matter about which one is actually stationary and which one is moving. — frank

On The Electrodynamics Of Moving Bodies? He talks about empty space. No mention of void in this English translation.

I'm not saying you are mistaken, just that it seems an odd translation/interpretation.

There's no void. — frank

There was before Newton.

Philosophy as physics without the maths.

Oh. I wonder if you had general relativity in mind.

I supose one might argue that the void disappeared when Newton introduced action at a distance. After that the void was never empty.

Survival only ever takes place within a context. What is fit, is what fits into it's environment. Cooperation, not competition, is paramount.

Not survival of the fittest, so much as survival of what fits.

Einstein used it — frank

How?

Curious, but a shame it's so obscure. Nothing as a subtype of all types.

,

Austin, especially in Other Minds, addresses "real".

But is it a real one? When you ask if it is real, what are you sugesting? No, it's a fake; it's an illusion; it's a forgery; it's a phoney, a counterfeit, a mirage... What is real and what isn't is decided in each case by contrast; there is no single criteria. — Banno

Perhaps we can deal with "nothign" in a similar way, by asking what it is that there is nothing of...

He has nothing in the bank; I've nothing in my pocket; there is nothing in a vacuum.

What there is nothing of is decided by what is absent.

Absolute nothing is a non-starter.

@sime?

Well, no. The correct thing to do is conclude that H is impossible; that there are things which cannot be computed.

...and the trouble with that is that there doesn't seem to be any obvious problem with Z.

So the problem must be H.

And, again, where does such a question occur in the example?

Seems to me to be in Z; the contradiction that is used to deny the assumption.

But I don't know what you think here.

In the other thread I suggested that the analogue would be "Will Program Z loop forever if fed itself as input?"

"So that you can ask silly questions."

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?

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

Yes, in the example argument above, Z is shown to imply a contradiction, and so is to be rejected.

That's the reason for rejecting the assumption.

I dunno. This all appears pretty basic. It still looks to ma as if you have not followed the structure of the argument.

I'm re-thinking it. Is it like triangles with four sides, or is it like $\sqrt{-1}$?

Yes this applies generally. — PL Olcott

To all reductio arguments?

Is it like a square circle? A triangle with four sides? A mere concatenation of words that can't be given form, cannot be constructed or worked with?

SO there is no identifiable flaw in the argument, yet it is wrong as a whole?

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.

Sure, all that. But even to say that is to presume that some sentences are true - " the hearer can all to easily interpret a sentence as saying something quite different from what the speaker meant", perhaps.

And yet there is a pop episteme which claims that there are no truths. Of course, no one here would say anything of the sort - we are all too sophisticated for that!

Here it is again:

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

I've bolded the assumption for you. It leads directly to the contradiction I've italicised.

Where's the flaw?

An agument cannot possibly be valid if it contains a fatal flaw. — PL Olcott

You keep saying that. Sure. Turing's argument is not an example of that. It is a reductio.

Reductio ad absurdum.

At least acknowledge that.

No, I don't have things in themselves in mind here. — Manuel

Oh, good.

So we agree that at least some of what science says is true. Turned out nice again, didn't it?

I, and most logicians, agree that

requiring the logically impossible is an invalid requirement — PL Olcott

and yet see the argument as valid.

Do you agree that the argument is a reductio? If not, what structure does it have?

any branch of knowledge is truth as it is revealed to us?

Are we talking about oysters again? You can't taste oysters without using your tongue, and so you can never taste oysters as they taste in themselves?

If not, then what?

...truth as it reveals itself to us... — Manuel

What's that, then?

Again, it just seems to me that you have misunderstood the structure of Turing's argument.

When an assumption leads to a contradiction, the assumption must be rejected.

• Assume that there is a solution to the halting problem.
• Show that this leads to a contradiction.
• Conclude that there is no solution to the halting problem.

Nothingness is inconceivable by definition. — Tom Storm

But that's not right, since here's a thread about nothingness.

Something is going on here, to do with nothingness. The folk posting here have something in mind, when they talk about nothingness.

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.

Truth and knowledge are very large concepts... — Vera Mont

So you can't have the very large stuff but you can have the small stuff? Then don't worry about the very large stuff.

Some sentences are true. Any epistemology that denies this is... fraught with contradiction.

There's little honour left in these fora.

Yes, you can't have a plane euclidian shape that is both a square and a circle. And yes, this is because these words do not describe or correspond to anything.

The sides of 's supposed square do not meet at right angles. Nor is it a plane euclidean shape.

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.

This thread is a continuation of Self Referential Undecidability Construed as Incorrect Questions.

Sometimes we think we know things that are not true. We can't know something that is not true. When we think we know things that are not true, we are mistaken.

But further, not all facts are empirical. Fallibilism isn't used in arithmetic - 1+1 isn't 2 until proven otherwise. Nor does moving the bishop back to the box show that the rules of chess are false.

SO facts are true. Well, there's that on which we might agree. and seem to know things that are not true. Tim is unhappy with small truths, wanting all or nothing.

Scientists, like everyone else, do make use of notions of truth. That dropped objects accelerate at around about 9.8 m/s², that plants need light to photosynthesis, such things are true.

Some methodologies turn into a hagiography of truth, only to find the mystery too great and reject their own god. But truth is a small thing. It's just what statements do.

It would be a puzzle if science were to "organise facts" that were not true. And abduction is a criminal offence.

I think you have some interesting stuff here, but you haven't demonstrated an error in Gödel or Turing.

it's a pretty cool result that leads on to Gödel's incompleteness and Turing's halting problem.