Thanks for your kind words. Tones has been generous with his time to me in the past so I can't complain, but his tone does sometimes hurt a little. (woe is me :P)

I didn't assume anyone is drunk.

Consider the statement

All bachelors are unmarried men.

That is watertight and contains truth as it’s self referential.

Do you claim that teatotallers are not drunktrollers ?

If so I am ready to destroy the above argument.

Do you claim that teatotallers are not drunktrollers ? — Deus

The word is 'teetotaler'. No teetotaler is drunk, therefore no teetotaler is both drunk and a troller.

Then there will exemptions to the first statement.

There could exist such a place where a teetotaller abstains from drinking on all occasions apart from when he unknowingly drinks alcohol because his wife can’t be trusted.

This then proves that a teetotaller is drunk and it follows that also a troller.

Your logic has been found wanting.

I can certainly write a program to output digits corresponding to 0.89[...]1. It's just that that program can never be executed to completion so it would never reach a moment where it would output a 1 digit. — keystone

Whatever you have in mind, it's not a program. If P is a program to print the entries in a denumerable sequence, then for each entry, there is step at which that entry is printed.

As in other threads, you're using technical sounding verbiage without regard for making sense with it.

He’s referring to the halting problem in relation to turings complete machine

a teetotaller abstains from drinking on all occasions apart from when he unknowingly drinks alcohol because his wife can’t be trusted. — Deus

Depends on the exact interpretation of a given definition. If taken literally in the sense of 'practices complete abstinence' then drinking alcohol even inadvertently makes one no longer a teetotaler at that moment. But I grant that ordinarily, probably most people wouldn't regard that as failure to maintain being a teetotaler; and I overlooked that possible situation. So, I'll give you not a full point for that one, but at least most of a point.

A pint did you say ? I will happily accept a Guiness

Whatever you have in mind, it's not a program. — TonesInDeepFreeze

I'm referring to a computer program. For example,

N=1
print(0.8)
while N+1: # see Comment 1 below
{ print(9)
N+=1
}
print(1)

# Comment 1: This is assuming that once you reach the largest possible number on the computer it returns 0 (effectively going full circle), which breaks the loop and prints a 1.

On an infinite computer it will never print a 1, but I can still write the program.

He’s referring to the halting problem in relation to turings complete machine — Deus

No he's not. The halting problem is not that there are programs that don't halt. But rather that there is no program to decide whether any given program and input will halt.

You don't know what you're talking about. You're just throwing out red herrings. A form of trolling.

It was an inference Tones, albeit an implied one at that. I saw where keystone was getting at.

If I am throwing red herrings it’s because i have surplus fish and you have earned yourself a fish. It’s got omega 3 and good for you.

And most certainly would make you less cranky and grumpy.

it will never print a 1, but I can still write the program — keystone

You can write whatever you like, but my point, as seen in context, is that it's not a program to print all the entries in the sequence.

Consider the ascending sequence of members of w+1 (omega plus one). That is a denumerable sequence with a last entry. But there is no program to write all the entries in that ascending order. On the other hand, it's trivial to have a program write:

1, 1/2, 1/4 ...

while 0 is not an output.

That's just a starker example of what you're doing. Yes, it's a program, and it outputs every successive halving. But 1 is not an output of the program.

You're trying to use "announcing numbers" to stand for two different things : emptying a room into the hallway and shifting occupants to successive rooms. It can't be both.

Under your scheme, announcing 0.9 creates the same problem for a finite hotel as an infinite hotel : any occupant of room 1 is now standing in the hall !

Proof that 0.891 = 0.9 : announcing either 0.891 or 0.9 leaves the infinite hotel in an identical state, namely 0.09 — Real Gone Cat

In the infinite hotel 0.89 = 0.9. Therefore shifting everyone up one room is equivalent to vacating room 1. And since vacating room 1 doesn't create more empty rooms, one should be suspicious about what is achieved in shifting everyone up one room.

In the finite hotel every number has a unique instruction. 0.9 means only one thing: vacate room 1. Infinite decimals are not required for the finite hotel.

The other thing is and something that Turing possibly missed is the use of parallel algorithms…which the halting problem could be solved.

Take the instance a program that aims to output an infinate number such as Pi … it will run and not halt.

However with well defined input paramters the problem of it is easily solved.

In the infinite hotel 0.89 = 0.9. Therefore shifting everyone up one room is equivalent to vacating room 1. And since vacating room 1 doesn't create more empty rooms, one should be suspicious about what is achieved in shifting everyone up one room.

In the finite hotel every number has a unique instruction. 0.9 means only one thing: vacate room 1. Infinite decimals are not required for the finite hotel. — keystone

I was just reading this thread, but it seems you have solved your own conundrum. In the infinite hotel the two are equivalent, as you yourself point out. So 0.9 recurring is equal to 1.

And in the finite hotel they are not equivalent, as you point out. So 0.9... with 9 repeated a finite number of times is not equal to 1.

That's just a starker example of what you're doing. Yes, it's a program, and it outputs every successive halving. But 1 is not an output of the program. — TonesInDeepFreeze

I understand your argument, but I think you didn't read my code (especially the commented line) so you don't realize that it can be run on any finite computer and when doing so it will always eventually output a 1. The larger the computer, the later it will output a 1 but that 1 is unavoidable. Of course, if the code were run on an infinite computer the program would never halt and therefore the 1 never comes.

eventually output a 1 — keystone

Then it will miss outputting one of the 9s.

You can't have cake and eat it too.

If it runs only finitely many steps but outputs the 1, then it skips an infinite number of the 9s.

If it runs without end, then it outputs each of the 9s, but never outputs the 1.

I was just reading this thread, but it seems you have solved your own conundrum. In the infinite hotel the two are equivalent, as you yourself point out. So 0.9 recurring is equal to 1.

And in the finite hotel they are not equivalent, as you point out. So 0.9... with 9 repeated a finite number of times is not equal to 1. — PhilosophyRunner

Right, and my conclusion is that the number system which was developed using finite intuitions breaks down when extended to infinity (the infinite hotel). And I want to suggest that this may be what's happening with math. The number system using numbers which was developed using finite intuitions (rational numbers) breaks down when extended to model the continuum (with real numbers).

You don't like that mathematics for the sciences doesn't comport with your understanding of impossible fictional realms. Yeah, that's a real dagger in the heart of the mathematics for the sciences.

Then it will miss outputting one of the 9s.

You can't have cake and eat it too.

If it runs only finitely many steps but outputs the 1, then it skips an infinite number of the 9s.

If it runs without end, then it outputs each of the 9s, but never outputs the 1. — TonesInDeepFreeze

Think of the counter within my program like an odometer that starts at 1 and has 5 digits. Eventually you will reach the 99,999 after which it increments up to 00000. At that point the program prints a 1.

The program executed on a finite computer will print a 1 and it will print it too early. I agree with you on this. But run it on a bigger computer and it will do a better job, printing the 1 after more 9's. Take it further and run it on increasingly bigger computers and get increasingly better approximations. But can you go the limit and run it to completion on an infinite computer? No. Nor do I believe that an infinite computer exists. But as someone who is fond of limits, I would have expected you to appreciate that the program has value even though we cannot literally go the limit.

You don't like that mathematics for the sciences doesn't comport with your understanding of impossible fictional realms. Yeah, that's a real dagger in the heart of the mathematics for the sciences. — TonesInDeepFreeze

I'm working in Hilbert's fictional realm. Are you saying that it's impossible? Also you're not in a position to critique my understanding of that realm since you haven't even tried to understand my opening argument.

Mathematics for the sciences? Try floating-point numbers. We don't use real numbers.

Printing it early is no trick.

the program has value even though we cannot literally go the limit. — keystone

".89[...]" is notation for a limit. And that limit is .9. And ".9[...]' is also notation for a limit. .9[...] = 1.

I'm working in Hilbert's fictional realm. — keystone

Set theory is abstract. It doesn't have hotels. To be more exact, I should say that from an imaginary analogy to set theory, you impose an incoherent interpretation. It's incoherent because you start out by describing a program to output values (presumably in a certain order) but it's not a program.

I exhausted loads of my time and patience with Thomson's lamp with you. You're making a variation of the same mistake here.

Mathematics for the sciences? — keystone

Yes, from the axioms of set theory, we derive the theorems of calculus.

Ridiculous.

Calculus was developed well before set theory came into the scene. Also in the field of mathematics it’s nothing more than a minor development/distraction

".89[...]" is notation for a limit. And that limit is .9. — TonesInDeepFreeze

Therefore instead of saying 0.891 he should simply say 0.9. — keystone

And that's exactly what the hotel manager concluded in my story. Your criticism of my story was of an inconsequential intermediate step. And even now, you focusing on the program is secondary. You want to argue without listening to my original argument.

Set theory is abstract. It doesn't have hotels. To be more exact, I should say that from an imaginary analogy to set theory, you impose an incoherent interpretation. It's incoherent because you start out by describing a program to output values (presumably in a certain order) but it's not a program. — TonesInDeepFreeze

The program isn't even a part of the story and the repeating term followed by a 1 was just an inconsequential intermediate step that was just an artifact of the decimal system. It's like you're criticizing a written argument by saying 'this i has not been dotted and this t has not been crossed so this paragraph makes absolutely no sense'.

If you see no value in thought experiments then you are without a very important tool.

I exhausted loads of my time and patience with Thomson's lamp with you. You're making a variation of the same mistake here. — TonesInDeepFreeze

It's you who's repeating the past by not listening to my initial argument. Instead you're nitpicking.

Right, and my conclusion is that the number system which was developed using finite intuitions breaks down when extended to infinity (the infinite hotel). And I want to suggest that this may be what's happening with math. The number system using numbers which was developed using finite intuitions (rational numbers) breaks down when extended to model the continuum (with real numbers). — keystone

But that is not what is happening here. Using finite intuition would not lead to thinking 0.9... = 1. So the maths that demonstrates 0.9... = 1 is not using finite intuition.

However it is you who is trying to analyze it using finite intuition, which is the source of confusion I think.

Besides, 0.9... is a rational number, so I don't understand your last sentence in this instance.

Let me explain in detail why your scheme is flawed. I'll only use a minimum amount of math, promise.

The hotel manager uses the technique of "announcing numbers" for two entirely different reasons. Sometimes, announcing a number removes an occupant from a room and does not specify where they go. Other times, announcing a number shifts occupants to successive rooms. But, imporantly, these algorithms are not the same.

But it's worse than that. Each of these algorithms only works for a tiny set of numbers. Consider a hotel with occupants in just the first two rooms (0.99). Only two rational numbers can be announced which shift occupants to new rooms : 0.891 and 0.081. And only three rational numbers remove occupants without shifting : 0.9, 0.09, and 0.99. But the set of rational numbers is infinite, so this cannot be the operation of subtraction defined for the rationals. For example, announcing 0.783 is meaningless, but subtracting 0.783 is perfectly valid.

You have mistaken the manager's actions for subtraction because your technique of announcing numbers just happens to give the same results as subtraction in these 5 cases. And you still had to give different processes for 0.891 and 0.081 on the one hand and 0.9, 0.09, and 0.99 on the other to force the outcomes to match subtraction.

In order for your technique to correspond with subtraction, you would need to describe a single algorithm that could handle all rational inputs. And then show a contradiction.

Crucially, announcing 0.89[...] and announcing 0.9 in the infinite hotel are NOT identical. One shifts occupants, the other removes the occupant in room 1 without specifying their destination. Interestingly, both leave the hotel in the same state : 0.09[...]

Remember, a broken watch tells the correct time twice a day.