I'll use '*' instead of the overbar.

0.89*1

is not defined.

There is no real number that has an infinite decimal expansion but with a final entry.

Your imaginistic scenario, not even itself approaching a mathematical argument, not even of alternative mathematics, is done. Argument by undefined symbolism is a non-starter.

You are typical of cranks who argue with undefined terminology and symbolisms. Using terminology and symbolisms in merely impressionistic ways.

/

continuously growing but always finite — keystone

On your own finitistic terms, at any point, the sequence is finite. 'continuously growing' is never witnessed. Only finitely many individual finite sequences.

I'm stuck at the first step. Why does "0.9" an intelligible announcement that everyone understands? I don't understand it. Do you, Keystone?

Keystone, since you understand that 0.9 means move from one room to another, then please tell us what the following expressions mean:

0.938K
330.five
"integer division by Olaf Gerdmuller"

@TonesInDeepFreeze
@god must be atheist
While I think my notation of putting a digit after the repeating term is an interesting way of potentially representing an infinitesimal, I understand why you might not like it. However, your criticism of it doesn't hurt the overall argument. I've rewritten the story below, but in binary where I don't have to resort to putting anything after the repeating term.

-------------------------------------


Here's how his number system works:

He keeps track of the occupancy state of the entire hotel with a single binary number: the occupancy number. Each digit describes the occupancy state of a different room whereby a 0-digit indicates that the room is vacant, while a 1-digit indicates occupied.

If he wants to vacate room 1, he needs to subtract 0.1 from the occupancy number so he announces 0.1 on the PA system.
If he wants to vacate room 2, he needs to subtract 0.01 from the occupancy number so he announces 0.01.
If only room 1 is occupied and he wants to shift that guest up one room, he needs to subtract 0.1 (to vacate room 1) and add 0.01 (to occupy room 2). In other words he needs to subtract 0.01 so he announces 0.01.
If there are only guests in rooms 1 and 2 and he wants to shift them up one room, he announces 0.011.
In general, if there are N guests in room 1 to N and he wants to shift them up one room, he announces 0.0 followed by N 1's. For example, if N=5, he announces 0.011111.

This system works perfectly for him for every N.

But one day his dream comes true and he gets hired at Hilbert's (actually infinite) Hotel and since his numbers system worked perfectly at the Potentially Infinite Hotel, he decides to use it at the fully occupied Actually infinite hotel.

A new guest enters and as usual the manager wants everyone to shift up one room. Because there are infinite rooms (i.e. N = infinity), he jots 0.01 (where underline is used here to represent repeating) on a piece of paper but before he gets a chance to announce that number the new guest points out the following:

The difference between 0.01 and 0.1 is vanishingly small, so small that you cannot name a number between them so they are equal. 0.01 = 0.1. Therefore instead of saying 0.01 he should simply say 0.1.

And so the hotel manager announces 0.1 on the PA system. As bolded above, the guest from room 1 vacates his room and frustratingly stands out in the hallway as he watches the new guest happily take his room. Unlike the paradox, nothing magical has happened in this story. After all, you can't get something from nothing.

The hotel manager quits his job and returns back to the Potentially Infinite Hotel. At least his number system works there.

I'm stuck at the first step. Why does "0.9" an intelligible announcement that everyone understands? I don't understand it. Do you, Keystone? — god must be atheist

Stick with decimal for my response. If the occupancy number is 0.9909 and he wants to shift the guests in room 1 and 2 up, he needs to perform an operation on the occupancy number to make it 0.0999. The difference between these two numbers is 0.891 so that's the number he would announce. Does this make sense to you?

Your imaginistic scenario, not even itself approaching a mathematical argument, not even of alternative mathematics, is done. Argument by undefined symbolism is a non-starter.

You are typical of cranks who argue with undefined terminology and symbolisms. Using terminology and symbolisms in merely impressionistic ways. — TonesInDeepFreeze

Hopefully the binary version of my story can put your concerns about undefined terminology and symbolism aside.

Have you understood my story and disagree with it or are you criticizing it without understanding it?

On your own finitistic terms, at any point, the sequence is finite. 'continuously growing' is never witnessed. Only finitely many individual finite sequences. — TonesInDeepFreeze

Are you saying that nobody is in a position to say that a process goes without end because they cannot witness the end?

I noted the first place in your original post that you spouted nonsense. Now you've revised. If I'm in the mood, I'll give you a second chance.

my notation of putting a digit after the repeating term is an interesting way of potentially representing an infinitesimal — keystone

That is more nonsense.

Keystone, your effort here is clearly appreciated by Tones. With a little bit of tweaking here and there you might be able to solve this infamous mathematical problem.

Just don’t let guys like Tones put you off as you can see he’s clearly the moody type but generous depending on the lunar cycle.

As for the problem posed by Hilbert himself I think it’s worth noting that his playfulness with the concepts of infinity or infinates is not something new but he has clearly presented mathematicians with a carefully constructed brick wall.

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

guys like Tones — Deus

There's only one.

Then I shall have to clone you an infinite amount of times to satisfy hilberts problem.

Please sign below

Wow, you're way off base. See my reply to keystone, above.

Again,

0.9 - 0.89 = 0.09 (You don't need 1 on the end)

and

0.9 - 0.9 = 0.09

So, 0.89 = 0.9

I am confused. At what point did I offer you any math for such a refutal ?

0.9 - 0.9 = 0.09 — Real Gone Cat

Genius.

Or to remove the decimal 9-9 = 0.9

You have truly distorted and confounded my conception of Math.

Next on the curriculum

1 + 1 = -5 or 3 or anything apart from 2

What "tweaking" of keystone's incorrect approach "solves" the problem?

Also, claiming Hilbert constructed a wall for mathematics is a strange take.

What "tweaking" of keystone's incorrect approach "solves" the problem? — Real Gone Cat

Essentially the use of non-linear equations could be useful.

A strange take indeed but that’s what math is …strange!

I'm using an underscore as the symbol to represent an infinitely repeated digit, as keystone has. Try reading it again.

Essentially the use of non-linear equations could be useful. — Deus

Example, please.

Sure, the simple application of a non-linear equation solves the countable infinity problem that was proved by Cantor and consequently the first argument of Hilbert in his presentation of the above paradox.

So where's the wall?

The one that Cantor build … infinity of infinities.

A redundancy in terms by applying Occams Razor

And I still don't see how you get from keystone's linear approach (numbers are only added or subtracted) to non-linear systems. Seems like a fair amount of "tweaking".

You think the phrase "infinity of infinities" is suspect? How so? Use math please.

I have used philosophy instead of the technical field of math as don’t know how to apply mathematical notations on this site.

Let me attempt anyway

Infinity + Infinity + infinity ♾ = ♾ Infinity

the countable infinity problem that was proved by Cantor — Deus

What specifically do you refer to ?

Let me help : There are an infinite number of rational numbers between 0 and 1. Also between 1 and 2. And so on. An infinity of infinities. And yet Cantor proved that the cardinality of the set of rationals is the same as the cardinality of the integers - they can be placed in a 1-to-1 relationship.

No problem there.

In ordinary set theoretic context, there is no object called 'infinity' that is an operand in an addition operation.

♾ — Deus

Is that supposed to be the leminscate?

Then this is also not new but the old re-formulation of Xenos Paradox