keystone

Going to higher dimensions helps explain my position and would more clearly demonstrate why 'more apparatus' improves my position. But yes, it would t...

Okay, fine. It's significantly different. I suppose I'm coming from the applied side of mathematics and I don't see how my view changes anything in my...

I suppose you would like me to paraphrase so you can judge my comprehension. Fair enough. Earlier on in the discussion, I incorrectly claimed that the...

@"TonesInDeepFreeze" To continue to clean up some of the language: A point in the context of my view is a k-point A line in the context of my view is ...

I appreciate you continuing this discussion. I'm getting plenty of value out of this dialogue. I don't think your infinite loop programming example of...

Actually, the algorithm used to perform arithmetic on the Stern-Brocot tree was originally written in Haskell (I rewrote it in Python) and it does jus...

@"TonesInDeepFreeze" You raise some good criticisms. I have reformulated my proposal to address them. The object that I want to talk about moving forw...

def endless_loop() while True: print("Looping indefinitely...") Return 1 I do plan to respond to all of your comments (it may take some time for me to...

So that I can respond it will be helpful if you clarify your position on programs by answering these questions I asked in my earlier message to you.

@"TonesInDeepFreeze" I just realized I incorrectly described the black lines in the S-B tree image. They do not correspond to paths from 0/1 to all th...

Consider the follow Python function: def endless_loop() while True: print("Looping indefinitely...") Return 1 Do you consider this a valid function? I...

@"TonesInDeepFreeze" As always, I appreciate your detailed responses. I believe in what I'm saying but IF it's untrue then you're right - according to...

I want to characterize all paths on the S-B tree by their destination. Infinite paths do not actually have destinations but neither does 1/x have a va...

Thanks for providing a detailed and thoughtful response. Let me re-share this image of the S-B tree (https://imgur.com/vWBO6U9) because it shows how I...

Fair enough. The author of the paper proved the algorithm to work specifically for the rationals (not reals). I imagine the proof is much more difficu...

All I said was that the computation when the inputs were infinite strings wouldn't complete. The SB algorithm can certainly accept infinite strings as...

Aside from never being able to complete the computation involving infinite strings, there is a scenario where the algorithm may appear to hang. This i...

It's clear you're providing useful comments but you've mentioned multiple topics/terms which I'm not familiar with. I've got some homework to do! Than...

Accepted. I suppose I changed the topic a little, hoping to show how arithmetic with real decimals (which troublesomely starts from the right of the s...

That's certainly a beautiful definition of the golden ratio.

@"Count Timothy von Icarus" I appreciate your responses but Tones captured my concerns in a more eloquent way than I could have put it so I'll refrain...

This is a chat forum, not a journal. We should be allowed to spitball here.

If you're talking about a number like sqrt(2) then I agree. However, for indescribable real numbers, I imagine that it would be less work to exhibit a...

I suppose I should use is instead of describe. "Other common definitions of real numbers include equivalence classes of Cauchy sequences (of rational ...

Are you suggesting that there's no way to completely describe an irrational number? Why does the infinite series have to remain rational? Oh...so woul...

Pi and the golden ratio are special in that sense. You can't do that with almost all other irrational numbers (unless you say that one of the line seg...

How do you finitely and completely describe these mathematical entities (irrational numbers)?

Do you think a Cauchy sequence of positive rationals can be used to describe the Golden Ratio? If so, whats the fundamental difference here? On the St...

Wikipedia says that the decimal representation of the Golden Ratio is 1.618033988749894... https://en.wikipedia.org/wiki/Golden_ratio Do you disagree ...

Hilbert's Hotel is a paradox because, in spite of being completely full, it can continue to make room for more guests without dislodging any of the cu...

But the whole point of Hilbert's hotel is that it can take in more guests. If it kicks guests out as it takes new guests in it's not actually able to ...

That's a big difference. In one case a vacant room is magically created and everyone is happy while in the other case there is no magic, the room is v...

I understand the claim that any number that infinitely repeats a finite sequence after the decimal point is a rational number. I know it's a basic and...

Firstly, thank you. It is clear that you understand how the hotel manager's number system works and I really appreciate that! You're right that there ...

It's about taking finite intuitions to the limit. In the hotel story, he takes his system which works with finite rooms and applies it to infinite roo...

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 artif...

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,...

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 real...

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 increme...

Right, and my conclusion is that the number system which was developed using finite intuitions breaks down when extended to infinity (the infinite hot...

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 ...

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 m...

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 assu...

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. (w...

Fair nuff. I agree that as a number it's nonsense...but I believe so are infinitesimals. This is the whole essence of my argument after all. I'm not g...

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

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

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 oper...

@"TonesInDeepFreeze" @"god must be atheist" While I think my notation of putting a digit after the repeating term is an interesting way of potentially...

Fair criticism