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 hold more.

In both cases, the occupant of room 1 must leave room 1. In the case of 0.01[...], they then move into room 2 (of course bumping everybody up), and in the case of 0.1, they just go home. Either way, after the announcement, the hotel now has 1 empty room and an infinity of occupied rooms. — Real Gone Cat

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 vacant room because someone is standing in the hallway. Not being able to distinguish between these two cases means that the hotel manager's number system is broken. No matter what he announces he has no idea whether his instructions are being followed as he intended. How does your comment resolve the issue?

You are unnecessarily confusing yourself. 0.9... IS a rational number. It is not that I think it is, rather it is.

Any number that infinitely repeats a finite sequence after the decimal point is a rational number. 0.9... repeats the finite sequence "9" infinitely, so is a rational number.

Irrational numbers, like Pi or the roots, don't have finite sequences repeating infinitely. — PhilosophyRunner

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 conventional idea. What I'm saying is that this claim rests on the notion of limits. Without limits, I don't think you can even prove that 0.9-0.9=0? And I'm arguing that limits describe unending journey's down the Stern-Brocot tree (which can be used as the framework for arithmetic), not destinations on the Stern-Brocot tree.

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. — Real Gone Cat

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 are a lot of announcing numbers that don't have meaning for him (e.g. 0.783). I started with decimal notation because of convenience/familiarity but have faced much criticism for the artifacts it introduced such as the idea of 0.001 being a number. Soon after my initial post I reframed it in binary to address these issues. In binary, every number between 0 and 1 has meaning in the hotel manager's number system. And just as we can show that there's a bijection between (0,1) and (-inf, inf), we can develop a system whereby every real number has meaning in the hotel manager's number system. Roughly speaking, here are the steps:

1) You provide any real number
2) I convert it to binary
3) Using the bijection, I find the correspond number within the range (0,1)
4) That number has a meaning in the hotel manager's system

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. — PhilosophyRunner

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 rooms. In math, we take our intuitions developed from numbers with finite digits and apply it to numbers with infinite digits.

Maybe this will help communicate my view: The fact that in decimal notation 1/3 has infinite digits is a red herring. 1/3 on the Stern-Brocot tree is represented as LL because we go Left-Left from 1/1. Every positive rational number can be represented on this tree using finite characters/digits. 1 is [] since it's the starting point. If we look at the numbers of LR, LRR, LRRR, LRRR, LRRRR, etc. we see that we're listing numbers that approach 1 and if we keep going we have LR. This corresponds to 0.9. IF we were to go the the limit and hit the bottom of this tree then LR=[] (i.e. 0.9=1), but there is no bottom of the tree, so they are not equal. LR is not a number, it describes an endless journey/process down the tree. Similarly, 0.9 is not a number, it represents an endless operation [0.9+0.09+0.009+...].
All the irrational numbers would appear at the bottom of this tree...but once again, there is no bottom of the tree. So when I think of pi, I think of an endless journey down this tree that begins with steps RRRLL. Rationals are destinations on the tree. Irrationals are journeys. I know LR and LR look similar but they're fundamentally different.

The reason why you think 0.9 is rational is because you believe it equals 1, which is indeed a rational number.

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.

".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.

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.

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.

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).

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.

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.

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.

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)

If I'm in the mood, I'll give you a second chance. — TonesInDeepFreeze

Fair nuff.

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

That is more nonsense. — TonesInDeepFreeze

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 going to defend infinitesimals nor am I interested in debating the validity of non-standard analysis. But I can certainly write a program to output digits corresponding to 0.891. 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.

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?

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?

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?

@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.

Yet you went on to ignorantly argue about it! — TonesInDeepFreeze

Fair criticism

Absolutely I do not agree.

'=' stands for equality. Period.

So, now I have to give you a free lesson from the first chapter of Calculus 1.

The infinite sum here is:

Let f(0) = 1. Let f(k+1) = f(k)/2

df. SUM[k e N] f(k) = the lim of f(k) [k e N]

thm. the lim of f(k) [k e N] = 2 — TonesInDeepFreeze

I have developed a paradox which (I think) shows the problem of using limits to define equality. Let's shelve this discussion until then. If you're up for it, I'll hold off on submitting a new discussion on that paradox until you're back as I'm sure you'll have an interesting opinion. Of course, I'm not expecting for any commitment from you whatsoever. It's a free world...this is just a nod to you, noting that I'd benefit in you seeing my post....

What in the world? Your comment about Peano systems is ludicrously ignorant. So I corrected you. There's no "looping back" by me. — TonesInDeepFreeze

I'm not in a position to argue that Peano systems are inconsistent so I'd like to set this aside for now. — keystone

It's amazing to me that cranks are FULL of criticisms to mathematics but they know nothing about it! — TonesInDeepFreeze

Look at the history, I've tried to end the discussion on this point but you keep looping it back it!

From the axioms, we prove that there is a unique x such that x has no members:

E!xAy ~y e x

Then we define :

0 = x <-> Ay ~ y e x.

And, informally, we nickname that "the empty set". — TonesInDeepFreeze

I'm going to let you have the last word on this point for now since I'm trying to keep the discussion focused on Thomson's lamp and continuous objects vs. points.

WRONG. Typical claim of someone who knows not even the first week of Calculus 1.

An infinite summation is a LIMIT, not a final term in the sequence, as the sequence has no final term. — TonesInDeepFreeze

So do you agree that the = in an infinite summation means something different than the = in a finite summation?

In that context, I don't mean 'system' in the sense of axioms and theory. I mean it in the sense of a tuple of a carrier set with a distinguished object and an operation, like an algebra. In that sense, 'consistency' or 'inconsistency' do not even apply. — TonesInDeepFreeze

I can accept it as an algorithm for generating the set, but not as a completed set....but we've been here before....

There is only one set that has no members. That it is called a 'set' is extraneous to the formal theory. The formal theory doesn't even need to mention the word 'set'. We could just as well say "the object that has no members". — TonesInDeepFreeze

It seems that this fundamental particle of set theory needs to be defined then.

'continuous' is defined in mathematics. I don't know what you mean by it. — TonesInDeepFreeze

How about instead of continuous object, I say topological manifold?

There is only an incoherent description of something that can't even be a fictitious or abstract model of anything, because it can't be the case that there is a final state that is a successor state where, for each state, there is a successor state.

Especially a finitist would see that immediately. For a finitist there is no such realm, and for an infinitist too. — TonesInDeepFreeze

It's almost like you're saying that constructing the whole from the parts provides an incoherent description. I do see that immediately.

Of course, one may adopt a thesis that mathematics should only mention what can happen with a computer (call it 'thesis C'). Then, go ahead and tell us your preferred rigrorous systemization for mathematics for the sciences that still abides by thesis C.

And one can reject thesis C. And there is a rigorous systemization of mathematics for the science that does not abide by thesis C.

I got on an airplane that flied well, getting me from proverbial point A to point B. Show me your better airplane. — TonesInDeepFreeze

Every formal theory begins with an intuition. I don't have a formal theory. I also don't think you want to discuss the intuition further since it's not formal...but I believe it is a better airplane. Based on all of the infinity paradoxes, I can't help but think that the current "rigorous systemization of mathematics for the science that does not abide by thesis C" is inconsistent. I cannot prove that formally, but I can discuss the infinity paradoxes.

But now matter how we define the set of natural numbers, starting element, the successor operation and the starting element, as long as it is a Peano system*, then we get distinct natural numbers. — TonesInDeepFreeze

I'm not in a position to argue that Peano systems are inconsistent so I'd like to set this aside for now.

The description is not coherent, since it posits that there is a last state for a process that does not have a last state. — TonesInDeepFreeze

I agree, but as I just mentioned, by this logic we also cannot say that 1+1/2+1/4+1/8+... = 2 since this implies that there must be a last term to the summation. In other words, if we just look at the first two columns of the table...

Step #, cumulative time
0, 1
1, 1+1/2
2, 1+1/2+1/4
3, 1+1/2+1/4+1/8
etc.

...time never reaches 2 minutes.

Set theory does provide a mathematical version of infinitely many steps. But not with a last step that is the successor to the previous step. — TonesInDeepFreeze

Does the geometric series have the last step as a successor to the previous step?

It is a fail to claim that Thomson's lamp impugns set theory. Indeed, if Thomson's lamp imgugns anything, it's the supertask that is described. Just as set theory does not assert that there exists such a supertask. — TonesInDeepFreeze

It seems like the clock that counts to 2 minutes is performing a supertask. Can we say that set theory cannot describe clocks?

Benacerraf (1962) pointed out [that the] description of the Thomson lamp only actually specifies what the lamp is doing at each finite stage before 2 minutes. It says nothing about what happens at 2 minutes, especially given the lack of a converging limit. — TonesInDeepFreeze

It sounds like by a similar reasoning we cannot say that 1+1/2+1/4+1/8+... = 2.

Removing the axiom of infinity from ZFC leaves a system inadequate for analysis. That does not imply that there can't be another system without the axiom of infinity that is adequate for analysis, just that that other system will not be ZFC\I (ZFC but without the axiom of infinity). — TonesInDeepFreeze

Ok, I get what you're saying now.

Then you replied that if set theory were inconsistent then set theory has that infinite sets are empty. And above you quoted me yourself instructing you that if set theory is inconsistent then still "infinite sets are empty" is inconsistent. — TonesInDeepFreeze

Okay, maybe I need to refine my statement. How about, if ZFC is inconsistent then you can prove that infinite sets are empty and you can prove that infinite sets are infinite?

Non responsive. You say there is no continuum, but in the imaginary world you describe, you have a ruler that you say is the continuum. Have cake or eat it. Choose one. — TonesInDeepFreeze

If I stop using the word continuum and instead say that a ruler is a continuous object does that sit better with you?

I did not debate the definition of 'set'. — TonesInDeepFreeze

I'm referring to your discussion with MetaphysicsUndercover where he says that a set by definition cannot be the empty set. This I do not want to debate here.

That you are "disturbed" doesn't change the fact that in set theory, distinctness of natural numbers doesn't require consideration of a continuum. You are just plain flat out wrong. — TonesInDeepFreeze

This conversation is going in so many directions I'm willing to set this point aside for now. Suffice it to say that I see nested sets of sets containing no objects in a similar way as I see geometry constructed from points - they're both creating something from nothing. I believe I'd be rehashing the same sort of arguments that I've already provided which you surely don't want to hear again, nor will you be convinced by it. After all, I feel much more comfortable talking about geometry than set theory so let's please drop this for now and I will acknowledge that as far as we know today, in set theory distinctness of natural numbers doesn't require consideration of a continuum.

We can add whatever math you want to my writeup...And still my point about the writeup stands. We have an infinite sequence. — TonesInDeepFreeze

For the table involving cumulative time, there is no row corresponding to a cumulative time of exactly 2 min. Does that mean that the table (and your set theoretic description) only describes the state of the lamp as time approaches 2 min?

Thomson's lamp is not a description of physical events. And it's not even model abstract set theory. Thomson's lamp does not show that set theory is inconsistent nor that set theory fails to provide mathematics for the sciences. — TonesInDeepFreeze

I know it's not physically possible due to the physical limitations related to flicking a switch but I think we can set that detail aside. In this fictitious realm, can set theory can be used to describe the thought experiment of Thomson's Lamp all the way to 2 min? If Set Theory can't allow the cumulative time to reach 2 min, it seems that Set Theory fails to provide mathematics for the 'sciences' of this fictitious realm.

where we need to assume that the real line is composed of infinite points — keystone

You asked me about finitely many points, not about potentially infinitely many points. Be clear. — TonesInDeepFreeze

I'm not sure if this is what you're referring to but I don't mention finitely many points. As you know, I don't think the real line is composed of points. However, I can see the confusion resulting from my loose use of terminology. When I say real line or continuum you interpret that as the real number line because that's how it used. So that's my bad. Anyway, I'm talking about a continuous object, for example a line, a string, a surface, etc. These continuous objects can be used to do all the math that we typically do (e.g. graphing curves) but they're not composed of points.

Let N = the set of natural numbers.

Let f be a function.

Let dom(f) = N

Let for all n in dom(f), f(n) = 1/(2^n)

So f(0) = 1, f(1) = 1/2, f(2) = 1/4 ...

0 is not in ran(f).

Let g be a function.

Let dom(g) = ran(f)

Let ran(g) = {"off", "on"}

Let for all r in dom(g), g(r) = "off" iff En(r = f(n) & n is even)

So g(1) = "off", g(1/2) = "on", g(1/4) = "off" ... — TonesInDeepFreeze

Okay, I see. I forgot the details of the Thompson's Lamp paradox. f(n) corresponds to the incremental time of light switching, not the incremental distance Achilles travelled. To me that is a moot point, but let's hold off on distance for the moment and focus on your complaint that I'm trying to fit set theory into my finitist perspective.

Do you agree that your formal definition describes the informal notion that there exists a complete table (having no last term) as described below?

Step #[n], incremental time [f], current state of lamp [g]
0, 1, on
1, 1/2, off
2, 1/4, on
3, 1/8, off
etc.

Also, if you look at the Wikipedia page (https://en.wikipedia.org/wiki/Thomson%27s_lamp) you will see a table which is more closely aligned with the paradox:

Step #, cumulative time, current state of lamp
0, 1, on
1, 1+1/2, off
2, 1+1/2+1/4, on
3, 1+1/2+1/4+1/8, off
etc.

Do you think that the incremental time table and the cumulative time table convey the same information, just in a different format?

ZFC uses a method of definitions such that no contradictions can be introduced through definitions. ZFC could be inconsistent, but not because of any definitions. And if ZFC were inconsistent then still so would be the sentence "infinite sets are empty". — TonesInDeepFreeze

In ZFC, is the equation 1+1=2 a definition, a theorem, or something else? My understanding is that if ZFC were inconsistent then one could prove both that the natural numbers are distinct and that they are equal.

So you dispute the continuum by posting a continuum. I take it that you consider that you need the stick to put the marks on. — TonesInDeepFreeze

I dispute a continuum composed of points. I take it that you consider the points to equal the continuum.

I don't think distinction between numbers (e.g. 1 and 2) can be made without accounting for the continuum that lies between them.
— keystone

Wrong. Look up the math sometime. — TonesInDeepFreeze

I know the standard construction, starting with natural numbers then integers then rationals then reals, etc. And often we say that the naturals are defined as nested sets of sets. I am disturbed by this approach but I know in another thread you are already debating the definition of a set so let's leave it at that.

Removing the axiom of infinity from ZFC leaves a system inadequate for analysis. That does not imply that there can't be another system without the axiom of infinity that is adequate for analysis, just that that other system will not be ZFC\I (ZFC but without the axiom of infinity). — TonesInDeepFreeze

Then maybe ZFC is inadequate for analysis. Once this discussion has concluded I'm going to start a new post with my argument supporting this view. Perhaps we can discuss this further at that time.

Distance is between points. That doesn't make points "nothingness". It doesn't make them nothing, let alone nothingness. — TonesInDeepFreeze

I see points as emergent from distance...but we've been here before...

I could easily switch roles with you, to play devil's advocate for, say, some given finitistic point of view critical of set theory. I could play that role. You couldn't do the same in reverse. — TonesInDeepFreeze

Care to try?

It’s not “my” theory. And likewise, your feelings are irrelevant to the argument it makes. So whatever. :cool: — apokrisis

I do appreciate the discussion we've had though. Thanks!!!

If I write N = {1, 2, 3, ...} it seems that N has infinite elements. But appearances can be decieving. If someone proved that 1=2=3=... then N actually only contains one element.
— keystone

You answered the question. You're not serious. — TonesInDeepFreeze

Of course, I'm not saying that the natural numbers are actually equal. I'm saying that natural numbers as defined in an inconsistent system can be easily proven to be equal. IF ZFC were inconsistent and IF someone proved that to be the case then wouldn't this be the celebrated conclusion? But IF that were the case, that doesn't mean that math is wrong, it just means that ZFC is not a good foundation for mathematics.

I bet it was yet another way for you to say that you like the idea of potential infinity. No, I haven't responded to you at all on that. I mean, the dozens and dozens of my posts now on display in at least two threads don't exist. — TonesInDeepFreeze

It's funny how you criticize me if I don't respond to some of your points but then you criticize me if you don't respond to some of my points.

And you egregiously obfuscate the terminology. 1/8 increments is not a continuum. You could at least give the consideration of not appropriating terminology in a blatantly incorrect way. — TonesInDeepFreeze

I'm saying that the wooden stick upon which tic marks are placed is the continuum.

The existence of the set of real numbers doesn't stop you from considering only a finite number of numbers for a given problem. — TonesInDeepFreeze

I don't think distinction between numbers (e.g. 1 and 2) can be made without accounting for the continuum that lies between them. The 1/8 tics on the ruler have purpose because in between them lies a space.

Smart people build a well engineered airplane but before they launch it they go through a non-scientific ritual of blessing the plane to ensure it will fly.
— keystone

What on earth are you talking about? — TonesInDeepFreeze

I'm referring to the objects of set theory being beyond our grasp.

I've answered that and answered it and answered it already. The answer is that the ordinary axiomatization of the mathematics for the sciences has an axiom that implies that there exist infinite sets. If we remove that axiom from the rest of the axioms, then we don't get analysis. Period. Final answer, Regis. Got it? — TonesInDeepFreeze

I get what you're saying, but I don't agree with it.

You have a framework. You don't have a hint of an idea how to make it rigorous, but that doesn't disallow that nevertheless it might suggest an intuitive motivation toward a rigorous treatment. On the other hand, other people don't share your framework and have different intuitions, and have made rigorous mathematics. It is poor thinking on this subject then to keep trying to put a different framework within your own. I've been saying this over many many posts. Do you see? — TonesInDeepFreeze

Sometimes there's not enough room for two conflicting ideas.

Points are not "nothingness". — TonesInDeepFreeze

It occupies zero space.

You're adding things into what I wrote that are not there. — TonesInDeepFreeze

I'm confused why you embedded a geometric series into the definition.

----------

@TonesInDeepFreeze: Earlier today I was seeing that you responded to my posts and I was looking forward to our continued conversation on this thread but based on your responses it's clear that this conversation has run its course. As you've mentioned multiple times, we're going in circles. We both think the other is not listening or being reasonable. In any case, I do appreciate that you gave me a lot of your time at composing well written responses. I sincerely thank you.

↪keystone

Some of your quote links are not going to the posts in which the quotes occur. — TonesInDeepFreeze

Oooh...I've been quoting wrong. Moving forward I'll do it the right way.

So the ontology is fundamentally complex. And hence not widely understood by folk....But good luck applying this kind of advanced systems logic to the simplicities of number theory. — apokrisis

"Behind it all is surely an idea so simple, so beautiful, that when we grasp it - in a decade, a century, or a millennium - we will all say to each other, how could it have been otherwise? How could we have been so stupid?"
John Archibald Wheeler

I'm just not feeling your theory. I don't see the point.

My focus is systems science. Which means all of the above really. But neuroscience in particular. — apokrisis

Very cool.

But if you just want to resist the concept of vagueness, that’s your lookout. I can only say it was about the single most paradigm shifting thing I ever learnt. — apokrisis

I'll keep it in my back pocket in case it will be useful to me in the future :D

How do you glue actually 0D points together to make a continuous line? How do you glue 0D width lines together to make a plane? — apokrisis

In my view 0D points are not objects so you don't glue them together. You don't make a curve from points. You don't make a surface from curves. You don't build the whole from the parts. In my view you go the other way around. You start with the whole and cut it up. Consider how we draw a line. We don't use pointillism until a line emerges. We draw a line and then put some tic marks on it to identify points. I don't see what part of my explanation you're not getting.

The issue to be resolved is how divisibility can co-exist with the continuum that it divides. — apokrisis

What exactly do you mean by this?