I"m happy you find meaning in this. — fishfry

I'm within epsilon. I no longer have any idea what we are conversing about. — fishfry

Unknown territory for me. No Wikipedia page I can find (among 26,000+), but perhaps it's under a different heading. You are full of surprises. Are you Niqui? :cool: — jgill

If you supply a two-sentence summary I'll read it. In Silicon Valley they call it your "elevator pitch." — fishfry

Oh now I have to converse about proof assistants? You know, if you've been picking up the lingo, that's great. Not of interest to me. It's impressive what they're doing. Just not an interest of mine. — fishfry

No such thing as infinite natural numbers — fishfry

Honestly you sound very crankish about all this. — fishfry

Why not just go learn some math. — fishfry

I'm not the guy for this any longer. — fishfry

Ok as far as it goes, but I have to suspend disbelief. — fishfry

Each row of the tree involves medians, which require ratios of integers and arithmetic of these ratios. So, your top down approach always involves bottom up procedures. You cannot correlate rational numbers with nodes without using expressions like a/b. Instead of simplifying, you are complicating something you assume. Just my opinion. — jgill

You missed the point of my asking you what the notation 1/1 means, in the absence of building up the rationals from the integers, the integers from the naturals, and the naturals from the axioms of set theory. Or even PA if you can do that. — fishfry

As far as the sense of what you're doing, it eludes me. Are you building the constructive real line? Lost on me. — fishfry

eiπ+1=0 — fishfry

I'm curious to know what that notation 1/1 means. In abstract algebra class I learned how to construct the rational numbers as the field of quotients of the integers. That's as bottom-up as you can be. So what is this 1/1 you speak of? — fishfry

If you fixed your notational issues I could quote your markup. — fishfry

Anyway. Yes I understand the difference. No I don't understand what POINT you are making about the difference.

simulation theorists, mind uploading theorists, "mind is computational," etc -- I think all these people are missing something really important about the world. So I think I'm a bit of an anti-construcivist!! — fishfry

Once you get a countably infinite infinity, you immediately get from the powerset axiom an uncountable infinity. — fishfry

But here are only countably many ways to talk about things. Leaving most of the world inexpressible. Now the constructivists are entirely missing that much of the world... — fishfry

You can always find a rational interval small enough to suit the needs of any computation you do, by analogy with always being able to find a suitably large but finite natural number when you need it for a computation.....We call that "finite but unbounded."....Is that a fair understanding of your point? — fishfry

To be actually infinite is far stronger. It's like putting out a good but rational approximation to a real number, versus "printing it all out at once" as it were. Having not just as many digits of pi as you need; but rather all of them at once. — fishfry

That's the magic of the axiom of infinity. — fishfry

Ok. I feel like we're about to go through this same exposition again. At least the notation's less confusing. — fishfry

I wish you wouldn't presume to speak for "a constructivist standpoint". — TonesInDeepFreeze

constructivism in the broadest sense does not disallow construction of infinite sets. — TonesInDeepFreeze

Cantor's proof that there is no enumeration of the set of real numbers is accepted by constructivism. — TonesInDeepFreeze

I don't have interest in going down another path like that with you. But I don't have to do that merely to correct certain misstatements and provide you with explainationss — TonesInDeepFreeze

Read the proof to its end. The union of the range of the function is an infinite union of disjoint intervals and that union is (0 1). — TonesInDeepFreeze

The use of 'can' there is merely colloquial. We may state it plainly: Any set of sentences is a set of axioms. More formally: For all S, if S is a set of sentences, then S is a set of axioms. — TonesInDeepFreeze

In another thread going on right now, it's been pointed out that there are uncountably many mathematical truths, and that most of them can't even be expressed, let alone proven. — fishfry

So what? My mathematical ontology is not confined to what's computational. Yours is. So you should study constructivism. It's pointless to try to discuss it with me... — fishfry

Please use a different notation. The notation (a,b) means something else. But you immediately have problems. What does "between" mean unless you define an order relation? — fishfry

But the specific mathematical statement you made earlier was incorrect. You'd do yourself a favor by recognizing that fact. — TonesInDeepFreeze

What you just said is an utter disconnect. That no finite partial sum is 1 in no way contradicts that (0 1) is an infinite disjoint union of intervals. — TonesInDeepFreeze

Let f be the function whose domain is the set of natural numbers such that:

f(0) = (0 1/2)
for n>0, f(n) = [(2^n - 1 )/2^n (2^n+1 - 1)/2^n+1)

The range of f is an infinite partition of (0 1). That is: the range of f is infinite; every member of the range of f is an interval; the range of f is pairwise disjoint, and the union of the range of f is (0 1). — TonesInDeepFreeze

Anyway, I gave a proof that you are incorrect when you claim that the interval (0 1) is not an infinite union of disjoint intervals, whether or not you want to take a minute to understand the proof. — TonesInDeepFreeze

No Cantor crank would ever have the self-awareness to know that he or she is a crank. — TonesInDeepFreeze

They are included in classical mathematics. — TonesInDeepFreeze

Not true. I published papers when I was active that never assumed infinity was actualized....so, your statement is not entirely correct. — jgill

So after all this time your interval notation does not not stand for its conventional meaning? — fishfry

• "Suppose I introduce a new concept called 'k-interval' to define the set of ANY objects located between an upper and lower boundary. Would you then consider allowing objects other than points into the set?"
• "Yes, the endpoints are rational, and the object between any pair of endpoints is simply a line."
• "Revisiting the analogy above, when I utilize an interval to describe a range, I am referring to the underlying and singular continuous line between the endpoints"
• "Yet, between each tick mark, there exists a bundle of 2ℵ0 points to which we can assign an interval."
• "I propose that we redefine the term interval from describing the points that lie between endpoints to describing the line that lies between endpoints."
• "One thing I need to make clear though is that when I write (0,4) I'm referring to a bundle between point 0 and point 4. I'm not referring to 2ℵ0 points each having a number associated with them. "

And also, if I could make this request ... can you write shorter posts? Short and to the point. — fishfry

I'm fine with cutting strings. You have never explained to me how this serves as a new foundation for math. — fishfry

If you have a set of rules (a bottom up concept) that let you know when two representations denote the same number, then why do you need the computer? Why not just accept that the rules themselves bring all possible numbers into existence already? — fishfry

I believe in the mathematical existence of the abstract objects they describe. — fishfry

There simply aren't enough algorithms to generate all the sets. There are countably many algos and uncountably many subsets of the natural numbers. — fishfry

I'd ask you to accept that rather than continually expressing disappointment with my posts. — fishfry

You said that numbers get instantiated when they appear in a computation. I asked you whether one number or several numbers get instantiated when various representations exist. Who determines that they act the same? Where is that process, that brings a number into existence? — fishfry

Is God watching all this and keeping track of everyone's version of each number? This seems like a cumbersome idea. — fishfry

I believe no such thing, what are you talking about? I believe in the axioms of ZF and not much else. They are purely a human artifact. — fishfry

It's hard to develop a theory of the reals without the axiom of infinity.....Even constructivists have infinite sets, not not noncomputable ones. — fishfry

But there is something to be said for infinitary math. Why shouldn't we enjoy having such a lovely theory of the infinite? What is the harm? — fishfry

You say, Start with a line. Make a cut. I don't know what these things are. — fishfry

You're just approximating the reals. — fishfry

And if it did, the contradictions would be repaired. People wouldn't stop doing infinitary math. — fishfry

I don't follow the analogy you're making. Cantor has underestimated or overestimated his discoveries? — fishfry

You have been making use of open intervals all along, haven't you? — fishfry

The noncomputable reals are telling us something. Infinitary math is telling us something. — fishfry

The history of math is expansive, never contractive. Nobody says, "Those complex numbers, they were a step too far." But they say that about infinitary math. — fishfry

I don't have an eye glaze factor when I'm coding, but I do when I do math. — fishfry

Are you getting frustrated? I'm sorry, I thought I was helping the best I can. — fishfry

I don't know why you are acting as if I'm not attending to what you say. I sense a difference of perspective that I'm not privy too. Everything seems fine at my end. — fishfry

One of the best ways you can respond to someone who brings a problem to you is to just ask them to explain it all to you in detail. — fishfry

There's nothing at stake for me here. I enjoy trying to relate your ideas with things I know in math, but there's never going to come a point where I "digest" this. — fishfry

Do these three computers each instantiate the existence of the same number 42? And how would you know? — fishfry

In finitism (rejecting the axiom of infinity) we can do a fair amount of number theory, but not analysis. You can't do calculus, you can't do physics. You can do finite approximations, but the underlying theory is infinitary. — fishfry

You should research that claim rather than just proclaim it. This is one of the reasons I am never going to "digest" your ideas. Many clever people have given these matters considerable thought. You should do a literature search on this idea to clarify your thinking. — fishfry

Can you see that grandiose claims made without sufficient background come down to untrained feelings and intuitions? Not that there's anything wrong with that. But it supports my belief that there is nothing to digest. — fishfry

I think you have a bad idea, not in the sense that it's absolutely wrong; but in the sense that you have a very naïve understanding of what's involved, so that it seems grandiose. — fishfry

Why? Infinitary set theory is perfectly clear of contradictions. Well, as far as we know. — fishfry

Are you a Cantor crank by any chance? — fishfry

The open sets were your idea. — fishfry

Did I post this Andrej Bauer article, Five Stages Of Accepting Constructive Mathematics? — fishfry

Someone could show me a diagram or list that was 100% correct and brilliant, and my eyes would still glaze. — fishfry

Such books don't axiomatize the principles used. And those books make use of infinite sets. — TonesInDeepFreeze

I think even constructivist and intuitionist set theories have a version of the axiom of infinity. — TonesInDeepFreeze

I'd like to know whether a "no complete, only potential, infinity" concept has been axiomatized in a way that would be to the satisfaction of cranks if they were ever to actually learn about such things. — TonesInDeepFreeze

You give me link to some unidentified video so that I would have to take my time to watch through to find out, or guess, what it is you want me to know about it. — TonesInDeepFreeze

Your line items are helpful to you, and that is the ultimate goal. Technically it doesn't matter whether I ever understand your ideas or not, as long as I am useful as a sounding board. So if you will take the glass half full approach to my not relating to your charts and graphs and lists, then you can feel free to keep posting them and my eyeballs will feel free to be glazed. — fishfry

Check out this guy. — fishfry

constructivism seems more reasonable...But intuitionism...that's just a little out there for my taste. — fishfry

You're not constructing the familiar reals? First time I'm hearing this. Maybe you're constructing the computable reals. Is that what you're doing? — fishfry

I'm pretty sure, but have no specific info about this, that people already decided you can't do analysis, that is calculus and the theory of the reals, without the axiom of infinity. But I could be wrong. I think if you could do analysis without the axiom of infinity that would be impressive. — fishfry

But infinities are one of the most fun and interesting part of math! I always liked infinities. I think I just don't understand the psychology of someone who doesn't like the axiom of infinity.

Tell me, what makes you interested in trying to do math without infinite sets? — fishfry

I would interpret that as your intuition that the open intervals with rational endpoints are a basis for the usual topology on the reals. All the open sets are unions (perhaps infinite) of open intervals with rational endpoints. But then again ... do you allow infinite unions and intersections of sets? Do you want to get rid of infinitary operations as well as infinite sets? — fishfry

Did I miss a post? Or do you mean the long list of definitions and principles that glazed my eyes a bit? — fishfry

Sorry I'm still confused. Did you mean the big list? — fishfry

Let me know when (or if) you have a system with formation rules, axioms and inference rules. — TonesInDeepFreeze

Depends on what you mean by "applied". Ordinary mathematics, even as used for basic applied interests such as speed and acceleration use ordinary calculus, which is premised in infinite sets. — TonesInDeepFreeze

Didn't say that, just got a little overwhelmed by all the line items. I'll take a look at it. — fishfry

But what about my point about constructivism? If you reject the noncomputable reals, you're a constructivist. — fishfry

Suppose I grant that you have some alternative construction of the reals. What of it? All models of the reals are isomorphic to one another. — fishfry

My point exactly. If you have an alternative view of the reals, nothing changes. — fishfry

I don't see what ideas you've challenged. — fishfry

It took weeks for me to understand your bundles, and just when I did, you took them away. — fishfry

"Binary numbers" aka real numbers can never be algorithms, since there are way too many of them. There are uncountably many reals and only countably many algorithms. — fishfry

Sigh. I am not getting much from this latest post. — fishfry

I'm not commenting on your other post — fishfry

If one proposes a mathematics without infinite sets, then that is fine, but the ordinary mathematics for the sciences uses infinite sets, which are not derivable from the rest of the set theoretic axioms, thus requiring an axiom. — TonesInDeepFreeze

For that matter, we don't physically experience breadthlessness, so breadthlessness is itself an idealization, just as infinitude is an idealization. — TonesInDeepFreeze

The notion of length is quite coherent. In context of the reals, length is a property of segments not of points. — TonesInDeepFreeze

Who said anything equivalent with "the continuum constructed in this fashion is paradoxically beautiful and only to be seriously discussed by the experts"? Specific quotes are called for. Otherwise the claim is a flagrant strawman. — TonesInDeepFreeze

It has been proposed in this thread that a sequence converges as n gets arbitrarily large. A sequence is a function. A function has a domain. If the domain is not infinite, then n cannot be arbitrarily large. — TonesInDeepFreeze

One is welcome to work it out in some other way. But then the natural question is: What are your primitives, formation rules, axioms and inference rules? — TonesInDeepFreeze

It was claimed that Russell's paradox is "still there". In what specific post-Fregean systems is it claimed that the contradiction of Russell's paradox occurs? — TonesInDeepFreeze

Back to the poster who claims to offer an alternative to classical mathematics: The word 'isolate' keeps coming up. What is a rigorous mathematical definition of 'isolate'? — TonesInDeepFreeze

It was claimed that the interval (0 1) is not an infinite union of disjoint intervals. It is false that the interval (0 1) is not an infinite union of disjoint intervals.Ostensively: (0 1/2) U [1/2 3/4) U [3/4 7/8) U ... — TonesInDeepFreeze

'Dedekind cut' is not defined in terms of 'gaps', nor 'executions', nor algorithms. 'Dedekind cut' is mathematically defined and we see mathematical proofs that the set of Dedekind cuts is uncountable. — TonesInDeepFreeze

set theorist Joel David Hamkins is investigating Infinite-time Turing machines. — fishfry

You seem to be unhappy that abstract math doesn't climb into the wiring cabinet and start patching cables. Do I have that right? I don't think you don't know what you're talking about. I think I don't know what you're talking about. I'm throwing out guesses. You don't like infinities, ok there's finitism. You want things to be algorithmic, ok there's constructivism. You want there to be a minimal positive real, ok that's computer arithmetic. None of it sticks. Why do you reject the doctrines you espouse? — fishfry

• Computers - Clearly physical computers are limited by our finite observable universe, but I think that we are also in agreement that the abstract computers of mathematics also cannot be infinite. They can be arbitrarily large but are nevertheless finite. I'm interested in abstract computers, not physical computers. 
• Infinities - I think it's impossible to model a continuum using a finite set of indivisibles. However, I'm proposing that we model a continuum using an evolving finite set of divisibles. As we make cuts, the finite set grows, but it remains finite. As we make joins, the finite set shrinks, but never becomes empty.
• Algorithmic - I don't want to use algorithms to construct indivisibles. I want to use algorithms to deconstruct (i.e. cut) divisibles (i.e. continua).
• Minimal positive real - I'm not saying that there exists a minimal positive number. Rather, I'm saying that we can generalize the output of algorithms by using a placeholder. So when I say that we cut (0,1) arbitrarily many times to produce an arbitrarily small number, I'm not saying that we can make infinitely many cuts or that there exists an infinitely small number. Rather, I'm saying that you can pick a positive number as large as you please and divide 1 by it.

Shouldn't the first principles be self-evident?
— keystone

Of course not...non-Euclidean geometry... — fishfry

If I stipulate that every mathematicians that ever lived is a bad person for doing whatever you think they did ... would it help? Can you step back a tiny bit and see that if every smart person who ever lived is a dummy acting from bad faith ... well, maybe it's you, and not them. — fishfry

Abandoning the entire bundle argument? So the real line is no longer made of a countably infinite union of overlapping open intervals, each characterized by a particular computable number it contains? I thought that was a pretty good thing to achieve agreement on. You are abandoning this now? — fishfry

As far as gaps go, they're important. The completeness property, aka the Least Upper Bound property, aka Cauchy-completeness, is the defining characteristic of the real numbers. Accept no substitute! If someone tries to sell you a model of the real numbers, ask them if it's complete! Mathematical shopping advice. — fishfry

Do you know the infinite complete binary tree? — fishfry

Ah, well that's like the active intelligence of intuitionism. — fishfry

Well you are a long way from making that many cuts when you start by denying even countable infinity!! Isn't that a little inconsistent? — fishfry

Very difficult to get a model of the real numbers while denying infinite sets. It's been tried, really. — fishfry

Jeezus. That ain't workin'. Maybe some control characters in there. And don't use ChatGPT, it rots your brain. Lot of foolishness floating about in the culture lately. — fishfry

Ok. I get that you feel enthusiastic about this. I'm on your side. I hope you can work out your ideas. I do think they are a little half-baked at the moment. That's an honest assessment. — fishfry

• When I say that 1/2 = 2/4, I mean that fractions 1/2 and 2/4 correspond to the same node on the SB-tree.
• When I say that 0.99 = 9/10 + 9/100, I mean that 0.99 and 9/10 + 9/100 are both algorithms which output the same fraction when executed.
• When I say that 0.99 ⥱ 99/100, I mean that the algorithm 0.99 outputs the fraction 99/100 when executed.
• With this view, positional number systems (such as decimal) describe algorithms on fractions, not fractions themselves.
• Only basic arithmetic is valid on fractions (specifically roots are excluded).
• Any set of fractions is incomplete (i.e. has gaps) so cannot be used in isolation to construct a line.

• When I previously said that 4 - 4/3 + 4/5 - 4/7 + 4/9 - ... does not halt, I meant that when executed, that algorithm will endlessly compute as it works towards outputting the complete sum all in one go (hence, infinite precision). However, I accept that what Turing would do is reformulate the algorithm to essentially compute a partial sum of the infinite series, output the resulting fraction of arbitrarily fine precision, and halt. I like Turing's approach so moving forward I'll treat an infinite series as this type of algorithm which halts. I acknowledge that this is a departure from classical mathematics which doesn't equate the infinite series with such an algorithm.
• It is meaningless to talk about the unique output of the algorithm 4 - 4/3 + 4/5 - 4/7 + 4/9 - ... because the output depends on the precision requirements.
• Instead of working with the equivalence class of all possible outputs from such an algorithm, I propose that we work with the algorithm itself.
• When I say that 9/10 + 9/100 + 9/1000 + ... !⥱ 1, I mean that that algorithm can never output the fraction 1.
• When I say that 0.9 ≃  9/10 + 9/100 + 9/1000 + ...  ≃ 1+ 0/10 + 0/100 + 0/1000 + ... ≃ 1.0, I mean that all 4 algorithms are asymptotically equal (i.e. they share the same limit).
• When I say that 9/10 + 9/100 + 9/1000 + ... is rational, I mean that that algorithm is asymptotically equal to a positional number algorithm whose fractional digits eventually end with repeating digits (namely, 0.9).
• When I say that 9/10 + 9/100 + 9/1000 + ... is natural, I mean that that algorithm is asymptotically equal to a positional number algorithm whose fractional digits are repeating 0's (namely, 1.0).
• When I say that an irrational has no positional number algorithm, I mean that it cannot be represented as a positional number algorithm whose fractional digits eventually end with repeating digits.
• A computable irrational, such as π, is an algorithm based on more complex arithmetic of fractions, such as 4 - 4/3 + 4/5 - 4/7 + 4/9 - ..., nevertheless being entirely represented with finite characters.
• I propose that non-computable irrationals are inaccessible and unnecessary, and that a real can only be a rational or a computable irrational.
• Following from my proposition, any set of reals is necessarily incomplete (i.e. has gaps) so cannot be used in isolation to construct a line.

• I propose that a line is a fundamental object, it is continuous breadthless length, and it is described by its endpoints. 
• I propose that we redefine the term interval from describing the points that lie between endpoints to describing the line that lies between endpoints. For example, the interval (-5,5) describes the line that resides between points -5 and 5.
• I propose that the only operation we can perform on lines (and continua in general) is to cut (or join) them AND I endeavor to argue that this operation is entirely sufficient. (Moving forward, as a matter of brevity, I won't mention joining).
• I propose that in 1D a cut entails the partitioning of a line. 
• A cut at a point entails cutting the line once, and labelling the midpoint that emerges with any fraction whose value is between the endpoints. For example, cutting (-5,5) by 2 results in (-5,2) U 2 U (2,5).
• A cut at a line entails cutting the line at two points, and labelling the midline that emerges with an open fractional interval. For example, cutting (-5,5) by (1,2) results in (-5,1) U 1 (1,2) U 2 U (2,5).
• To understand how reals fit into this picture, it is beneficial to see them as Cauchy sequences of fractional intervals, and the corresponding algorithm (when executed) outputs a fractional interval arbitrarily deep into the sequence, and halts.
• For example: 0.9 ≃ 1.0 ≃ (9/10,11/10), (99/100,101/100), (999/1000,1001/1000), (9999/10000,10001/10000), ...
• A cut of (-5,5) at 0.9 entails a cut at one of the intervals in the Cauchy sequence, which necessarily is a cut at a line.
• Instead of selecting a specific interval in the Cauchy sequence, the cut of (-5,5) at 0.9 can be generalized as a cut by (1-ε, 1+ε), where we can replace ε with a small positive number and 1-ε and 1+ε are fractions. The resulting system can be generalized as (-5, 1-ε) U 1-ε U (1-ε, 1+ε) U 1+ε U (1+ε,5).
• Any set of points is incomplete (i.e. has gaps) but the points are just one part of a continuum-based system. The system also includes lines, and as a whole the system is continuous.
• I propose that calculus is the mathematics real cuts on continua.

I'll slog on a little longer. — fishfry

It would help if you'll engage with my key point tonight, which is that you've been misunderstanding the nature of halting with respect to computable numbers. Can you see that 1/3 = .333... is computable, because the program "print 3" halts in finitely many steps for an n, giving the n-th decimal digit of 1/3? — fishfry

• I agree that the program you describe halts, however, I'll focus on Turing's version, by agreeing that the program that computes 0.333... to an arbitrarily fine precision halts.
• I agree that by definition that's what it means for a number to be a computable number, so by definition 0.333... is a computable number.
• I agree that by that definition pi is a computable number.
• I agree that no program can compute pi to infinitely fine precision.
• You might even agree with me that no program can compute 0.333... to infinitely fine precision.

After all, you say your top-down view starts with the real line. But I say, I don't know what the real line is. How do you know there is any such thing unless you construct it from first principles? — fishfry

Ok. So far, after all this, what I understand of your idea is that the real line consists of a countably infinite set of overlapping open intervals, each containing a computable number. So far so good? — fishfry

You are delusional. Could it be that you are the one who's confused, and not mathematicians? — fishfry

Cranky. Grandiose claims not backed by anything coherent. — fishfry

What are you doing that, when I quote your numeric examples, the quote text comes out in a column? — fishfry

What? You know, none of this makes any sense. — fishfry

What? There's no difference with respect to algorithms. Consider 1/3 = .333... — fishfry

def fraction_to_base(numerator, denominator, base):
    result = "0."
    remainder = numerator

    while remainder != 0:    
        remainder *= base
        digit, remainder = divmod(remainder, denominator)
        result += str(digit)

    return result

If there is a difference between 1.0 and 1.00000... you are off on your own. I can't hold up my end of this. Nothing you write is correct. — fishfry

Yes. I would like to distinguish between real numbers and real algorithms.
— keystone

Of course, because they are entirely different things, and there are a lot more real numbers than algorithms. — fishfry

You've just made all this terminology up. — fishfry

Then you give me no reason to care. You are not going to "solve QM" with your line of discourse. — fishfry

I think I am nearing the end here. You just are not making any sense...Not feelin' it tonight...I don't see where this is going. I might be doing you a disservice by encouraging you...Can we turn the page? — fishfry

I look forward to a breakthrough in your quest. But I am very old and have multiple medical conditions, so I may not be around. Smooth sailing, fellow explorer. — jgill

For those in the profession who do not deal with transfinitisms and set theory or foundations it's likely they would agree. When I say that a sequence converges to a number as n goes to infinity I simply mean n gets larger without bound. I don't think I have ever spoken of infinity as a number of some sort, although in complex variable theory one does speak of "the point at infinity" in connection with the Riemann sphere. But I am old fashioned. — jgill

Exploration is the soul the subject, but one does not explore the heart of Africa by strolling around city park. Sorry ↪keystone — jgill

You are doing Engineering math. — fishfry

Computable numbers, which have algorithms, or are identified with their algorithms, or are found by executing their algorithms. Not sure which of those you mean but they're all about the same. — fishfry

But each computable number is the number that WOULD be computed if you finished executing the algorithm, but you can't; so each computable number is a number inside a little interval. Have I got that right? — fishfry

• $1.\overline{0}$ is a real algorithm.
• $0.\overline{9}$ is a real algorithm.
• $1.\overline{0}$ - $0.\overline{9}$ = $0.\overline{0}$, which is a real algorithm.
• $1.0$ is a rational number.

That would give you a countable set of open intervals whose union is the real numbers, including the noncomputables. But you'd never have to "identify" a noncomputable. And in fact each of the endpoints (c \pm \frac{1}[n}(c \pm \frac{1}[n} are themselves computable. — fishfry

But they are not in general doing me much good. What if the overabundance of diagrams was increasing the likelihood I'd quit? You can see that under that hypothesis, you are acting against your own interests by battering me with diagrams...just be judicious in how often you include them in posts. — fishfry

So far I get that your system involves little intervals centered at the computable numbers. — fishfry

Are we on the same page here? I really feel that we are. — fishfry

Russell's paradox and QM as well? Please, show me how this is supposed to work. — fishfry

Ok. So as far as I get this: The real numbers are made up of a bunch of open intervals centered at the computable reals. Is that right? And FWIW I think your truncated algorithm idea will give the same reals as my plus/minus 1/n intervals. — fishfry

Given a line segment, points in this object are purely potential, non-existent until a device is used to "isolate" them. Is that about it? If so I doubt any practicing mathematician would be interested. But math philosophers might be. A lot depends upon where you go from here. Just my opinion. — jgill

You don't believe in the real numbers, how can you manipulate them? — fishfry

But if your approximation only needs to be to the minimum distance in a system of computer arithmetic, then you're doing computer arithmetic. — fishfry

You think every real number can be arbitrarily approximated by an algorithm. That's false. — fishfry

What if none of your figures make sense to me? — fishfry

Your latest uses these epsilon quantities, which you've defined as the minimum possible length in a given physical computer. So you are doing computer arithmetic. Not that there's anything wrong with that! But it seems to me that's what you're doing. — fishfry

What do you think is wrong with the current philosophical foundation? And why would a mathematician care? — fishfry

Do you understand that this is the first time that you've told me what you're doing? — fishfry

• "I'm familiar with these methods [of building reals from the empty set]. I believe there is a bottom-up and a top-down interpretation of them. I'm not satisfied with the orthodox bottom-up interpretation of them"
• "Pi is just as important in the top-down view as it is in the bottom-up view. However, as with many other things, it just needs a little reinterpretation to fit into the top-down picture."
• "Considering epsilon's role in calculus, let me just say that with some reinterpretation, calculus can be elegantly integrated into the top-down perspective without the need for infinitesimals."

But the real numbers are categorical. Any two models are isomorphic. So you are not going to be able to produce a "better" model of the real numbers. One representation, construction, or description gives you exactly the same set of real numbers as any other. — fishfry

What about computer arithmetic, fixed and floating point representations, smallest and largest possible values?...So you are doing normal math except within the limits of a finite computational space. If not fixed/floating point, something else. But computer arithmetic regardless. — fishfry

• Step 1: Manipulation of real numbers
• Step 2: Computation based on rational numbers (approximations)

You cannot telescope down to pi on computer-limited representations of numbers. If you mean that your number pi is actually a little interval around pi with approximation bounds given by the limitations of your computer representation, I'm fine with that. — fishfry

Don't see the point though. — fishfry

If you reject the noncomputable reals, what you have is the constructive real numbers, and the calculus based on them is called constructive real analysis. — fishfry

I'm genuinely sorry I can't be of more help. — fishfry