In that thread they state that "any set of sentences can be a set of axioms." I want to distinguish between what is (i.e. actual) and what can be (i.e. potential). It is tempting to actualize everything and declare that there are uncountably many mathematical truths. However, I would argue that these truths are contingent on a computer constructing them. When I speak of finite necessary truths I'm referring to the rules of logic itself. — keystone
I'm trying to establish a view of calculus which is founded on principles that are restricted to computability (i.e. absent of actual infinities). You don't have to abandon your view of actual infinities to entertain a more restricted view. Perhaps we can set aside the more philosophical topics and return to the beef. — keystone
The term 'line' comes loaded with meaning so to start with a clean slate I'll use 'k-line' to refer to objects of continuous breadthless length (in the spirit of Euclid). I'll use <a,b> to denote the k-line between a and b excluding ends and <<a,b>> to denote the k-line between a and b including ends. If b=a, then <<a,b>> corresponds to a degenerate k-line, which I'll call a k-point and often abbreviate <<a,a>> as "a". I'll call the notation <a,b> and <<a,b>> k-intervals. — keystone
The systems always start with a single k-line described by a single k-interval (e.g. <-∞,+∞>). A computer can choose to cut the k-line arbitrarily many time to actualize k-points. For example, after one cut at 42, the new system becomes <-∞,42> U 42 U <42,+∞>.
The order relation comes from the infinite complete trees.
Are we at a place where we can we move forward? — keystone
Ok. I feel like we're about to go through this same exposition again. At least the notation's less confusing. — fishfry
I'll try to do a better job this time. But first, one other area of confusion has been the distinction between infinite and arbitrary as it relates to an algorithm's design vs. it's execution. A Turing algorithm for constructing N, is designed to output a set of m elements, where m can be arbitrarily large. By this I mean that the algorithm itself sets no limit on the size of its output; rather, the size of the output depends on the execution (i.e., the chosen 'precision' based on available resources). Please note that I'm not saying that m is a particular number, nor is it infinity. Instead, when talking about the algorithm itself, m serves as a placeholder for a value that is determined only upon execution of the algorithm. Upon execution, m is replaced with a natural number and the output is a finite set. In a similar vein, when I speak of ε being arbitrarily small, I am using it as a placeholder to describe an algorithm. Upon executing that algorithm, ε is replaced with a positive rational number that is small, but by no means the smallest. Is this clear? — keystone
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
That is a fair understanding of my point but I do want to highlight one thing: it's not always about the computation. If I want to focus on algorithm design (and not execution), I can keep ε's floating around. The ε's only need to be replaced when I execute the algorithm and perform the computation. Fair? — keystone
Yes, actual infinities are beyond computation. — keystone
It does seem to be a bit magical. I'd like to avoid magical thinking if at all possible. — keystone
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
I haven't made a point yet. I just wanted to clarify this as I previously found it a stumbling block in our conversation. — keystone
I'm a computational fluid dynamics analyst, so I naturally approach things from a simulation perspective. — keystone
In the context of a potentially infinite complete tree, to me it makes sense to talk about potentially (countably) infinite nodes and potentially (uncountably) infinite paths. In this sense, the paths have more potential than the nodes. I don't think any beauty is lost in reducing infinity to a potential. — keystone
As QM suggests, something funny happens when we're not observing the world. I consider this to be the magic of potential. Anyway, this is fluffy talk about potential...let me get to the beef. — keystone
I'd like to distinguish between a fraction and a real. The fraction description is finite (e.g. 11 — keystone
which can be represented as an algorithm that generates the Cauchy sequence of fractional intervals: (910,1110),(99100,101100),(9991000,10011000),(999910000,1000110000),… — keystone
(
1
,
+
∞
)
. — keystone
eyes glazed? — keystone
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
Please allow me to respond in the context of the SB-tree. Fractions correspond to nodes. Reals correspond to arbitrarily long paths (well, almost but providing clarifying details would bloat this post). There's no point to introduce natural numbers, integers, or rational numbers as disagreement would ensue. I would say they are all fractions but you would likely say they are all reals. — keystone
The cut at fraction 1/1 is fully captured at row 1 of the tree. — keystone
The algorithm corresponding to the cut at real 1.0 generalizes how the cut would be captured at any arbitrary row beyond row 1 (well, to be precise I should really use ε_left and ε_right instead of just ε). Finally, the execution of the cut at 1.0 happens on a particular row once the computer chooses values for the ε's. What should be clear is that none of this happens at the bottom of the tree. This is an entirely top-down approach. — keystone
I think it's just that Latex does not get used properly in quotes. — keystone
I'm rewriting my last post in plain text and using the notation I recently proposed.
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I'd like to distinguish between a fraction and a real. The fraction description is finite (e.g. 1/1), whereas the real description is infinite (e.g. 1.0, which can be represented as an algorithm that generates the Cauchy sequence of fractional k-intervals: <9/10, 11/10>, <99/100, 101/100>, <999/1000, 1001/1000>, <9999/10000, 10001/10000>, ...
Because fraction descriptions are finite, a cut at a fraction can be planned and executed all in one go. A cut of <-∞, +∞> at the fraction 1 results in: <-∞, 1> ∪ 1 ∪ <1, +∞>.
Because real descriptions are infinite, a cut at a real must be planned and executed separately.
The algorithm to cut <-∞, +∞> at the real 1.0 is generalized as: <-∞, 1-ε> ∪ 1-ε ∪ <1-ε, 1+ε> ∪ 1+ε ∪ <1+ε, +∞> where ε can be an arbitrarily small positive number.
In the spirit of Turing, the execution of the cut of <-∞, +∞> at the real 1.0 could have us replace ε with 1/10 as follows: <-∞, 9/10> ∪ 9/10 ∪ <9/10, 11/10> ∪ 11/10 ∪ <11/10, +∞>
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One thing that I've failed to get across is that I'm not outlining a procedure which will be used to construct infinite numbers. These systems I'm outlining, such as <-∞, 1> ∪ 1 ∪ <1, +∞>, are valid systems in and of themselves. Finite systems such as these are all we can ever construct in the top-down view. — keystone
eiπ+1=0 — fishfry
Please allow me to respond in the context of the SB-tree. Fractions correspond to nodes. — keystone
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
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
EDIT: After posting this I realized that there might be some confusion about Niqui Arithmetic. I have since posted another message entitled NIQUI ARITHMETIC. Please read that first. — keystone
Peano arithmetic can be formalized in Coq. — keystone
Similarly, Niqui arithmetic on the SB tree, which builds on Peano arithmetic, has been proven in Coq. There's an unquestionable structure to natural numbers and fractions that we both agree on. What we disagree on is the ontology related to these necessary truths. You believe that Peano arithmetic applies to infinite natural numbers, — keystone
whereas I believe it applies to arbitrary natural numbers. By this, I mean that Peano arithmetic corresponds to an algorithm designed to take as input any arbitrary pair of natural numbers and output the expected natural number. My ontology does not require the existence of any number. I only need numbers when I want to execute the algorithm, and I only need two numbers at that, not an infinite set. — keystone
\Although the above focuses on Peano arithmetic, the same applies to Niqui arithmetic. While the actual computations of Niqui arithmetic involve the manipulation of symbols or electrical signals, an elusive structure emerges in our mind when studying the algorithm—the SB-tree. Nobody has ever envisioned the complete tree, but we have seen the top part, and when I say 1/1 occupies a particular node, that top part is all I need to see. I don't need to assert the existence of an unseen complete tree; after all, it is merely an illusion that helps us understand the underlying algorithm (Niqui arithmetic). — keystone
I'm trying to establish parallel ontologies: Actual vs. real. At this point, we have actual numbers (fractions) and real numbers. We have actual points (k-points corresponding to fractions) and real points (k-lines corresponding to real numbers). This distincting is rather bland in 1D but it becomes much more consequential in 2D when establishing a foundational framework for geometry and calculus. — keystone
The Philosophy Forum appears to be quirky. I tried quoting this multiple times, sometimes including the spaces surrounding it, sometimes not, and about half the time it puts a column of 1-character lines. — keystone
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
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
Niqui arithmetic: Niqui's method allows you to take as input a symbol and a pair of locations in an unlabelled tree and it returns a corresponding location in the tree. It does not presuppose any mathematics other than Peano arithmetic. — keystone
My interpretation: It just so happens that if you label the nodes of the tree according to Stern-Brocot then those symbols correspond to the familiar operators of arithmetic.
Why this is important: If one can informally say that Peano defined the natural numbers according to discrete ordered positions along a line then that is no different than me saying that Niqui defined the fractions according to discrete ordered positions along a tree. — Ludwig V
I mentioned this only to suggest that my view may not be pre-axiomatic. I think Peano arithmetic is very important and Niqui took that to the next level. I also don't care to converse about proof assistants. — Ludwig V
Sorry, I meant to say "an infinite number of natural numbers" as in "ℵ0
ℵ
0
natural numbers". I can see how this was misleading because when I later wrote "arbitrary natural numbers" I was referring to placeholders that can be populated by any natural number you come up with. — Ludwig V
I'm just looking at things from the perspective of a computer. A computer doesn't access infinite sets, it always works with the finite set of finite inputs provided to it - so why not only assert the existence of those inputs (and whatever abstract objects it actually manipulates to deliver an output) and see how far this restricted math can go? — Ludwig V
What I'm proposing is not entirely philosophical. — Ludwig V
I think we got stuck in the weeds because I began to justify how fractions can exist in my view but that justification doesn't interest you. — Ludwig V
I think temporary suspension of disbelief is probably the best path forward so that we can jump to the good stuff before you decide to quit...or have you already decided... — Ludwig V
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
Did you try to make sense of my 'elevator pitch'? I wasn't communicating nonsense. It wasn't even my work I was talking about... — keystone
Epsilon, eh? Will a few figures push you over the edge? That's where I need to go to move this forward... — keystone
Does the axiom of identity mean Ludwig V = keystone ?
Just curious. :smile: — jgill
Does the axiom of identity mean Ludwig V = keystone ?
— jgill
LOL I don't think so but I see what you mean. — fishfry
If you click on the link to the quotations in your message, you will find yourself here:-I'm within epsilon. I no longer have any idea what we are conversing about. — fishfry
That is my message. It is on the "Infinite Staircase" thread, and does not include any of the passages attributed to me in your quotations. So I have no idea who wrote them.I thought that might be your answer. Perhaps we shouldn't pursue the jokes, though. — Ludwig V
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