Many people, often labeled as "infinity cranks," argue that actual infinities are riddled with contradictions. — keystone
That's why I ask if you're a Cantor crank. I just want to know who I'm dealing with. — fishfry
Certain areas of mathematics, like combinatorics, are sufficiently distant from foundational issues and actual infinities. These areas transcend the label of 'classical' mathematics. — keystone
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
I feel like you dislike me or what I represent. I can’t debate classical mathematics to your level of formality, and you don't seem interested in my ideas — keystone
I don't think we'll agree on terms. For example, in another message to you which you ignored I explained that I want to think of the infinite series 9/10 + 9/100 + 9/1000 + ... as a Turing computable algorithm, which can output arbitrarily precise partial sums but never output a 1. I get what you're saying, but in this sense, your function will never output intervals which will union to (0 1). — keystone
I don't believe any Cantor crank shares my perspective, if someone wants to label me a Cantor crank — keystone
this distinction wouldn't interest you since it relates to the ideas I'm proposing. — keystone
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
You clearly have a lot of knowledge — keystone
I believe the main issue is that new topics are added more often than old ones are removed, leading to bloated posts. I'll not respond to a few of your comments to address this request. — keystone
Label the original string (-inf,+inf). — keystone
Cut it somewhere. Label the left partition (-inf,42). Label the right partition (42,+inf). Label the small gap between the strings 42. Now you have a new system: (-inf,42) U 42 U (42,+inf). But you seem to get hung up on those intervals number being continuous even though I'm saying that those intervals describe continua - abstract string in this case. — keystone
Moving forward, instead of writing "computer+mind", I'm just going to write "computer".
I believe that true mathematical rules exist independently of computers. These rules are necessary truths and finite in number. If one assumes they describe actually existing objects, such objects must exist beyond our comprehension, as no computer could contain them. — keystone
However, if we assume that mathematical objects must exist within a computer, then not all mathematical objects can actually exist and it becomes a matter of a computer choosing which objects to actualize. — keystone
Please allow me to use the SB-tree as something concrete to talk around. I acknowledge that any infinite complete tree will do. — keystone
We outline the rules for constructing the SB-tree and can mentally construct it to an arbitrary depth. Everything we ever actually construct is finite. Why insist on believing in the computationally impossible — the existence of the complete SB-tree? — keystone
You've said that the reals correspond to unending paths down the infinite complete binary tree, so indeed, there are potentially
paths that cannot be algorithmically defined. This doesn't mean the rules for constructing the tree are incomplete; it simply means there are paths computers can never traverse. Computers cannot exhaust these rules. — keystone
Or here's how I see it. When I see the tree, I do not see paths and nodes. Instead I see a continua at each row, being cut by the numbers at each row. For example, I see the top two rows of the SB-tree as:
Row 1: 0 U (0,1) U 1 U (1,+inf)
Row 2: 0 U (0,1/2) U 1/2 U (1/2,1) U 1 U (1,2) U 2 U (2,+inf)
...
With this view, I would rephrase the conclusion as follows: computers cannot completely cut continua. Computers cannot exhaust cutting. Actually, I would go one step further and assume that computers are all that's available, so I would simply say that continua cannot be completely cut. But we know that already, you'll never cut a string to the point where it vanishes. — keystone
Sometimes I push back as a form of defense. Nevertheless I'll try and be more mindful of this. I'm very appreciative of our conversation. Thanks! — keystone
Here are quotes from my earlier posts. You don't have to read all bullets as they all say the same thing. I'm just trying to highlight that the confusion is not for lack of me trying.
"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. " — keystone
No Cantor crank would ever have the self-awareness to know that he or she is a crank. — TonesInDeepFreeze
Sounds like your professor just didn't like foundations — fishfry
You might think so from what I said, but he was young and pretty enthusiastic about teaching the subject. We had numerous worksheets that eventually led to the construction of the exponential function. So, his comment at the end came as a bit of a surprise. — jgill
But the specific mathematical statement you made earlier was incorrect. You'd do yourself a favor by recognizing that fact. — 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
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). — keystone
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
You don't have to keep repeating your point; I understand it. — keystone
Since the complete output of the function cannot be generated all at once — keystone
It seems you're trying to point out flaws in my viewpoint by identifying how it differs from yours. — keystone
If you want to challenge my perspective effectively, it would be more impactful to identify actual contradictions or limitations within my own ontology rather than highlighting its differences from yours. — keystone
For example, Cantor's proof that there is no enumeration of the set of real numbers is accepted by constructivism — TonesInDeepFreeze
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
Within the context of my view, we can talk about algorithms designed to construct infinite sets (as in your example) but we cannot talk about the complete output of such algorithms. — keystone
Cantor's proof holds value within the context of my view. — keystone
So be it. — keystone
Constructivism is broader than intuitionism. Intuitionism is one form of constructivism. — TonesInDeepFreeze
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