Many people, often labeled as "infinity cranks," argue that actual infinities are riddled with contradictions. — keystone

What makes them cranks is not that they don't accept that there are infinite sets nor that they find the notion of infinite sets nonsense or fatally problematic. What makes them cranks is that their arguments about classical mathematics are ignorant, ill-informed, misrepresentational and astoundingly irrational. Sure, it's great that classical mathematics can be critiqued, but the ignorance, misinformation, misrepresentations and irrationality of the cranks is noxious

That's why I ask if you're a Cantor crank. I just want to know who I'm dealing with. — fishfry

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

Certain areas of mathematics, like combinatorics, are sufficiently distant from foundational issues and actual infinities. These areas transcend the label of 'classical' mathematics. — keystone

They are included in classical mathematics. They may be developed in set theory, without using the axiom of infinity, but they are still included in set theory. Moreover, with mathematical logic we have the formalizations of primitive recursive arithmetic that can be interpreted in set theory, and PA that can be interpreted in set theory, and set theory except with the negation of the axiom of infinity, which is inter-interpretable with PA.

You clearly have a lot of knowledge, and many of your posts, including your recent ones, are informative and well-intentioned. However, sometimes I feel bad after reading your posts. Even though I have a thick skin for criticism (as seen with fryfish’s posts), I still sense that fryfish likes me despite their criticisms. On the other hand, 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, so I’m not sure what we have left to discuss.

Regarding the hyperlink, I admit I was rude. I was upset and wanted to throw some rudeness back at you. I should go back and switch the link to be a Rickroll for good measure.

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

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).

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

I've been very open about my views on Cantor, actual infinities, my informal training, and my motivations. While I don't believe any Cantor crank shares my perspective, if someone wants to label me a Cantor crank, that's their prerogative.

They are included in classical mathematics. — TonesInDeepFreeze

I need to rephrase my statement as I was using 'classical mathematics' in an unorthodox sense. I should have said bottom-up mathematics instead of classical mathematics. Combinatorics transcends the distinction of bottom-up and top-down mathematics. Again, this distinction wouldn't interest you since it relates to the ideas I'm proposing.

Whatever I say will be flavored by my top-down view which you're not interested in. I think I don't have anything for you.

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

It doesn't matter whether I like you. For that matter, I can't have any fair opinion of you as a person aside from this extremely narrow context of posting. I have no reason to doubt that you are a decent and likable person away from posting. On the other hand, yes, I very much dislike your posting modus operandi.

It's not only a matter of discussing to a degree of formality. Rather, it's that you say a lot of things that are incorrect. But, yes, you handwave through just about everything.

As to my interest, I took a whole lot of time a while back to go through all the details of your proposal at that time. My participation was indeed generous. But even as I adapted to your many revisions, it ended up in a dead end where your proposal was still hopelessly vague and reliant on sophistical ambiguities. And even if I don't have the time and interest to engage yet again the broad handwaving scope and details of your musings, it is still eminently reasonable to point out particular clear falsehoods and misunderstandings you post. For example, rather than take a minute to understand my refutation of your false claim about (0 1) and to understand my proof, you fuss that I don't seem to like you.

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 didn't respond to your notion of an algorithm, since it doesn't vitiate that (0 1) is an infinite disjoint union of intervals. Again, an example of your modus operandi. You make a false claim, but then complain that it's not refuted because of some other red herring about some other notion you have.

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. You argue on the basis of your kinda sorta associations about two different things rather than be responsible to make actual logical connections.

I don't believe any Cantor crank shares my perspective, if someone wants to label me a Cantor crank — keystone

For the record, I did not say you're a 'Cantor crank'.

this distinction wouldn't interest you since it relates to the ideas I'm proposing. — keystone

More childishness from you. From the fact that I'm not interested in going over all your stuff all over again, it is not entailed that I rule out being interested in any ideas you might mention. And again your childish penchant for turning blame around. You admit that you misused 'classical' but still manage to blame me anyway.

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

Bringing your original comment back...

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

What I'm trying to convey is that f(0) U f(1) U f(2) U ... doesn't fit into any computer, just as complete infinite series do not fit on a computer. What makes sense computationally is f(0) U f(1) U f(2) U ... f(N) when N can be an arbitrarily large natural number. Just as Turing focused on computations of partial sums when considering computable reals, I want to focus on computations of partial unions when considering infinite unions. Within the constraints of computation, there is no partial union of intervals from f which corresponds to (0,1).

You clearly have a lot of knowledge — keystone

My knowledge of mathematics, logic and philosophy is quite meager. But I do have a good grasp of certain basics and an intent not to misstate them (at least within reasonable informal explanations).

Of course, we understand that computations are finite.

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

When were you in a grad math program? — jgill

'75 -'77.

Sounds like your professor just didn't like foundations.

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

Ok thanks. I realize that I myself write long posts.

Label the original string (-inf,+inf). — keystone

Define your notation, since you already told me this is NOT an open interval in the real numbers.

Did I not complain bitterly enough about this in my last post?

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

I don't know what your notation means and you are not going to tell me.

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

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. See

https://thephilosophyforum.com/discussion/15304/mathematical-truth-is-not-orderly-but-highly-chaotic

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

I don't think anyone believes that, not even the constructivists. Maybe they do. It's pointless to argue constructivism with me, I know nothing about it.

Please allow me to use the SB-tree as something concrete to talk around. I acknowledge that any infinite complete tree will do. — keystone

I don't know why. It's not helpful to me at all.

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

I don't think it's productive for you to try to talk me out of my mathematical beliefs. I believe in the axiom of infinity and the higher transfinite cardinals.

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

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, since I have already made good-faith efforts to get interested in constructivism, without getting interested.

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

Whatever. I can't argue these points. Computers are woefully inadequate to express mathematical truth.

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

You're welcome. I just can't argue constructivism. There use to be some constructivists on this board. Long gone.

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

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?

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

Some do.

I've never seen one. Every one of the hundreds and hundreds of cranks I've seen lacks the self-awareness to understand the ways in which they are a crank.

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. :cool:

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

Foundations are a bit of a backwater. My grad school had an excellent math faculty but no interest in foundations. They had one professor who was an eminent young set theorist. He didn't get tenure even though he was becoming quite famous. He quit math and went into medicine. Quite a loss for set theory.

Looking back I wish to hell I'd worked harder. I was depressed or having some grad school blues, never had very good study habits, didn't develop any. Oh well. Regrets of the past.

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

This is my statement you're referring to: "I can write (0,1) as the union of arbitrarily many disjoint intervals. However, I cannot write (0,1) as the union of infinitely many disjoint intervals."

In your response, you did not explicitly present the union. Instead, you described a function that takes a natural number as input and outputs an interval. At no point did you actually show a union of infinite intervals in its entirety.

I understand that, classically, your function is interpreted as identifying infinite intervals that exist simultaneously. However, I do not accept that perspective. I believe that you need to construct the objects you are discussing.

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

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.

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

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.

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

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.

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?

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).

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

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.

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

From my perspective, we can only discuss objects that can be explicitly constructed. Since the complete output of the function cannot be generated all at once, it is meaningless to talk about the range of the complete output. However, I acknowledge that the standard view assumes that you can discuss the range of the function whereby the range unions to (0 1). You don't have to keep repeating your point; I understand it. I'm just viewing this from a constructivist standpoint, and from my perspective, my statement holds: you cannot explicitly write the union of infinite disjoint intervals.

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

Again, we're approaching this from different ontological perspectives. It seems you're trying to point out flaws in my viewpoint by identifying how it differs from yours. 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.

You don't have to keep repeating your point; I understand it. — keystone

You hadn't said that you understand the point, so the point deserved repeating.

Since the complete output of the function cannot be generated all at once — keystone

But you don't have to keep repeating that point, as it has many times been recognized that the there is no finite listing of an infinite set. Even more simply than that there is no finite listing of the members of the infinite union of intervals, we may observe that even more basically there's no finite listing of the set of natural numbers.

I wish you wouldn't presume to speak for "a constructivist standpoint". You do seem to be along the lines of constructivism (along with a notion of potential infinity) but there's a lot more to constructivism than you know about, so I think it invites error when you speak on behalf of constructivism.

For that matter, it is not a given that my argument is not constructive. I constructed an infinite set in the sense that I used only intuitionistic logic to prove the existence of a particular, named set. If I am not mistaken, constructivism in the broadest sense does not disallow construction of infinite sets. It only requires that an assertion of the existence of a set with a certain property is only allowed when at least one particular, named set is proven to have that property. I claimed that there is a set that has the property of being an infinite disjoint union of intervals such that the union is (0 1). And I proved the existence of a particular, named such set. However, granted, the proof requires the axiom of infinity, which may not be considered constructive, since it merely asserts that there exists a successor inductive set without adducing a particular one. However, the axiom of infinity is equivalent to the claim that there exists a set whose members are all and only the natural numbers, which is a particular, named set.

For example, Cantor's proof that there is no enumeration of the set of real numbers is accepted by constructivism. From any enumeration of a set of real numbers, we construct a real number that is not in the range of that enumeration. Constructive.

It seems that you heard about constructivism and your reaction is "Goody, now I can put my own wonderful alternative framework under the banner of a cool, authoritative school of thought" but without actually understanding what all is involved in constructivism.

It seems you're trying to point out flaws in my viewpoint by identifying how it differs from yours. — keystone

That is incorrect. In the instance about "can", I merely provided you the information that mathematics doesn't need to use "can" but rather can use "is".

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

As I've said about four times already, months ago I spent a quite generous amount of time and energy following up the finest details in your proposal, but that ended up in a bust with your continually shifting equivocations, handwaving and contradictions. Now it seems you're proposing yet another revision. I don't have interest in going down another path like that with you.

Meanwhile, when I do mention certain individual misstatements, you repeat your whining that I am not engaging the full glory of your wonderful alternatives. But I don't have to do that merely to correct certain misstatements and provide you with explainations, even though too often you evidence that you lack the maturity and restraint from self-grandiosity to truly think about the explanations.

And I haven't claimed a particular ontology, so "your ontology" is inapposite.

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

I am getting in halfway into the chat. Is "constructivism" here used in the context as related (not synonymous) to Brouwer's intuitionism, or something else?

Constructivism is broader than intuitionism. Intuitionism is one form of constructivism. I don't opine as to what other poster's notion of constructivism is, except that it would not be correct to claim that arguments such as Cantor's argument that there is no surjection from the naturals to the reals are not constructive. Similarly, not correct to regard my recent argument about a certain union as not constructive.

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

I think it's fair to say that my top-down view likely fits under the constructivism umbrella, but my view does not represent constructivism as a whole. It is a fair request that I represent my view, not constructivism as a whole.

constructivism in the broadest sense does not disallow construction of infinite sets. — 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. Rather, in the spirit of Turing, we can only talk about the partial output of such algorithms, which necessary is a finite set.

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

Cantor's proof holds value within the context of my view.

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

So be it.

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

Classical mathematics itself first formulated that there is no algorithm that prints all the members of an infinite set and halts.

Cantor's proof holds value within the context of my view. — keystone

How nice. My point is that it is constructive.

So be it. — keystone

Let "so be it" be.

Constructivism is broader than intuitionism. Intuitionism is one form of constructivism. — TonesInDeepFreeze

Alright, so it was what I was thinking. Just confirming.