Condensed:

1. sime's claim of what Brouwer meant is the OPPOSITE of the basic notion of potential infinity, and sime has shown no source that supports his claim.

2. sime is plainly incorrect that Cantor's notion of absolute infinity is the same as sime's, and sime's argument about ZFC in this context is incorrect since ZFC dramatically DIFFERS with Cantor on the matter.

If a student asked you to explain "what is a non-terminating process?" what would your reply be, and how would you avoid running into circularity? — sime

Look in virtually any introductory textbook or set of lecture notes on computability. And we don't need a notion of 'potential infinity' to explain the notion of non-termination. The classical treatment of computability is replete with the notion of non-halting. For example, it is a well known simple fact that a program to list the natural numbers does not halt.

Moreover, it is the responsibility of proponents of the notion of 'potential infinity' to provide the needed definitions to support the notion; not the responsibility of people who don't rely on the notion. The fact that you are mixed up about this subject shouldn't entail that you try to patch that up by supplying incorrect and incoherent claims and attribute them to Brouwer and the intuitionists. You continue to say that Brouwer and intuitionists understand potentially infinite sequences to be "eventually finitely bounded". Yet, after multiple requests, you fail to provide a source where Brouwer or anyone said that.

I cannot think of any way of explaining what is meant by a non-terminating process, other than to refer to it as a finite sequence whose length is unknown. — sime

Classically, it can be explained as not one finite sequence but as a sequence of finite sequences. Or, non-classically, to avoid having an infinite sequence of finite sequences, as anyone can read in virtually any article about 'potential infinity', even at the most basic level: For any finite sequence, there is a finite sequence of greater length, but there is not an upper bound to the lengths of such finite sequences. That is the OPPOSITE of saying that there is a finite bound on the lengths.

"non-terminating" processes do in fact eventually terminate/pause/stop/don't continue/etc. — sime

One can posit that no physical process continues without termination. But, as I've asked you again and again, please cite where Brouwer said that a potentially infinite sequence is "eventually finitely bounded". More generally, if you have no Brouwer source to point to, then you should not conflate your claim that an ideal process is not realized physically so a non-terminating process is "eventually finitely bounded" with Brouwer who, as far as we know, never advocated that a potentially infinite sequence is "eventually finitely bounded", especially as the notion of potential infinity is the OPPOSITE.

Where do you find "eventually finitely bounded" in Brouwer, or even any secondary source, on potential infinity? Please cite a specific passage.
— TonesInDeepFreeze

It is a logically equivalent interpretation of Brouwer's unfinishable choice sequences generated by a creating subject — sime

You use terminology in such a sloppy yet grandiose way:

(1) Logical equivalence is a special notion. You haven't shown any "logical equivalence".

(2) By mere fiat you declare a logical equivalence. Still, you do not cite anything Brouwer said that even suggests (let alone is "logically equivalent") to "eventually finitely bounded", especially as it is the OPPOSITE of the notion of potential infinity.

Martin[-]Lof's "The Mathematics of Infinity" — sime

I will look at it. I am not well versed in type theory and category theory.

Classical mathematics and Set theory conflate the notions of absolute with potential infinity, — sime

ZFC could not possibly conflate the notions since set theory doesn't even have a notion of 'potential infinity' nor does set theory mention 'absolute infinity'.

Again, 'absolute infinity' is a notion of Cantor that is not used in ZFC. You persist to use 'absolute infinity' in your own personal sense (for you, 'absolute infinity' is the notion that an intensional definition can specify a set that cannot be finitely listed), which is very different from Cantor's use of the term.

And even with your sense, though ZFC does define sets that are not finite, that is not "conflating" with some other notion ('potential infinity') that ZFC does not even address.

in computer science, where a rigorous concept of potential infinity becomes needed, and where ZFC is discarded as junk. — sime

A rigorous definition would be good for those who use the concept. But ZFC does not use it. And ZFC does provide a rigorous axiomatization for classical computer theory. If you want a rigorous non-classical computer theory, then it's your job to make it rigorous; your lack of doing that is not a fault of ZFC.

And I guess there are some people in computer science who regard ZFC as junk, but that is not at all any kind of consensus or, as far as I know, even a large contingent. You don't legitimately get to speak on behalf of "computer science".

Cantor does mean what i mean in so far that his position is embodied by the axioms of Zermelo set theory: — sime

Wow. You are so wrong, and so obviously so. A combination perhaps of ignorance and intellectual dishonesty.

The very purpose of Z is to not include axioms that would allow an absolute infinity. Why don't you get a book on introductory set theory and inform yourself on this subject?

The Law of Excluded Middle is not only invalid, but false with respect to intuitionism. — sime

Yes, LEM is not a logical validity in intuitionism. That doesn't contribute to claiming that Cantor's sense of 'absolute infinity' is the same as your own personal sense of it. (Also, there is a technical question about what 'false' in a semantic sense means for intuitionism. In finite domains, LEM is TRUE in intuitionism. Of course, we would have to look at specific intuitionistic set theories to see whether LEM is false in any given model.)

The axiom of regularity (added in ZFC) prevents the formulation of unfinishable sets required for potential infinity. — sime

I have no idea what you have in mind with the notion of the axiom of regularity "preventing unfinishable sets". ZFC has no predicate "unfinishable sets", so the axiom of regularity couldn't allow nor "prevent" anything about. It is incorrect to say that ZFC makes determinations on notions that are not even expressed in ZFC.

And the axiom of regularity doesn't contribute to a claim that what Cantor meant by 'absolute infinity' is what you mean by it.

For example, in {1,2,3 ... } where "..." refers to lazy evaluation, there is nothing wrong with substituting {1,2,3 ...} indefinitely for "..." — sime

I don't know what that is supposed to mean. But, to be clear "..." is not in the language of set theory, not even as extended by definition, but rather it is informal notation that can be eliminated with actually rigorous notation.

no honest mathematician knows what is being asserted [with the axiom of choice] beyond fiat syntax when confronted with an unbounded quantifier. — sime

The axiom of choice is intuitive. One is free to reject it, but you are incorrect to say that mathematicians don't know what the axiom asserts.

And the axiom of regularity doesn't contribute to a claim that what Cantor meant by 'absolute infinity' is what you mean by it.

The Axiom of Extensionality : According to absolute infinity, two functions with the same domain that agree on 'every' point in the domain must be the same function. — sime

No, according to what YOU mean by 'absolute infinity'; it's not what Cantor meant.

And the axiom of extensionality doesn't contribute to a claim that what Cantor meant by 'absolute infinity' is what you mean by it.

Even more basically, even though ZFC does capture a great deal of Cantorian set theory (but not including Cantor's 'absolute infinity'), that does not entail that what Cantor meant by 'absolute infinity' is what you mean by it.

So I guess you mean type theory couched in category theory. I don't know enough about that to evaluate your claims about it, but there are some general points to mention.

First, what textbooks or lecture notes form the basis of your notions of category theory and type theory, especially explaining their relationship? I would like to be able to see a systematic treatment so that I can see the full context rather than a standalone, unsupervised article in a pot luck encyclopedia.

Category theory can also be couched as a non-conservative extension of ZFC (viz. ZFC+Grothendiek_universe). So it makes no sense to denounce ZFC as you do while using a theory that presupposes ZFC. And various of your comments (in another thread) about ZFC show that you don't understand ZFC, nor intuitionism, so I have little reason to believe that you understand the even more complicated subjects of category theory and type theory.

In other threads, and now in this one, you use 'inf' as if it stands for something. But now you switch to 'N' for the set of natural numbers. 'N' is not problematic, since it does refer to a particular object, viz. the set of natural numbers. But I don't know of any texts that refer to infinity itself as a mathematical object (other than 'infinity' as a point on the extended real line and things like that, and the notation 'to inf', which resolves without 'inf' as a noun). If you would cite such texts then I might be able to look them up.

Simply:

Where do you find "eventually finitely bounded" in Brower, or even any secondary source, on potential infinity? Please cite a specific passage.

Again, I'm not asking about your own notions; I'm asking where you got your idea as to what Brower said about it.

And you might hold off snidely telling me to "reread the article(s)" when you still don't cite specific passages in an article and the article you mentioned does not at all support your claim.

It would be nice if you addressed my points directly aside from (or at least in addition to) whatever other meandering about various subjects you have in mind.



Instead of defining 'absolute infinity' you give various notes on your ideas about it.

'absolute infinity' is a noun. For mathematics, the definition of a noun requires the definiendum (a noun phrase for defining a noun) on the left and a definiens (a noun phrase for defining a noun) on the right.

So a definition of 'absolute infinity' would be of this form:

absolute infinity = [insert a mathematical noun phrase here]

/

in mathematics absolute infinity corresponds to interpreting the intensional description of a total function or algorithm as being synonymous with an exact limitless extension — sime

There you still don't provide a definition, but you take 'absolute infinity' to refer to a certain kind of "interpretation". Here are the problems for you:

(1) You say, "in mathematics". The ordinary mathematical literature does not use 'absolute infinity' in the sense you do. As I've pointed out to you several times, 'absolute infinity' was a notion that Cantor had but is virtually unused since axiomatic set theory. And Cantor does not mean what you mean. So "in mathematics" should be written by you instead as "In my own personal view of mathematics, and using my own terminology, not related to standard terminology". Otherwise you set up a confusion between the known sense of 'absolute infinity' and your own personal sense of it.

Accepting that there is a common notion of an intensional definition as opposed to an extensional definition, we still see that 'an extensional definition' just boils down to the definiens being a finite list. But a set is infinite iff it has no finite listing. So there is nothing gained from saying 'absolutely infinite' rather than merely saying 'infinite'. The definitions:

x is finite iff x is 1-1 with a natural number*

x is infinite iff x is not finite

Then your sense of 'absolutely infinite' is just 'infinite'.

Absolute infinity refers to a semantic interpretation of a mathematical, logical, or linguistically described entity — sime

Then syntactic infinitistic mathematics is not itself liable.

Absolute infinity arises when an analytic sentence is mistaken for an empirical proposition. — sime

What example do you offer of a mathematician or philosopher of mathematics making that mistake?

total function — sime

What total function do you offer as an example? 'total' is better thought of as a 2-place relation. A function is total on a set iff the domain of the function is that set. So what exactly do you have in mind regarding totality? And, for example, as to the the infinitude of the set of natural numbers, what total function do you think is improperly employed?

/

Again, the article says nothing that is tantamount to describing potentially infinite sets or sequences as "finite entities of a priori indefinite size" or as "finite entities" or having "length [that] is eventually finitely bounded".
— TonesInDeepFreeze

It is practically equivalent to the common definition of potential infinity as being a "non-terminating" sequence that is never finished but occasionally observed after random time intervals. — sime

No it is not "practically equivalent". You ADDED "finite", "a priori indefinite", "finite entities" and especially "eventually finitely bounded". Those are not mentioned, neither literally or practically, in the article's explanation of the notion of potential infinity (except 'finite' is mentioned regarding the finitude of each natural number). Generally, the article does not say that potentially infinite sequences are finite sequences. Even for an intuitionist, a finite sequence is not a potentially infinite sequence, notwithstanding that only a finite portion of a potentially infinite sequence is constructed at any given point. To really stress the point, especially, the article says nothing like "eventually finitely bounded".

The important thing regarding the common definition of potential infinity is that in order to obtain a value or extension, the process constructing the sequence must be "paused" after a finite amount of time. — sime

Please cite a source where that is mentioned as part of the "common definition".

when dx is declared to be infinitely small or x to be infinitely large. — sime

You said that your claims about the notion of potential infinity are supported by the article about intuitionism. Now you're jumping to non-standard analysis. You would do better to learn one thing at a time. You alrady have too many serious misunderstandings of classical mathematics and of intuitionism that you need to fix before flitting off from them.

/

* That is circular with my definition of 'is a natural number' in another thread, but that can be overcome by using a different definition of 'is a natural number' such as 'an ordinal that is well ordered both by membership and the inverse of membership').

the inductive definition of the natural numbers in type theory.

1. [...] g(0).
2. [...] g(s+1) = improve g(s)
3. Define a [...] fixed-point g(inf) = improve g(inf) — sime

I've never seen that in type theory or elsewhere. it seems to make no sense. Please say where you have ever seen that as type theory?

* There's no mention of anything type theoretic there.

* It's not a definition (inductive or otherwise) of 'is a natural number' nor a definition of 'the set of natural numbers'.

* 'g(0)' needs to be followed by '= T' for some term T.

* 'inf' in a context such as this does not refer to a mathematical object. Multiple times in other threads, I told about your misunderstanding related to this.

/

This does make sense (transfiinte recursion), though it's not a definition of 'is a natural number' or 'the set of natural numbers):

g(0) = c
g(s+1) = g(s)
g(k) = F(g restricted to k) for F a class operation and k a limit ordinal

For a definition of 'is a natural numbers' we don't need induction or recursion. We only need:

n is a natural number iff n is a finite ordinal

For a definition of the set of natural numbers' in set theory we have:

D is inductive iff (0 is in D and for all x, if x is in D then xu{x} is in D)

the set of natural numbers = the least inductive set

https://www.templeton.org/news/is-there-a-god-shaped-hole-at-the-heart-of-mathematics? — Photios

That article egregiously misstates Godel.

"Godel was famous for proving mathematically that all mathematical systems are incomplete"

No, Godel proved that some systems (or a certain kind) are incomplete. It is well known that there other systems that are complete.

"Godel also believed that the universe would be incomplete, and inconsistent."

Where is Godel supposed to have said said? It doesn't even make sense. What are complete and consistent, or not, are formal systems, of which "the universe" is not one. Also, If a system is inconsistent then it is complete. That is well known to any first semester student in mathematical logic.

"Godel’s ontological proof uses mathematical logic to show that the existence of God is a necessary truth."

The proof uses not just ordinary methods in mathematical logic but also modal logic with certain assumptions.

"Godel’s own incompleteness theorems proving the limited and unprovable nature of all mathematical endeavors."

Endeavors are not provable or not. Rather, formulas are provable or not relative to a system. And there is no formula such that for all systems the formula is not provable.

"1. finitely specified, 2. large enough to include arithmetic and 3. consistent, then it is incomplete."

No, It should be: 1. recursively axiomatizable, 2. expresses enough arithmetic to formalize the Godel sentence (or sometimes said as "expresses a certain amount of arithmetic" or "is an extension of Robinson arithmetic", et. al), 3. is omega-consistent (later generalized to 'consistent' by Rosser).

We await Metaphysician Undercover's rigorously presented new mathematics that realizes what he considers "the correct way to divide space and time according to physical reality" and avoids the many problems he imagines in current mathematics.

ππ isn't exact if you use rational numbers — Agent Smith

Again, what is the definition of 'exact'?

Pi is an irrational number. That doesn't make it not exact.

with 1/3 you poses all the information even if you can't write 3s forever — TiredThinker

With pi you have all the information you need to write its digits for as long as you want to write them.

Pi however can't be fully known. — TiredThinker

What is the definition of "fully known"?

Pi is fully known to be a unique real number with an exact definition.

It must be more of a concept than a certain thing? — TiredThinker

Every mathematical object is an abstract concept and not a physical object.

I'll let him know you're available. And cheap - just two bits a digit and two digits max.

The poster has deleted any form of the puzzle from his original post

There are no passages in the article that support your misrepresentations. If you think there are passages that support you, then you can cite them specifically.

Again, the article says nothing that is tantamount to describing potentially infinite sets or sequences as "finite entities of a priori indefinite size" or as "finite entities" or having "length [that] is eventually finitely bounded".

You fabricated those.

And what was your point arguing about "absolute infinity", which is a notion that has virtually been out of play since axiomatic set theory? For that matter, as I've already asked, what is your definition of "absolute infinity"?

some ambiguity — Philosophim

He said there are no tricks. I don't know whether ambiguity should be taken as a trick.

the middle number is 2, and the first and last must be the same character — Philosophim

The puzzle doesn't stipulate that "how many" must be answered with a numeral.

It's like if someone asked, "what is 1+1?". The answers, "it's an even number" or "it' >1" are true but not the actual answer, "2". — DavidJohnson

If someone answered it is "1+1" (everything is itself) or "4-2" or "sqrt(4)" then those would all be correct answers. But not correct if the question is what single numeral represents 1+1.

But with this puzzle, it's open ended.

How is it not misleading to talk at all about URLs now when URLs have nothing to do with it?

If URLs were relevant and not merely incidental, then you would have mentioned them the first time. I understand that you are now quoting verbatim just in case there is any shading of the wording that might be relevant, but surely it has nothing to do with URLs, right?

Anyway, I'd like to know at least in what passage of your old version or new version is the clue you mentioned.

That's not what I'm saying. — DavidJohnson

That's exactly what you said.

ZFC is of course inadmissible for the purposes of this discussion — sime

For potential infinity. Though, in ZFC, we can describe sequences of finite sets.

the I,S and T axioms, divorced from the problematic axioms of ZFC — sime

Problematic for some people, not for others. And I am pretty sure that if you take ZFC out from under IST you're left with a theory that accomplishes quite little.

appear to be a relevant fragment of some formalization of potentially infinite logic. — sime

Has anyone tried?

Linear Logic — sime

It looks very interesting. But you couldn't seriously propose it as a way for college freshmen to learn calculus.

Your whole line of argument has sputtered. As well as you still have not addressed that you terribly misrepresented Brouwer. And I still don't know why in the world you go on about "absolute infinity", which is a notion pretty much entirely unused since Cantor was superseded by axiomatic set theory.



.

intensional definition of pi — sime

The definition specifies a certain definite object. — TonesInDeepFreeze

Set theory is extensional. A set is determined solely by what are its members (what its extension is). But you are using 'extensional' in the sense of there being a finite listing. Well, in that sense no infinite set has an extensional definition. The very definition of 'is infinite' is 'does not have a finite listing'. There's nothing special about pi in that way. Even a rational number (which is an equivalence class of integer pairs) does not have a finite extension.

It is only exact in an ideal sense. — emancipate

Like mathematics generally.

I find 3.14 is sufficient for my practical purposes. — Metaphysician Undercover

Well that settles it then. Next time Elon needs some calculations to land a craft, he should just call you for your results rounded to two decimal points.

So you contend that previous answers, though correct, do not suffice because they have variations, while the desired answer has no variations, and that there is a "clue" that has been missed.

In what line of your edited version is the clue?

He would be tarred and feathered if he left that out though it's relevant. He wouldn't have.

It can't have anything to do with URLs, because the original post, before it was edited, described the puzzle without mentioning URLs and supposedly solvable by "pure logic alone"..

It's a good thing that whoever determined the solution to this puzzle is not among the referees of the P v NP contest. Someone would give a correct solution and the referees would say, "Sorry no million dollars, not even a cigar, it's not what we had in mind."

ufour — pfirefry

The third character is 'o' not 'u'.

1

What is not correct here?:

noneplussevenn

I wrote

n
oneplusseven
n

first character = n
third character = n
different characters = eight = one plus seven

noneplussevenn

I wrote

n
oneplusseven
n

first character = n
third character = n
different characters = eight = one plus seven

Hamlet is a character in a famous play.

Hamlet2Hamlet

or

Hamlet7Hamlet

2

The first and third characters are the blank character.

000

None of the characters are different from one another, so there are 0 "different" characters.

You simply skipped the point that mathematical definitions are never circular or tautologies. — TonesInDeepFreeze

You skipped it again.

That article does not describe potentially infinite sets or sequences in a manner that is properly paraphrased as "finite entities of a priori indefinite size" or as "finite entities".

Moreover, most remarkably, in another thread:

https://thephilosophyforum.com/discussion/comment/653236

you say, "whose length is eventually finitely bounded". Again, there is nothing in the SEP article that states anything like that. — TonesInDeepFreeze

You skipped that point twice now. Maybe you'll address it later, but as it stands, you fail to substantiate your claim as to what Brouwer or any intuitionist or any constructivist or any mathematician or philosopher ever said.

obscuring matters even further, e.g the hyperreals . — sime

Since hyperreals are formalizable in set theory, they are formalizable in internal set theory too.

You mean without absolutely infinite sets, presumably. — sime

Why in the world would I couch anything in terms of "absolutely infinite"? The notion of "absolute infinity" does not occur in classical mathematics since Cantor was superseded by Z set theory, as I had already alluded to:

Who specifically do you have in mind as mentioning "absolute infinity" since Cantor's work was superseded by formal set theory? — TonesInDeepFreeze

And you skipped that question. Would you please answer it?

You keep talking about "absolute infinity" but in absence of any reference to it in mathematics after Cantor, and in absence even of your own definition of it.

approximating an ideal, abstract and causally inert mathematical derivative. — sime

Mathematical objects are in general abstract. And knowing what you mean by "causally inert" awaits knowing your definition.

[a potential infinitesimal] is the reciprocal of a potentially infinite number — sime

A philosophical view, such that we are already familiar with, on the notion of 'potential infinity' is of course welcomed. And I have little doubt that it has been formalized somewhere. But you again skipped my point that it is only by seeing a particular formulization that we can compare its ease of use with classical mathematics. From the (admittedly not deep) reading I have done, generally alternatives to classical mathematics are much more complicated to formulate.

Moreover, you mention internal set theory, but internal set theory is an extension of ZFC, so every theorem of ZFC is a theorem of internal set theory. So, if one rejects any theorems of ZFC then one rejects internal set theory. Moreover (as I understand), non-standard analysis and internal set theory make use of infinite sets, and though (as I have read) there is an intuitive motivation of 'potentiality' in internal set theory, I do not find 'potentially infinite' defined in those papers or articles I have perused. Let alone that you are now combining non-standard analysis with potential infinity without reference to where that is in any mathematical treatment (possibly there is one, but you have not pointed to one).

Then, aside from the mathematical and philosophical subject, we now are talking about heuristics in the form of pedagogy. Nothing you have mentioned so far appears to be any more pedagogically promising than the classical definition of 'the derivative'. Indeed, there is no reason to think that a high school or college student would not have a much harder time grasping non-standard analysis combined some way with "potential infinity" than grasping the usual notion of a derivative. Nor have you pointed out any instance in which scientific or engineering calculations would be better enabled by your seemingly personal graft of non-standard analysis with a notion of 'potential infinity'.

outdated formal traditions that still remain dominant in the education system. — sime

Infinite sets come into play in Calculus 1. What pedagogy would you propose for people to find derivatives without infinite sets?

* What writings by intuitionists are fairly rendered as describing potentially infinite sets or sequences as "finite entities of a priori indefinite size" or as "finite entities" of any kind?
— TonesInDeepFreeze

SEP's article on intuitionism is a useful introduction for understanding the notion of Brouwer's tensed conception of mathematics — sime

That article does not describe potentially infinite sets or sequences in a manner that is properly paraphrased as "finite entities of a priori indefinite size" or as "finite entities".

Moreover, most remarkably, in another thread:

https://thephilosophyforum.com/discussion/comment/653236

you say, "whose length is eventually finitely bounded". Again, there is nothing in the SEP article that states anything like that.

/

absolute infinity isn't a semantically meaningful assignment to a mathematical entity — sime

Who specifically do you have in mind as mentioning "absolute infinity" since Cantor's work was superseded by formal set theory?

Moreover, my earlier remark stands and should not be overlooked:

* 'infinity' as a noun does not ordinarily have a mathematical definition, though 'is infinite' does. A mathematical definition is never circular nor a tautology. — TonesInDeepFreeze

You simply skipped the point that mathematical definitions are never circular or tautologies.

The semantic notion of absolute infinity (whatever that is supposed to mean) — sime

Whatever what is supposed to mean? "The semantic notion of absolute infinity" or "absolute infinity"? Anyway, in either case, who specifically do you have in mind as mentioning "absolute infinity" since Cantor's work was superseded by formal set theory?

existence of non-standard models that satisfy the same axioms and equations without committing to the existence of extensionally infinite objects. — sime

What specific non-standard models and specific "axioms and equations" do you have in mind?

What is your rigorous mathematical (or even non-rigorous philosophical) definition of "extensionally infinite"?

Science and engineering continues to work with classical mathematics , as well as classical logic, due to their vagueness, simplicity and brevity — sime

Formal classical mathematics is exact in the sense that there is an algorithm to check whether a given finite sequence of formulas is or is not a proof. Any alternative to classical mathematics that also has that attribute would need to be also evaluated for simplicity of the formulation of the system.

intensional definition of pi — sime

The definition specifies a certain definite object.

appealing to the uncertain contigencies of practical experiments with finite resolution. — sime

That is not at all how 'pi' is defined.