I think Russell is saying that you don't put modal operators in front of terms, only in front of formulas. So you could have:

NEx(Kx & Ay(Ky -> y=x) & Bx)

or

PEx(Kx & Ay(Ky -> y=x) & Bx)

But he also requires that

NEx(Kx & Ay(Ky -> y=x) & Bx) holds only if Ex(Kx & Ay(Ky -> y=x) & Bx) is logically true.

/

where to put the diamond that indicates the thing that is K only exists contingently? — Banno

Be careful. The diamond ('P' in my notation above) says that the statement is possibly true, but it doesn't say that the statement is not necessarily true.

Copleston seemed not to be familiar with the thinking behind Russell's point that he eschews 'necessary' and 'contingent' as adjectives to describe entities as opposed to propositions. But I am curious what Russell meant by "a logic that I reject".

some pseudo-philosopher posters are limited in their thinking to finite physical Reality : no place for metaphysical Ideality. Consequently, intimations of anything outside the physical/material system of Cause & Effect amounts to blasphemy against their personal belief system (their creed). — Gnomon

In this thread? Which posters do you claim are limited in their thinking to only physical reality? And what have they posted that allows you that claim, and to the extent that they have claimed that disagreement is anything like blasphemy?

Formalization is the way for an idea to be treated seriously. — keystone

Most importantly, it is the way to confirm (to yourself or to anybody) that you have actual mathematics free of any hand-waving.

huge investment of time and money — keystone

Not a lot of money. A few good books.

If ZFC is consistent then there's no cake and eating it too. — keystone

Right. No Kate and Edith too.

No, set theory shows how the paradoxes with the naive notion of sets are avoided.
— TonesInDeepFreeze

We've been down this road already. — keystone

Yes, and down that road we arrive at a concept free of the paradoxes of naive set theory.

Given that everything fits nicely together for you and in your view the paradoxes are addressed, I can see how you're not motivated to pursue a potential infinity solution. — keystone

What? I've said over and over and over that I am open to learning about alternative mathematics - including strong finitism and, constructivism and intuitionism. I don't know enough about them, but I know vastly more about them than you do. And I even went through a lot of posts in this thread alone to consider your own proposals. Your snipe is ridiculous.

Every time you blatantly lie about me, your integrity shrinks and shrinks. But you just can't resist ...

Let's not debate my motivations. — keystone

While you lie about mine.

Let's leave it at that. — keystone

Let US? No, I don't need to take direction from you as to whether I comment or not.

Interesting dialogue at that link.

Famously, it's a given that sophistry, intellectual dishonesty, downright factual dishonesty and confusions born of ignorance are rampant on the Internet. But it's especially disconcerting that that would be the case on a forum about philosophy itself, and even more especially when mathematics is misrepresented, misconstrued and mangled in the discussions. I believe it is worthwhile to mention instances of that as they appear. And it is worthwhile to engage when posters dispute such mentions, as that serves to highlight the record of such posting, as illustration of the problem, and as elaboration in explaining specifics of the problem. Each instance is an object lesson. Unpacking is worthwhile.

And, in and of itself, posters should not feel obligated to desist in defending themselves nor to desist in flagging when posters post lies, falsehoods, misrepresentations, ignorance, confusion, fallacies, speciousness and other forms of intellectual dishonesty and intellectual irresponsibility.

And I believe it is worthwhile to resist the essentially censorious insistence that people should not follow up on certain aspects of a discussion. Discussions naturally go down subpaths that deserve discussion as components of a larger subject. So when mathematics is brought into a discussion it is eminently worthwhile to talk about whether a poster's presentation of mathematical aspects is coherent, informed or true.

In a discussion about political philosophy, if a poster makes certain claims and interpretations regarding the U.S. Constitution as part an argument toward a philosophical view, then it is eminently worthwhile to discuss whether those claims are true and those interpretations coherent and fair.

In a discussion about philosophy concerning cosmology, if claims and interpretations about general relativity are brought in, then of course there may be follow-ups.

In a discussion about first cause and infinite regress, if a poster makes certain claims and interpretations about mathematics, then of course there may be follow-ups. And that is poignantly true when a poster is spreading confusion, misunderstanding and falsehood on the subject of mathematics. It is definitely worthwhile to flag that, and, even better, to supply correct information and explanation.



That post is so densely, richly packed in error, speciousness and hypocrisy that a lot of unpacking awaits an ambitious unpacker.

But first, yes, a risible moment:

TonesInDeepFreeze
Sometimes all you can do is laugh and walk away. — Aristotle
— Banno
Amen! — Gnomon

Nice. Bannon suggests that Gnomon has been so laughable that it's best just to walk away from him (or her). Then Gnomon says, "Amen!"

I don't know what set Tones off on his "Gnomon said" rant. — Gnomon

(1) 'rant' is again argument by characterization. It might as well be paraphrased as "My posts are reasoned commentary; your posts are rants". It's an instance of a generalization of an insight George Carlin articulated so brilliantly in his wonderfully trenchant routine:

https://youtu.be/JLoge6QzcGY

[2:02 - 2:22 on the point, though the whole clip is quintessential comedy]

It's absurd and egregious for Gnomon to characterize my posts as "rants", presumably in contradistinction with his posts. Moreover, Gnomon's own posts included ad hominem claims, would be mind reading, about people's motives.

(2) Gnomon says he doesn't know what started the "Gnomon said" track. Thus, Gnomon lacks awareness even of what he(or she) said. The track started when Gnomon falsely and bizarrely claimed that I had added something to a quote. Of course it is reasonable for me to refute that false claim and then to continue to refute Gnomon's stubborn and incorrect insistence that I share fault with him (or her) anyway.

And it is a worthwhile lesson highlighting brazen intellectual dishonesty, to the point of a poster trying to prevail in a claim even if it means denying a plain bald fact as to what the subject and predicate are in a simple sentence.

And the track originated in Gnomon declaring that jgill's comment was "Irrelevant!" when jgill's comment was not irrelevant; and generally presuming to say for other posters what aspects of the discussion they need to regard as not apropros, especially when Gnomon him(or her)self talked about those aspects before I even posted in the thread, and doubly especially as the original poster him(or her)self first introduced that aspect and later explicitly made challenges to other posters for exactness about that aspect.

(3) In greatest generality: If Poster P and Poster Q sequentially reply back and forth, then it is incorrect and unfair to say or to insinuate that just one of them is too persistent.

Gnomon didn't say or mean whatever knocked the chip off his shoulder. — Gnomon

(1) No chip here. On the other hand, it's tempting to say that there is probably a chip behind insisting that posters should not follow up on certain aspects of a conversation, and possibly behind categorically declaring "Irrelevant!" to a poster's report of his study rather than allowing that what one personally is interested in does not subsume all that is relevant, let alone possibly saying "Tell us more about those studies".

(2) I stated explicitly what I dispute in Gnomon's posts to me. For Gnomon to say he doesn't know is nuts. Suggesting that my dispute with him (or her) came out of nowhere as some kind of lashing out from having a chip on my shoulder is wrong.

(3) And it was Gnomon who first disputed me in this thread, not vice versa. Again, the hypocrisy.

I certainly had no intention to insult him, or to debate the technicalities of higher math with him. — Gnomon

I didn't claim Gnomon insulted me. But after I asked Gnomon to please indicate that bolding is added when he (or she) adds bolding to my quotes, Gnomon finally replied by childishly adding gratuitous bolding to that very quote of me asking him (or her) not to do that; as far as I can tell, his (or her) point was merely to scoff at my request, which was a reasonable request to respect a universal convention that one notes that emphases have been added to a quote. Gnomon was childishly insulting in that instance. He (or she) showed his (or her) stripes. Then he (or she) follows with the protestation, "Aw shucks; I ain't said nothing against Tones; I've just been minding my own business" [not a quote]. Like I said, laughter is is not the involuntary response provoked here.

I was about to mention that he's gnawing on an imaginary bone, with no nutritional value. But such a light-hearted tongue-in-cheek remark — Gnomon

Dig it when someone says "I'm not going to say X" thereby saying X. It's sophomorically sneaky. Politicians and lousy writers on politics especially like to do it.

I'll take [banno's] advice to just laugh quietly and walk away. [...] — Gnomon

That would be advice that Gnomon laugh at him(or her)self.

And it's delicious irony that Gnomon says he (or she) will walk away when he (or she) just then did NOT walk away. If banno's advice where actually taken to heart, then Gnomon would have indeed just walked away rather than posting more digs against his (or her) interlocuter.

PS___Since he's bursting at the seems — Gnomon

[italics original]

I don't know whether it was intended to spell 'seams' as 'seems', but as to 'bursting', I am thorough point by point; I provide ample explanations and adequate steps in my arguments, so that they are convincing and to evince that they are transparent; and I reiterate points, especially refutations, that dishonest posters try to pretend have not been made as they deploy the tactic of SKIPPING.

I'll let Tones have the last word : fill-in the blank [ . . . . . . . ] — Gnomon

I guarantee that that is not true. Fill in the blank [ . . . . . . . ]

I didn't say that I'll only consider formalizations. I have been interested in the earlier proposals though not formalized. Rather, I said that I'm not inclined now to study your latest revisions.

I wanted to bounce my pre-formalized idea off of someone to see whether it was worth me investing in formalizing it. — keystone

Somehow, I don't believe you. To formalize you'd have to know what formalization IS. Be honest: Learning what goes into an axiomatic formulation is not a goal for you.

the proofs come from the axiom — keystone

So, hopefully, you understand now that there's no "cake and eating it too" about the S-B tree and Cauchy sequences in set theory, or generally in set theory having both finite algorithms and infinite sets.

unless I can prove the axioms to be inconsistent there's no point discussing my musing. — keystone

Perhaps you meant 'consistent' there. First you have to have primitives, formation rules, inference rules, and axioms. Then you can address whether the axioms are consistent. But it's not required to prove their consistency.

While I would have deeply appreciated you trying to truly understand what I'm trying to say, I fully acknowledge that it is reasonable for you to not want to invest the time into it. — keystone

I understood what you said in the earlier proposals. And I showed you the respects in which it was incoherent until eventually a couple of coherent proposals did emerge (though still clouded with certain stubborn misconceptions you've had).

when discussing the standard position on an intuitive level many paradoxes arise. — keystone

No, set theory shows how the paradoxes with the naive notion of sets are avoided.

The standard position doesn't gel with our intuitions — keystone

Axiomatic set theory is quite intuitive to me. I listed the axioms for you. I find each of them to be eminently intuitive.

Laughter is not the involuntary response provoked here.

In that post, Gnomon makes a number of statements about mathematics qua mathematics. Attempting defintitions (thery're botched) and making claims. About mathematics. Yet Gnomon begrudges discussion of mathematics in relation to a topic couched by the original poster who him(or her)self mentioned mathematics and even explicitly as technical. But that's just typical poster hypocrisy and incoherence.

You did it again: You bolded in my quote without indicating that the bolding was not original.
— TonesInDeepFreeze
Yes. I emphasized the subject of the sentence, to show where you missed the point of the original statement. Apparently, that didn't have the desired effect. But I'll continue to do it again, if you continue to misinterpret my meaning. — Gnomon

You say you'll do it again. And you did just do it again. The bolding you put in was not mine.

And you did it just to prove that you can - a childish, petulant act.

You are wrong.

(1) Even people who radically disagree at least respect the honesty of not misrepresenting a quote to make it appear the quoted person emphasized words that he did not emphasize. That is just BASIC common sense and intellectual honesty. I have seen even the worst Internet creeps - demagogic, specious opinion writers, bloggers, and forum posters who at least indicate where they have added emphases in quotes. But you insist on putting yourself beneath even that minimal convention of honesty.

(2) I didn't misinterpret anything. I took your statement literally as you wrote it. A school kid could diagram the sentence to show its subject and predicate. As I said, if you meant something different, then that's not my fault. And I have not insisted that you did NOT mean something different - only that it's a plain fact that it's not what you literally wrote, therefore jgill and I were not incorrect to respond as we did to what you literally wrote.

(3) Even IF (which is not the case) I misinterpreted you, there is a distinct difference between even willful misinterpretation and misrepresenation (which are not acceptable) and willfully fabricating emphases in quotes. The latter is just pure blatant dishonesty above even arguable disingenuousness.

(4) As to misunderstanding, you intentionally STRAWMANED me, as I noted in one of my previous posts.

Meanwhile, all the points I made in previous posts still stand.

In the words of Paul McCartney, "let it be". — Gnomon

In my own words, "You're vastly better at quoting songs than you are at making sense."

Do the general questions listed above have mathematical solutions? — Gnomon

Seems that I didn't include my reply to that.

It is not clamed that mathematics solves those philosophical problems. It is only claimed that mathematics may inform solving the problems.

Anyway, again (because this fact keeps getting SKIPPED by you) in context of this thread, it was invicta who invoked mathematics to defend her(or his) argument, and not with a proviso that that was merely metaphorical, and later explicitly challenged for "exact" mathematics on a mathematical point and then explicitly for a "mathematical" reply on a mathematical point.

Of course, the Greeks were mathematical pioneers, but they had not formed a mathematically satisfactory account of the notion of infinitude. That does not entail that one should not apply modern concepts and knowledge to questions ages old.

And, of course, the Greeks never said that Pi is a circle.

This also should not have been SKIPPED over:

philosophical questions about contingency and necessity, causation and explanation, part/whole relationships (mereology), possible worlds, infinity, sets, the nature of time, and the nature and origin of the universe.
— Gnomon

Of course those are informed by mathematics and science.

contingency and necessity. That is informed by modal logic, which is a study in formal logic very closely related to mathematical logic.

mereology. Also studied in formal logic.

possible worlds. Again, informed by modal logic. Also, analogous to semantics for intuitionistic logic for intuitionistic mathematics.

infinity. The notions 'is infinite' and 'points of infinity' are informed by mathematics.

sets. Informed by set theory and class theory, which are mathematics and are themselves foundations for mathematics.

the nature of time. I don't know about 'the nature of', but the subject of time is, of course, informed by mathematics and physics.

the nature and origin of the universe. questions about the universe are of course addressed by cosmology, which is informed by mathematics.

It is curious, at best, to me that a person would want to dogmatically declare that philosophy should not be discussed in cross-context with [added: certain] subjects. Especially when the original poster her(or him)self introduced mathematical aspects and not merely metaphorically. On the contrary, intellectual curiosity, intellectual creativity and open mindedness invite cross-study/conversation, not shutting it down. — TonesInDeepFreeze

You did it again: You bolded in my quote without indicating that the bolding was not original.

I have asked you at least three times not to do that, but instead to indicate that the bolding was added.

Clearly, by this point you are egregious.

One more time: If you add bold in a quote of mine, then note that the bold was added.

that is not my claim. It's your erroneous interpretation, but not my intention. — Gnomon

It's literally what you wrote. So, what you wrote is not what you meant. Of course, one may always correct what one wrote to align with what one meant. But it's not jgill's fault nor my fault to have replied to exactly what you literally wrote.

Instead, it is "philosophical concepts" that the predicate modifies with "not addressed". — Gnomon

Wrong. The word 'are' begins the predicate.

Subject:

"Infinite Regress" and "First Cause"

Predicate:

are philosophical concepts that are not addressed by Mathematics

This could not be more clear as basic English. 'are' begins the predicate.

The sentence says that "Infinite Regress" and "First Cause" ARE (1) philosophical concepts and (2) not addressed by mathematics.

Jack and Jill are happy kids who like to play.

The subject is Jack and Jill. The predicate begins with 'are' and the predicate is 'are happy kids who like to play'.

I hope you understand by now.

I admitted above that the sentence construction could be misconstrued --- by someone with a pre-conception. — Gnomon

Blame the reader for a "pre-conception". No, the two readers here read your sentence at face value, literally what you wrote.

And even IF you had written the sentence so that 'philosophical concepts' were part of the subject:

"Infinite Regress", "First Cause", and philosophical concepts are not addressed by mathematics

then still, it is the case that mathematics (and science) may be used, and is (are) used, to inform philosophy.

It is only dogma that mathematics should not be discussed in this thread and that, even more generally, mathematics should not be discussed in addressing certain philosophical questions. Not only is that arbitrary self-serving dogma, but it is deleterious to the benefits of cross-study and an expansive discussion about various subjects.

And your would be controlling dogma is untenable anyway as to this discussion, since the original poster first brought mathematics into the discussion (and not just "metaphorically", and indeed the original poster did not qualify that the mathematics she or he invoked was merely metaphorical) and later the poster explicitly challenged ANOTHER poster to be mathematically "exact" on a point of mathematics and then explicitly challenged another poster to provide a "mathematical" answer on a point of mathematics.

You have SKIPPED my adducing that fact at least two times now.

And it's the "Infinite Regression"*1 argument, not the definition of "Infinity", that is in question. — Gnomon

I never said that the definition of 'infinity' is in question with that particular comment of yours. Whatever you meant by "Infinite Regress" and "First Cause" is up to you. But they are the subject of the sentence you wrote, not 'philosophical concepts'.

impassioned mathematical side-track — Gnomon

That you go, argument by mere characterization.

I haven't been impassioned, nor was jgill impassioned.

And to say it's a 'side-track' is question begging, since whether mathematics is apropros to the subject is what we've been disagreeing about.

Your position is blatantly untenable:

It is invicta who first introduced mathematics into the discussion. And then invicta later even challenged another poster to justify a point mathematically. And then invicta explicity challenged for "mathematical" response.

And you SKIPPED the points I made about mathematics and philosophy interacting.

I suspect that we are actually in agreement about the math of PI & Infinity, but perhaps not about the philosophical concept of a pre-big-bang First Cause. — Gnomon

I haven't stated any opinion on first cause. And, by the way, 'big bang' is ordinarily understood to be a subject of physics and cosmology, a scientific notion, so it too is informed by mathematics.

invicta seems to be insisting on a colloquial usage of "infinity" — Gnomon

Please quote what you regard as invicta saying that she or he means only a colloquial sense.

while you are insisting on technical definitions. — Gnomon

That is a major STRAWMAN.

(1) I have not insisted that usage be confined to only the technical definitions. I offered the technical definitions so that there would not be pointless disputation in regards the mathematical aspects of the conversation.

(2) I did not address invicta's notion of infinity. Rather, I addressed his claim that Pi is a circle. And later his arguments about inscribing the expansion of Pi in a circle.

You have imposed on me things I did not say. Perhaps due to YOUR preconception. Philosopher, heal thyself.

I suppose you would like me to paraphrase so you can judge my comprehension. — keystone

I just wanted to know whether you understand.

I incorrectly claimed that the S-B paths converged to a limit. — keystone

I like nodes better than paths for this.

In set theory, every denumerable sequence of nodes converges to a limit.

In k-musings, there is no limit for the sequences to converge to, and there are no denumerable sequences anyway.

But the rest of your paraphrase is good.

I think this may be an important point to you because you are stressing the importance of completeness to calculus. — keystone

It is core to standard analysis. I don't claim that there can't be viable alternatives to standard mathematics.

But I stressed the lack of such limits in k-musings because you kept posting as if those limits exist in k-musings.

What I'm suggesting is that by starting with uncountably infinite objects (corresponding to real numbers) you are effectively starting with the 'bottom of the tree'. And that agreeing to the former and not the latter is wanting your cake and having it too. — keystone

What is the 'bottom'? What are the 'former' and 'latter'?

A continuum defined by numbers — keystone

^^^ A structure isomorphic with the continuum may be made with non-numbers. Anyway, in set theory, every object is a set.

[in k-musings] numbers defined by a continuum. The ordering of numbers in this [k-]system does not need to be complete. — keystone

You don't have a system. You have some ideas.

And you still misunderstand what I posted. You have it backwards. In set theory, defining the ordering does not require proving completeness. Rather we define the ordering and then prove completeness.

Would you agree to either of the following?
1) A continuum is defined completely by numbers.
2) A line is made up entirely of points. — keystone

(1) I don't know what sense of 'defined' you mean. I said what the continuum is:

c (the continuum) is the set of reals with the standard ordering

The set of reals is the carrier set for c.

A continuum is any structure (a carrier set and an ordering) isomorphic with c.

And a continuum may have a carrier set whose members are not any kind of number.

*** You seem to have a notion that we have to distinguish numbers. No, every object is a set. There's not even a definition of 'is a number'. Though there are definitions of 'is a natural number', 'is a rational number', 'is a real number', etc. But we don't need the word 'number' there. For that matter, we could instead say 'is a zatural', 'is a zational', 'is a zeal'. There's no special force in saying 'number'.

(2) In set theory, 'point' and 'line' can be defined (we don't have to take them as primitives such as in axiomatic geometry). A line is a certain kind of set, and its members are called its 'points'.

I don't know the purpose of this exercise.

I don't think there's a need to define the limit of an algorithm. — keystone

What? Here's the context:

Are my proposed algorithms that different from Cauchy sequences?
— keystone

Indeed they are! I EXPLAINED this. I don't understand what you don't understand in my explanation.

(1) An algorithm is finite. A Cauchy sequence is denumerable. And an equivalence class of Cauchy sequences has the uncountable cardinality of the set of equivalence classes of Cauchy sequences.*

* I think that sentence is right.

(2) There are only denumerably many algorithms, but uncountably many equivalence classes of Cauchy sequences.

(3) Cauchy sequences have a limit. But if we somehow defined the limit of an algorithm, then that would be infinitistic (unless some actual rigorous workaround could be formulated). — TonesInDeepFreeze

You asked whether algorithms are so very different from Cauchy sequences. One of the differences I mentioned is that Cauchy sequences have limits, but even IF we defined limits of algorithms (of the kind of algorithms that approximate irrationals), then they would be infinitistic (so they would not comport with your finitism).

Obviously, I'm not suggesting that you countenance limits for algorithms. I'm only answering your query about how algorithms are different from Cauchy sequences.

very number-centric view. — keystone

Oh, get out of here already with that nonsense. See ^^^ and *** above.

Your "line", the k-line, has NOTHING on it, as YOU said. So 'continuous' is not even applicable. And there is no infinite set of cuts on the k-line that comes after all the rows. You just now admitted that.
— TonesInDeepFreeze

Over and over you repeat the same point, as if I'm not understanding you. I understand what you're saying. — keystone

I repeated it because you repeated contradictions of your own stipulations. I just went with what you literally wrote when you laid out the proposal. Then you say there's some other explanation. At this point, I'm not interested. I carefully read your earlier proposal - I noted each of your stipulations and definitions. Then later what you WROTE (notwithstanding what you might have MEANT) contradicted that. It's inconsiderate to ask a reader to not be able to take each stipulation and definition as having some constancy - to have to continually start all over again to keep up to your later explanations as to what you meant when you didn't write what you meant originally. I'm done with that.

PERSONAL MOTIVATION

What attracted me to the S-B tree in this thread is that we can take reals to be sequences of nodes. Unlike with equivalence classes of Cauchy sequences, we can see particular Cauchy sequences that we can use to define each particular real. (You wanted to use paths instead, but either nodes or paths should work.) Then I was interested in how that might be developed to derive the needed notions of ordering, addition and multiplication.

Then you changed your proposal to taking generating algorithms themselves as the reals. That interested me too, since, if I'm not mistaken, it is a notion in the subject of computable analysis, which I don't know enough about but piques my curiosity. And, again, that raises the question of how to define the ordering and addition and multiplication.

Then you added more apparatus that doesn't seem to me to improve the more basic and original goal that was not being addressed. Then you went further about "higher dimensions". I'm not sufficiently interested in whatever that's about to invest time and energy on it, while instead my curiosity is with the original questions of defining ordering and the operations.

But, of course, you should continue to post whatever interests you, notwithstanding my own disinterest in it.

INTUITION / FORMULATION

I don't think there's just a single roadmap to creating mathematical theories. But my guess is that a mathematician first has an intuition. Then she develops that intuition - in both depth and extent. Then she figures out how to formalize the ideas and to prove the important theorems.

So while the mathematician is still in the pre-formalized stage, deepening and extending the intuitions, she is putting herself into a kind of "intellectual debt". That is, the mathematician eventually is going to have to "pay" for the intuitive commitments with the hard cash of formalizing them.

When we formalize, it's usually along these lines:

We state the syntax of the primitive symbols, then the terms (nouns), the predicates (adjectives), formulas and sentences (statements), inference rules (logic), axioms (basic premises), and deduction (proof). Then definitional axioms (definitions) are added and theorems (the mathematical content) are deduced (proven).

Also, we state a formal semantics that provides for the meaning of the syntactical objects and also provides a means for proving that certain sentences are not theorems.

In the late 19th and early 20th centuries, different mathematicians developed ideas about how intuitions about 'number', 'is infinite', etc. could be formalized. Most of those intuitions among mathematicians, even when differing, offered an essential consensus. This eventually led to ZFC set theory as the standard theory. (But we only need (Z+DC)\regularity.)

But there was dissent. From finitists (stricter than Hilbertian finitism that still allows use of infinite sets as formally handled), constructivists and intuitionists, and predicativists. For the most part, those mathematicians were not very much concerned with formalizing their alternative mathematics. However, eventually much of alternative mathematics has been formalized. And the range of alternatives has wonderfully burgeoned. Now there's a truly amazing, densely populated gamut of alternative mathematics, and it's been formalized. And there's reverse mathematics, which figures out how to have the desired theorems but from weaker axioms.

(Z+DC)\REGULARITY (a set theory)

With (Z+DC)\Regularity we can formulate mathematics including number theory, analysis, topology, geometry, abstract algebra, graph theory, computability, probability, statistics, game theory ... on and on ... and mathematical logic itself.

(Z+DC)\Regularity addresses formalizing analysis this way:

The logic is first order predicate logic with identiity.

The only primitive is 'is a member of'.

The axioms are:

Extensionality: For any sets x and y, they are the same set if they have the same members.

Schema of Separation. For any "formalizable property" P, for any set x, there is the set of all members of x having property P.

Pairs: For any sets x and y, there is the set whose only members are x and y.

Union: For any set x, there is the set of all members of members of x.

Power Set. For any set x, there is the set of all subsets of x.

We prove the existence of a unique set that has no members, called '0'.

We prove that for any set x, there is the set whose only member is x, called '{x}'.

We prove that for any sets x and y, there is the set whose members are all the members of x and all the members of y, called 'xuy'.

Infinity: There is a set w such that 0 is a member of w, and for any set x, if x is a member of w then xu{x} is a member of w.

We develop the reals this way:

We define 'is a natural number'

We prove that there is a set whose members are all and only the natural numbers.

We define 'equivalence class' (per an equivalence relation).

We define 'is an integer' as 'is an equivalence class of natural numbers'.

We define 'is a rational' as 'is an equivalence class of integers'.

We define 'converges'.

We define 'is a Cauchy sequence (of rationals)'.

We define 'is a real' as 'is an equivalence class of Cauchy sequences'.

We define '<', '+', '*' for reals.

We define 'is a complete ordered field'.

We prove that the reals with <, +, * is a complete ordered field.

We define 'is isomorphic with'.

We prove that all complete ordered fields are isomorphic with the reals.

We define 'the continuum' as 'the reals along with <'.

Then we develop differentiation and integration to provide mathematics for things like speed, acceleration, etc.

SET THEORY and the S-B TREE

It is crucial to recognize that the S-B Tree is also itself developed in set theory. Thus, in set theory, we can construct and deduce from the S-B tree while also having all of the developments I described above.

So, in set theory, there is both the tree that doesn't have a final row or "row infinity" and the continuum. This is not having our cake and eating it too. Whatever we have comes from proofs from the axioms. The axioms are productive enough to proof the existence of many things including: the continuum, the S-B tree, finite algorithms, etc.

k-MUSINGS

You are in an intuition stage. If you ever followed through to write some mathematics, then you would confront the debt you're accumulating and pay it off with rigorous formulations. But, in the meantime, one still needs discipline to not just mouth a bunch of incoherent mental picture stories. Even with intuitions, one would like not to commit to informal contradictions (unless one wants to base the proposal in a paraconsistent logic). Which is to say, crankery is a dead end.

— Gnomon

That is nothing less than bizarre for you to say.
You wrote:
"Infinite Regress" and "First Cause" are philosophical concepts that are not addressed by Mathematics — TonesInDeepFreeze

The bolding is not mine. You've quoted me a few times as you've added bolding. I already asked you to note that the bold or emphasis is added. Are you not familiar with that convention?

I assumed the subject of the sentence was obvious. — Gnomon

Yes, the subject is:

"Infinite Regress" and "First Cause"

The predicate is:

are philosophical concepts that are not addressed by Mathematics

That is a claim that "Infinite Regress and "First Cause" are (1) philosophical concepts and (2) they are not addressed by mathematics.

jgill mentioned his own studies in which mathematics do address those concepts.

Interesting that you immediately pounced to declare "Irrelevant!" when jgill shared his knowledge that there is mathematics concerning the subject of your sentence.

"Infinity" is a legitimate mathematical topic, but "Infinite Regression" is an ancient philosophical conundrum. Hence, to get into mathematical technicalities is irrelevant to questions about a world-creating act of Causation. — Gnomon

That is dogmatic. It is not precluded that mathematics may inform philosophy. It is not precluded that different areas of study may illuminate one another.

Moreover, separating out my earlier part in this discussion, the mathematics I mentioned was not meant to confirm or dispute any philosophy, but rather to offer standard definitions to the extent that the discussion did go into mathematics. But then also to flag the blatantly false claim that Pi is a circle.

Indeed, invicta has gone on to challenge other posters in explicit mathematical terms. I pointed this out to you twice, but you SKIP it, I surmise because for you to recognize it would be to recognize that your claim is false: It is false that it was only other posters who wanted to follow up on the mathematical aspects of invicta's postings while invicta merely wished to use mathematical terminology in a metaphorical way.

Here is is yet again; this time explicitly challenging in a mathematical context:

What proof is there of 1/Pi…creating irrational diameter ?

Mathematically speaking ? — invicta

[bold added]

invicta has been mixing mathematical terminology and mathematics subjects with his personal philosophical musings. It is quite appropriate for people to check the mathematical usage and mathematical claims. It is not incumbent on anyone to limit discussion to your own judgements as to what is and what is not relevant, especially when the original poster has explicitly challenged other posters as to mathematics qua mathematics.

And you don't need to say 'technicalities' over and over, as if the fact that mathematics is technical disqualifies it here. It's mathematics, which, of course is a technical subject.

An Uncaused Cause or Prime Mover is a Platonic/Aristotelian notion of Metaphysics, not Physics, nor Math. [...] Or do you think philosophical questions can be solved mathematically? — Gnomon

I hope you don't propose that as a dichotomy.

Of course, one may propose that mathematics may inform the metaphysical questions while not claiming that mathematics solves the questions.

An Uncaused Cause or Prime Mover is a Platonic/Aristotelian notion of Metaphysics, not Physics, nor Math. — Gnomon

Then it would be a good idea for anyone who wants to divorce that subject from math to not bring math into it.

Moreover, again, different fields of study may illuminate one another. Much of philosophy has been informed by mathematics and by science.

Anyway, again, it was invicta who invoked mathematics, so naturally people will comment on it.

A self-drawn circle is the theory to be "trashed", not the definition of Infinity. — Gnomon

Whatever other posters have in mind, I haven't said anything about 'self-drawn circle'.

And, at this point it is egregious that you keep SKIPPING the point I have made a few times already:

I posted the mathematical definitions of the terminology 'is infinite' and 'points of infinity' so that, to the extent when 'infinity' is mentioned in a mathematical context or context that mixes mathematics with philosophy, we may avoid pointless terminological contestation. Moreover, to highlight the crucial distinction between the adjective concept 'is infinite' and the noun concept 'infinity'.

Apparently, in the face of such sniping, Invicta bailed on his own thread. — Gnomon

You don't have a basis to infer the reasons for invicta's posting frequency.

For those interested in the actual topic of this thread — Gnomon

Of course, one may be interested in invicta's philosophical gravamen while also being interested in the mathematical notions he mentions.

philosophical questions about contingency and necessity, causation and explanation, part/whole relationships (mereology), possible worlds, infinity, sets, the nature of time, and the nature and origin of the universe. — Gnomon

Of course those are informed by mathematics and science.

contingency and necessity. That is informed by modal logic, which is a study in formal logic very closely related to mathematical logic.

mereology. Also studied in formal logic.

possible worlds. Again, informed by modal logic. Also, analogous to semantics for intuitionistic logic for intuitionistic mathematics.

infinity. The notions 'is infinite' and 'points of infinity' are informed by mathematics.

sets. Informed by set theory and class theory, which are mathematics and are themselves foundations for mathematics.

the nature of time. I don't know about 'the nature of', but the subject of time is, of course, informed by mathematics and physics.

the nature and origin of the universe. questions about the universe are of course addressed by cosmology, which is informed by mathematics.

It is curious, at best, to me that a person would want to dogmatically declare that philosophy should not be discussed in cross-context with other subjects. Especially when the original poster her(or him)self introduced mathematical aspects and not merely metaphorically. On the contrary, intellectual curiosity, intellectual creativity and open mindedness invite cross-study/conversation, not shutting it down.

What proof is there of 1/Pi…creating irrational diameter ? — invicta

Let C = 1.
Pi = C/D.
So 1/Pi = D
Toward a contradiction, suppose D is rational.
So there are integers n and m (m not equal to 0) such that D = n/m.
So Pi = m/n.
So Pi is rational.
But P is not rational.
So a contradiction.
So it is not the case that D = 1/Pi is rational.

Mathematically speaking ? — invicta

As opposed to what? Philosophically speaking, artistically speaking, athletically speaking?

/

Anyway, your claim that Pi is a circle is false.

/

a line with irrational extension would snake its way when drawn — invicta

I don't know what you mean by 'irrational extension' or 'snake its way when drawn'. But this is the case:

The decimal expansion of any real number is denumerable. The set of points on a circle is uncountable. So there is an embedding of the decimal expansion into the circle (with points in the circle ordered in the obvious way of "left to right" from highest point to lowest point and "right to left" from lowest point to highest point).

But there is an embedding of the decimal expansion into lots of kinds of geometric figures. There is an embedding of the decimal expansion of Pi into a triangle, into a hexagon, into a line... And there's nothing special about Pi in that regard. The decimal expansion of any real number can be embedded ("arranged") into a circle, or triangle, or hexagon, or line...

Pi is not a circle. But the decimal expansion of Pi can be embedded into a circle. If you find philosophical significance in that, then okay. But then why not find philosophical significance also in the fact that the decimal expansion of Euler's constant can be embedded into a triangle? Or what greater philosophical significance with one over the other?

These stand:

It is just not the case that one has to accept everything invicta says as merely metaphorical when [invicta] [...] presses others to be mathematically exact. — TonesInDeepFreeze

Of course. And a symbolic metaphor for a thing is not that thing. When invicta says that Pi is a circle, and even presses others to be mathematically exact in disputing that claim, it is unreasonable to disallow that [invicta] is [...] speaking about mathematics not merely [...] non-mathematical musings. — TonesInDeepFreeze

/

One can arrange digits of any number on any figure you please. One could arrange digits of Euler's constant on a triangle. That doesn't make Euler's constant a triangle. Not even a metaphor for a triangle. One can arrange digits of Pi along a hexagon. So Pi is no less a hexagon than it is a circle. And if one says, as does invicta, that Pi IS a circle, then a circle is a hexagon. — TonesInDeepFreeze

But you also said, "Infinite Regress" and "First Cause" are philosophical concepts that are not addressed by Mathematics".
— TonesInDeepFreeze
Again, I did not say what you attribute to me. The "not addressed" is your imaginary addition to what I said. — Gnomon

That is nothing less than bizarre for you to say.

You wrote:

"Infinite Regress" and "First Cause" are philosophical concepts that are not addressed by Mathematics — Gnomon

And I said that you wrote: "Infinite Regress" and "First Cause" are philosophical concepts that are not addressed by Mathematics".

Then you say that you did not say it and that it is an imaginary addition to what you wrote.

You did write it; it's not an imaginary addition:

"Infinite Regress" and "First Cause" are philosophical concepts that are not addressed by Mathematics — Gnomon

/

This whole off-topic series of accusations & counter-accusations is what I was referring to as "the shallow end of philosophical debate". — Gnomon

You said that jgill's comment was irrelevant. I pointed out that it is relevant. And then you rejected that, on the bizarre basis that you didn't write ""Infinite Regress" and "First Cause" are philosophical concepts that are not addressed by Mathematics".

Shallowness is not preferred, but neither is incoherence. jgill's comment was not shallow and it is preferred that it be defended against your exclamation "Irrelevant!", and it is preferred that a thread not be so incoherent that a poster claims that another poster fabricated a quote that was not fabricated.

There is no "addition" to my quote of you. On the other hand, a few times you've added bold to quotes by me, without indicating that the bold was not original. That's not a huge deal, but it is inaccurate and it's apropos to mention it in contrast to your bizarre claim that I added to a quote of you.

Disagreements about terminology are unnecessary. Discussants can instead acknowledge the clear definitions in mathematics: — TonesInDeepFreeze

As to accusations, I haven't made any. On the other hand:

Tones, apparently you didn't read the OP, — Gnomon

False accusation.

bear-trap Banno — Gnomon

That's not an accusation, but putting a pejorative moniker on an interlocuter is less than high minded. I don't mind that kind of thing, but pointing it out goes to highlighting that you are mentioning shallowness while you also contribute to it.

what has incensed some posters in this thread — Gnomon

That is tantamount to an accusation that other posters have posted in anger. I said I don't which posters you claim to be incensed; but you still don't say.

they hope to demolish by turning a broad philosophical question into a narrow technical definition. — Gnomon

That is an accusation. And it's not clear that it's true. I'm not convinced that anyone has acted as if philosophic claims must be reduced to technical claims. But when philosophical claims do invoke technical matters, then it is wholly legit to talk about how those technicals are being treated.

/

Let's get back to sharing opinions on general philosophical questions, not specific mathematical technicalities. [with smiley face in original] — Gnomon

In other words, I am being asked not to post on an aspect of the conversation that interests me. Yet, I have explained quite well why that aspect is important. Invoking mathematics into a philosophical argument deserves not mangling that mathematics. Posting incoherently about the mathematics is a set up for degraded discourse from the start.

At best, the poster should say from the start: "I use a lot of mathematical terminology. It is not to be construed that it has anything to do with the ordinary mathematical meanings. Rather, it's just my own personal use of technical language. And while I don't stay with ordinary meanings found in mathematics, or even in everyday use, I don't define my personal use either. I expect that readers put aside their own understanding of the terminology, but still see what I personally mean though undefined. Any confusions therefore are not my fault but the readers' fault."

So, yeah, anyone is free not to comply with your suggestion that we stick only to the aspects of the subject you want us to stick to. Again, when philosophical questions or arguments are couched with mathematical claims, then it is not just appropriate, but it is very helpful, to discuss whether the mathematical aspect is being well treated.

The open-ended OP --- an essay question, not true/false --- regarding opinions about "First Cause", was of mild interest to me, but not the nitty-gritty facts of mathematical infinities. — Gnomon

And other people are interested in the mathematical aspects that were mentioned. In particular, I contributed a table of definitions so that, at least, discussants don't need to get bogged down in disagreements about mathematical senses.

Surely, you don't think a discussion should be limited to only the aspects that interest you?

Note that in the Ouroboros symbol, the snake that seems to be recreating itself, actually has a head and tail, a beginning and end. — Gnomon

Let's stick with the philosophy and not get bogged down in the nitty-gritty facts about ancient symbols and illustrations. [sarcasm, of course] Oh, but wait, maybe posters should not talk about the mathematics mentioned in relation to philosophy, but, on the other hand, there should be no preference that Gnomen not talk about ancient illustrations mentioned in relation to philosophy. [more sarcasm, of course]

/

Referring to invicta as "he" :
"Unfortunately, he continues to argue with Banno about interminable terms that have no bearing on the original post -- just digging himself deeper into the shallow end of philosophical debate." — Gnomon

Just to be clear, you are quoting yourself, not me.

I'm not sure what your point about the pronoun is*. Just to be clear, you first used male pronouns for invicta.

* Though I would agree, especially since 'invicta' is feminine, without knowing the poster's gender, that it is presumptuous to use male pronouns. But one, such as yourself (and me and others, later), might be forgiven for that lapse.

You are still confused. You still SKIP the MAIN points I post. You SKIP over the explanations about how you're mathematically wrong (not wrong to eschew standard mathematics, but wrong about the implications of your OWN framework). Often enough, your notions are incoherent. You make ridiculous jejune arguments (see posts above). And you lie about me. You're a sinkhole.

I believe though is that if this approach ever gets formalized it's going to use a lot of similar language as Cauchy sequences. — keystone

So what? Lots of things use similar language, but say RADICALLY different things.

I think it's a matter of perspective by what one means by 'that different'. — keystone

What? I listed the CRUCIAL, ESSENTIAL ways in which they are different. Rather than recognize that, you cop out with "it's a matter of perspective what one means by 'that different'."

The k-line becomes important when exploring higher dimensions. — keystone

You haven't even figured out the first "dimension". But carry on, though I will very likely not be subscribing.

To continue to clean up some of the language: — keystone

I'm not inclined to indulge you with a formulation that is more complicated than it needs to be. I offered you a more simple outline. You can follow up on it if you like.

Your use of 'line' is only a figure of speech. It's not a line. It has nothing on it
— TonesInDeepFreeze

k-lines are associated with k-functions that describe their infinite potential. — keystone

Wow, you just turn on a dime away from what you say previously. I've explicated enough. Go back and read my posts and think about them rather than driving right over them.

AMAZING obtuseness right here:

Maybe one day you will see set theory as the mathematics of the bottom of the S-B tree…the bottom which you (rightfully) claim doesn't exist. Perhaps it is you who wants to eat your cake and have it too. — keystone

And this answers the question: No you did NOT understand the post mentioned in my post above.

I went out of my way to distinguish between standard mathematics and keystone musings.

In standard mathematics, there ARE objects to serve as limits. I gave an exact example of that. In standard mathematics, with infinite sets, there IS a limit to the sequence of successive finite approximations of Phi. But in keystone musings, without infinite sets, there is NOT. I get to say, "Phi is the limit", because set theory proves there IS such a limit. You do not, because your framework PRECLUDES that infinitistic limit.

Your admonishment about this is a product of you getting completely backwards.

But at least it's good to have an answer: No, you did not understand that post.

Meanwhile, I've asked you three times now whether you understand this post:
https://thephilosophyforum.com/discussion/comment/806060
But you still say not a word about it.
— TonesInDeepFreeze

Sorry, I thought I was answering this question indirectly but let me be more clear. The successive outputs of a k-algorithm do not converge to any object. Ever. The S-B algorithm does not terminate (or to someone who believes in actual infinity - there is no bottom of the S-B tree). — keystone

It's good that you've conceded that there is no convergence to an object, that the algorithm does not terminate, and there is no last row. And it seems that the post helped you to that. But I don't know actually know what role the post had in that. I was asking whether you understand the post, which includes the various aspects of its explanations. Knowing your answer would let me know how much communication is taking place here.

apparently you didn't read the OP — Gnomon

It's not apparent, because it's false.

I would have to be an idiot to make the "claim" you pin on me above. — Gnomon

I can't help you there. You made the claim:

"Infinite Regress" and "First Cause" are philosophical concepts that are not addressed by Mathematics — Gnomon

jgill replied:

Wrong. I and others have studied infinite regress in detail — jgill

You replied:

Irrelevant ! — Gnomon

But jgill's reply is relevant, by saying that there is mathematical study that addresses infinite regress.

some posters are treating invicta as an idiot — Gnomon

For the record, whether or not other posters have done that, I am not one of them.

What I actually said was that his OP was not a scientific or mathematical assertion — Gnomon

But you also said, "Infinite Regress" and "First Cause" are philosophical concepts that are not addressed by Mathematics". Then, when jgill mentioned that there is mathematics that addresses infinite regress, you declared that to be irrelevant. Obviously, it is relevant to reply to "X is not addressed by Y" by saying that one has specifically studied Y addressing X.

He even asked if "anyone wanna trash this theory?". Would anyone in his right mind ask that of a mathematical fact? — Gnomon

You wrote:

Obviously, what has incensed some posters in this thread is the supernatural implications of the OP. Which they hope to demolish by turning a broad philosophical question into a narrow technical definition. — Gnomon

invicta starts with some non-mathematical musings but then also connects that with mathematical subject matter.

(1) I asked before, who do you claim is incensed?

(2) Whatever may nor may not be the case with other posters, I have no interest in demolishing his non-mathematical musings.

(3) invicta himself invited his post to be "trashed". I mentioned that in connection with your protest that posters are hoping to demolish the "supernatural implications" of his post.

(4) When a poster connects non-mathematical musings with mathematical subject matter, it is reasonable to point out where the poster is seriously confused about that mathematics.

(5) It is a not mere a "narrow technica[lity]" to mention that, for example Pi is not a circle.

when others began to make an issue of the PI/infinity concept, I simply pointed out that it was used in a metaphorical context, not as mathematical fact. So, get off his back. — Gnomon

For the record, whatever may or may not be the case about other posters, I made no comment on the acceptability of invicta's notion of Pi vis-a-vis the notion of infinity. Instead, I just gave the relevant mathematical definitions if anyone might care about the mathematics, and a suggestion that, as far as the mathematical aspects of the conversation, disagreements about definitions are not necessary.

As to metaphorical couching, invicta moves between non-mathematical musing and mathematics. At any given point it may be difficult to know whether he's being merely metaphorical or intending to be mathematical. But at certain points, he is explicitly mathematical, as he even challenges ANOTHER post to raise HIS game to mathematical precision:

Pi is exactly the ratio of circumference to diameter. It is not infinite — Banno

Ok then mister, please give me the exact value of Pi — invicta

Pi is not a circle — Banno

Of course it’s a circle what is the value of the line when you’ve performed the calculation circumference/diameter…it’s Pi of course. — invicta

It is just not the case that one has to accept everything invicta says as merely metaphorical when he himself presses others to be mathematically exact.

If you want to get technical, PI is indeed an infinite series of numbers — Gnomon

Most technically that is not the case. Depending on the approach, Pi is an equivalence class of Cauchy sequences of rationals, or Pi is a Dedekind cut. Perhaps one may also attempt to define 'is a real' as reals being denumerable sequences, but that is not common. (2) Pi also is the sum of an infinite summation. That is, Pi is the limit of a sequence.

a circle -- no beginning or end -- is sometimes used symbolically as a metaphor for infinity — Gnomon

Of course. And a symbolic metaphor for a thing is not that thing. When invicta says that Pi is a circle, and even presses others to be mathematically exact in disputing that claim, it is unreasonable to disallow that he is himself speaking about mathematics not merely his non-mathematical musings.

/

That graphic of digits of pi arranged in a circle is probably lovely, but I don't know what one is supposed to infer from it.

One can arrange digits of any number on any figure you please. One could arrange digits of Euler's constant on a triangle. That doesn't make Euler's constant a triangle. Not even a metaphor for a triangle. One can arrange digits of Pi along a hexagon. So Pi is no less a hexagon than it is a circle. And if one says, as does invicta, that Pi IS a circle, then a circle is a hexagon.

I am still curious what examples of steps (5) through (8) there are. I've never seen it.

Addition to an earlier post by me:

what has incensed some posters in this thread is the supernatural implications of the OP. Which they hope to demolish by turning a broad philosophical question into a narrow technical definition. — Gnomon

I don't know who you claim are the "some posters", but at least I have not "hoped to demolish" the poster's notions about "supernatural implications". If the poster thinks it is supernaturally implied that Pi is a circle, then I don't have much to say about that except that it is worthwhile to mention that P is not a circle, from which one can take whatever supernatural implications one thinks there are to take.

And as to "demolishing", the poster himself invited:

Anyone wanna trash this theory? — invicta

Irrelevant ! — Gnomon

Here's what jgill responded to and his response:

"Infinite Regress" and "First Cause" are philosophical concepts that are not addressed by Mathematics
— Gnomon

Wrong. I and others have studied infinite regress in detail, as infinite compositions or iterations. — jgill

His response is quite relevant to your claim that infinite regress is not addressed in mathematics.

Good to know that I don't have to scramble to read an entire article you've linked to just to know what you're claiming.

Anyway, again, nothing I've said depends on claiming that two-valued logic is sufficient to model human reasoning. So I don't know the relevance of your comment to me that two-valued logic is not sufficient.

In your definition of the set of minimal elements, you mistakenly left out a quantifier over 'x'.

The formula should be:

M(F) = {m e F | Ax(x e F -> ~ x proper subset of m)}

Then the rest of the definition of 'is finite' is good.

Yes, as you might know, in a 1924 paper, Tarski stated the definition and proved its equivalence with other definitions, as discussed in Suppes's great intro text book 'Axiomatic Set Theory'.

It also turns out that Tarski's definition is equivalent to:

S is finite <-> S is 1-1 with a natural number.

Usually when someone posts a link, it is taken as a suggestion to visit that link. So, without you saying what specifically you wanted me to take from the article, I would have detoured to read and study an article of which all you mean to say is what you posted anyway.

And nothing I've said depends on claiming that two-valued logic is sufficient for modeling human reasoning.

ght to to use a colloquial meaning of "infinite" in a philosophical proposition, without being challenged to present a mathematical justification. — Gnomon

It wasn't just philosophical. There is a mathematical context also, whether primary or secondary. Especially the assertion that Pi is a circle is mathematical.

incensed — Gnomon

I don't know who you think was incensed.

I didn't mention Dedekind infinitude, only because I had too much to cover already.

x is infinite <-> x is 1-1 with a natural number. [call this 'Tarski infinite']

x is Dedekind infinite <-> x is 1-1 with a proper subset of x

I don't like the use of the lemniscate as you do, because it invites conflating the point of infinity in the extended reals with a cardinal. Also, we don't write 'card(S) = leminscate' to say that S is infinite. As you mention, there are infinite sets of different cardinalities, so it can't be the case that there is just one object (named by the lemniscate) that all infinite cardinalities are equal to.

Tarski came up with another concise definition that can be shown identical to Dedekind's. — jorndoe

Tarski's definition and Dedekind's definition are not equivalent in ZF but they are equivalent in ZFC.

I've responded to all your main points and nearly all your secondary points. And I answered the exact questions you asked me.

Meanwhile, I've asked you three times now whether you understand this post:

https://thephilosophyforum.com/discussion/comment/806060

But you still say not a word about it.

What's up with that?

I'm getting plenty of value out of this dialogue. — keystone

I've supplied some pretty good posts.

I don't think your infinite loop programming example of me not listening was a fair representation — keystone

The infinitude of it was a joke to express that it feels interminable. More than fair in that way.

The idea behind this proposal is that the fundamental object is the line, not the point. — keystone

The simple version is programs. Bringing in a concept of an initial object that is determined solely by -inf and +inf and then pseudo-intervals is extraneous. Everything you need is captured by:

"There is an algorithm that successively outputs the S-B rows. And there are algorithms to output successive S-B finite paths, each path ending in a node. We take the comp-reals to be those algorithms, calling them real-ithms."

This algorithm can be described as R RL[...] and I call it phi. — keystone

Exactly. The algorithm is comp-Phi. That's what I said. And the algorithm only needs to print successive rows - not worrying about cuts and pseudo-intervals. My guess is that you wanted to gussy it all up so that it seduces us to think that in a prettier getup there's some kind of vanishing point horizon mirage-limit. But at least now (hopefully, sincerely and stably*) you've backed off from claiming that.

* Let's set the stopwatch to see how long before you relapse and try to smuggle it in dressed up differently again.

But doesn't that mean that standard calculus depends on there being no gaps on the line? — keystone

What? I just said, "gaps" is not defined by you. The mathematics meanwhile is that there is a continuum. A continuum requires that the ordering is complete, meaning that for every bounded set, there is a least upper bound.

Infinite sets are so deeply embedded in your thinking that you're not even willing to imagine the possibility that points are not fundamental. — keystone

You don't get to incorrectly say what I'm willing to imagine. I don't even have notion of "points are fundamental", let alone that I won't imagine that it's not the case.

I'm happy to see any mathematics or formal logic - no matter how different from set theory and standard logic - as long as there are axioms, rules of inference, and definitions such that we can rigorously verify whether something is an axiom, an application of an inference rule or a definition. Or any algorithm, hopefully that can be conveyed as a Turing machine.

Please don't make pronouncements about what I am willing to imagine.

In my proposal — keystone

I understand your proposal. I am just saying that you don't need all those gooey toppings. Plain vanilla does the job just as well.

Are my proposed algorithms that different from Cauchy sequences? — keystone

Indeed they are! I EXPLAINED this. I don't understand what you don't understand in my explanation.

(1) An algorithm is finite. A Cauchy sequence is denumerable. And an equivalence class of Cauchy sequences has the uncountable cardinality of the set of equivalence classes of Cauchy sequences.*

* I think that sentence is right.

(2) There are only denumerably many algorithms, but uncountably many equivalence classes of Cauchy sequences.

(3) Cauchy sequences have a limit. But if we somehow defined the limit of an algorithm, then that would be infinitistic (unless some actual rigorous workaround could be formulated).

Why don't you already know this?

Is my proposed line that different from the real number line? — keystone

Your use of 'line' is only a figure of speech. It's not a line. It has nothing on it; it's a placeholder only - as YOU said. It's not a line in the sense of geometry or analytic geometry.

Already, we have confusion because you use a word in an utterly personal way and it gets conflated with the actual mathematical sense. So you should call it 'the k-line' so that it doesn't get mixed up again with 'line' in the mathematical sense.

And at no output does the cutting remotely resemble the continuum. First, at every output, there are only finitely many cuts and thus only finitely many rationals described. Second, there are no irrationals described. That is VERY different from the continuum that has both rational reals and irrational reals and altogether not just finitely many, but uncountably many, and proving a continuum.

And even if we took the UNION of the rows (which you can't do, because that is infinitistic) we still would have only the denumerable set of rationals described and none of the uncountably many irrationals.

Is my proposed line not continuous? — keystone

What? Are you TROLLIING me? Your questions are so ignorant and stupid that I can't help but suspect that you are.

Your "line", the k-line, has NOTHING on it, as YOU said. So 'continuous' is not even applicable.

And there is no infinite set of cuts on the k-line that comes after all the rows. You just now admitted that.

Standard analysis achieves length by having uncountably many points. — keystone

Wrong. You don't know ANYTHING about this. You don't even know high school mathematics.

Length is the absolute value of a difference. Even without irrationals, we have length with just rationals. Uncountability is not required to define length. Sheesh!

Is length not also achieved by having pseudo-intervals? — keystone

I don't know. You can do the arithmetic to see whether differences restricted to only those between rationals all on a row work out as desired.

But even if it worked out, calculus needs more than just lengths. It seems you don't know what calculus is. Do you?

we must remember that calculus came before set theory — keystone

So what? It used infinitisitic methods. Set theory provided axioms to make those methods rigorously derived from axioms.

I'm only proposing a different foundational underpinning. If you don't think that's philosophy then sure. — keystone

I respect and encourage philosophical frameworks for various notions of finitism. But your own mathematical proposal inspired by your particular finitism is incoherent. You are have a massive mental block that doesn't allow you to understand the basic illogic in your thinking. You keep wanting to have both only finite objects but also objects that exist only as provided as an end of an infinite process, while refusing in different forms to recognize that there is no such end hence no such objects.

/

I haven't tried to formalize your latest idea, but the rough sketch I come up with is this:

There is a primitive object, called 'the k-line'.

There are two more primitive objects, called '-inf' and '+inf. They are ordered so that -inf is less than +inf.

There are two more primitive objects: R and L.

The k-line but also associated with -inf and +inf is the base row.

There is an algorithm, call it the 'k-S-B algorithm', that generates rows, starting with the base row, then to the next row that is row 0, ad infinitum. The k-S-B algorithm recursively exhausts all "turn decisions" of R and L.

The k-S-B algorithm also associates each row with a set of fractions and an ordering on them, and the fractions are grouped in "cuts" which provide "pseudo-intervals".

So, a row is the k-line, with associated fractions, along with associated cuts and associated pseudo-intervals.

A real-ithm is an algorithm that executes non-terminatingly and each successive output is a finite sequence of Rs and Ls depending on a sequence of "turn decisions".

A k-real is a real-ithm.

NOTE:

We don't need the k-line. It is extraneous to capturing the information we want. We can just say a row is the set of cuts.

We don't need cuts. They are extraneous to capturing the information we want. We can just mention the fractions and their ordering.

I think the reason you want all that is to give the illusion that it amounts to a kind of pseudo-"continuum". But it doesn't. Essentially it's a big red herring. Toss out the red herring and simplify as I showed you, which is basically what you proposed yesterday.

NOTE:

There is no final row.

No real-ithm outputs a denumerable sequence.

Only computable reals are described.

A continuum cannot be described (as I explained a few posts ago and again here).

I shoulda gleaned it meant the golden ratio. I was distracted by fact that coincidentally I used 'G' and 'R' for something different.

What does 'the GR' stand for there?