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.

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.

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. — TonesInDeepFreeze

Going to higher dimensions helps explain my position and would more clearly demonstrate why 'more apparatus' improves my position. But yes, it would take time to try and understand what I'm saying and time is limited. You're right, I've taken this discussion beyond the original question and it is reasonable for you to not want to come along with me. You've already been generous with your time. Thanks.

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. — TonesInDeepFreeze

I agree. I suppose as this conversation evolved I wanted to bounce my pre-formalized idea off of someone to see whether it was worth me investing in formalizing it. Although I'm disappointed, I acknowledge that it is reasonable for you to not want to discuss it until it is formalized.

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. — TonesInDeepFreeze

Yes, the proofs come from the axiom and unless I can prove the axioms to be inconsistent there's no point discussing my musing.

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. — TonesInDeepFreeze

I agree with the first sentence. I also agree that many times I have not been clear. 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.

You seem to have a notion that we have to distinguish numbers. No, every object is a set. — TonesInDeepFreeze

I think I understand what you're saying. Start with a small set of axioms and everything else follows. Numbers follow. Points follow. You do not want to discuss this on an intuitive level and want to stick with formalities. This is not how one would explain what should be a simple concept to a grade schooler but that's fine if that's how you want to approach it. And I think that's how you have to approach it because when discussing the standard position on an intuitive level many paradoxes arise. The standard position doesn't gel with our intuitions and so we must stick with the formalization. But you and I have been down this road before.

I don't know the purpose of this exercise. — TonesInDeepFreeze

If you are not willing to discuss basics on an intuitive level then there is no purpose of this exercise.

----------------------------

All in all, it's clear that we want to have different discussions. You want to talk in terms of formalities and I want to talk in terms of intuitions. Neither of us are able to talk on the other person's level. While I'm extremely disappointed that this conversation has come to an end, I once again want to thank you for your time and insights. As I mentioned before, I've gotten great value from this discussion. Thank you.

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.

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. — TonesInDeepFreeze

Fair enough.

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. — TonesInDeepFreeze

Formalization is the way for an idea to be treated seriously. Learning how to get there would require a huge investment of time and money. If the idea seems very promising and my finances were just right I would pursue it. Let's not debate my motivations.

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. — TonesInDeepFreeze

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

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

We've been down this road already.

Set theory is quite intuitive to me. I listed the axioms for you. I find each of them to be eminently intuitive. — TonesInDeepFreeze

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.

-----------------

Let's leave it at that. Thanks again!

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.

It's odd that a thread on the S-B expansion arises in this forum. S-B is an outlier in number theory - the Wiki page is classified as low interest. And since a primary interpretation of its generation are continued fractions of a special type, one immediately loses easy addition and subtraction - which decimal expansions of the reals have.

The value of continued fraction representations is that chopping them off at various levels give rational approximations to what one is expanding. This is easily seen when expanding a real number, like the Golden Ratio. But where it is of greater value is expanding a complex or real function as a continued fraction, providing rational functions (one polynomial over another) as approximations to the expanded functions.

$Tan(z)=\frac{z}{1-\frac{{{z}^{2}}}{3-\frac{{{z}^{2}}}{5-\ddots }}}$

Although I have little knowledge of this kind of number theory (S-B) I feel the line of inquiry expressed here would be of little interest to the mathematical community. But I could be wrong.

Edit: After doing an internet search for "area" in S-B defined in the Wiki article on Farey sequences, I probably am wrong about the interest shown in S-B by mathematicians. Embarrassingly so as I find that two former colleagues of mine have included it in their book. :yikes:

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.