Keeping track of where the discussion stands:

I wrote:

It is not problematic that it is the limit of a sequence of rationals but is not one of the entries in that sequence. — TonesInDeepFreeze

That still stands, notwithstanding your reply (which ends with a lie).

never being able to complete the computation involving infinite strings — keystone

Exactly.

Meanwhile, with the other common definitions, we do define addition and multiplication of real numbers and that is not blocked by the fact that computations do not accept infinite sequences as inputs.

what is a point if not a node on the tree? — keystone

Unless instructed otherwise, I would take 'point' and 'node' as synonymous in this context.

Sqrt(2) does not converge towards any node the tree. — keystone

If we are redefining 'is a real number' as 'is a path in S-B' (I would prefer 'is a sequence of nodes on a path in S-B'), then of course such a sequence does not converge to sqrt(2), since sqrt(2) is a sequence and not a node.

it appears to converge to a node that exists at 'row infinity' — keystone

Right, it doesn't converge to any node on the tree. But with the definition

a real number is a path on the S-B tree

convergence is no longer relevant in this context, since a real number is path and not a node to which a sequence converges.

And about "row infinity":

Of course, there is no 'row infinity' which is why I relate it to a mirage. — keystone

Exactly, you don't have a mathematical basis, so you resort to merely figurative, undefined, subjective language. One should not mind figurative language used to convey intuitions about mathematics, to help us get a "mind's eye" grasp of certain concepts. But when the figurative language ends up not backed by actual mathematics then it is fitting to respond, "Okay, that might be interesting, so get back to us when you've worked out the math."

There is an inconsistency in claiming that both (1) the real number line exists and (2) 'row infinity' does not exist. — keystone

No there is not. You're lying. Inconsistency is the derivability of the conjunction of a statement and its negation. You can't show any such derivation.

The real line is constructed in our mathematical theory. And also, in our mathematical theory, the S-B tree does not have a "row infinity". If you claim that is inconsistent, then PROVE it. And if you can't, then you should desist from lying about it.

Your speciousness and intellectual dishonesty here is similar to the previous threads with you.

And it's even WORSE in this thread, because in the other threads, the discussion was about thought experiments, which are informal analogies about mathematics, while in this thread, we are talking about an exact mathematical object.

The crux of this is that there uncountably many reals but only denumerably many names.

But it's difficult to reply point by point to your post, because it's too tangled and knotted up. It would be much better to just start from the beginning. That would be to move step by step through a textbook treatment of this subject in mathematical logic.

It's like when the cables among a lot of electronic components are so tangled and knotted that you can't tell what is connected to what, so you have to just unplug everything and then reattach all the cables in a methodical way.

But I'll address a few points anyway, reiterating some of what I've already said:

First, just to be clear: 'countable' doesn't meant 'finite'. Rather, 'S is countable' means 'either S is finite or S is 1-1 with the set of natural numbers'. And then 'S is denumerable' means 'S is 1-1 with the set of natural numbers'. So there are finite countable sets and infinite countable sets. And a denumerable set is an infinite countable set. And, 'S is uncountable' means 'S is not countable'. Lastly, 'uncountable' doesn't just mean 'infinite'. Yes, if S is uncountable, then S is infinite, but also S is not 1-1 with the set of natural numbers.

"PLUGGING IN"

Because we are grappling with the notion that there are uncountably many real numbers but only denumerably many names, we need to be more exact in what we mean by 'plug in'. This gets pedantic, but it's necessary:

An equation is a syntactical object. So when we substitute a constant symbol for a variable, we are not "plugging in " a number. Rather we are plugging in one symbol (a constant symbol) for another symbol (a variable).

The constant symbol STANDS FOR a real number, but it is not itself a real number.

Recognizing that fact helps to dispel bafflement about the fact that there are uncountably many reals in the solution set but only denumerably many substitutions we can make for the variable.

BOTTOM LINE

There are uncountably many real numbers.

So the solution set for

x+1 = 1+x
(where '+' is defined as the addition operation on the set of real numbers)

is an uncountable set.

But there are only denumerably many names, so there are uncountably many unnamed real numbers, so there are uncountably unnamed real numbers in that solution set.

And, since there real numbers that don't have names, there are no names for those real numbers to plug into the equation.

FURTHER EXPLANTION:

SYNTAX

Ordinary mathematical languages have a denumerable (countably infinite) set of symbols.

The syntactical objects are the terms ("names") and formulas ("statements").*

Every equation is a formula.

Every term and formulas is a finite sequence of symbols.

With only denumerably many symbols, there are only denumerably many finite sequences of symbols. So there are only denumerably many terms and denumberably many formulas.

So there are only denumerably many names and denumerably many equations.

The terms with no free variables are the closed terms.

Every closed term can be abbreviated with a constant symbol per a definition for that constant symbol. For example, we can provide a formulation that is a definition for the constant symbol 'pi':**

Ax(x = pi <-> Ecmd(c is a circle & m is the circumference of c & d is the diameter of c & x = c/d))

And there are only denumerably many constant symbols.

The formulas with no free variables are the sentences.

A theory is a set of sentences closed under deduction.

By 'theorem of a theory' we mean a sentence that is a member of the theory.

Usually, with a theory we also mention an axiomatization of that theory. So a theorem is a sentence provable from those axioms.

In our usual mathematical theories (i.e., any of the usual extensions of Z set theory), we have this formulation that is a theorem:

{x | x is a real number} is uncountable

/

SEMANTICS

For a given mathematical language, we provide the "meaning" for the terms and formulas through the method of models. A model is a function from the set of symbols:

To the universal quantifier, the model assigns an non-empty set, which we call the 'universe for the model' or 'the domain of discourse' for the model.

To each n-place operation symbol, the model assigns an n-place function on the universe.

(a constant is just a 0-place operation symbol)

To each n-place predicate symbol, the model assigns an n-place relation that is a subset of the universe.

(a sentence letter is just a 0-place predicate symbol)

The model does not assign anything for the variables. It shouldn't, because variables are not supposed to have a fixed designation. But we can make a separate assignment for the variables and then we have a model plus an assignment for the variables.

Then, for closed terms, the model assigns members of the universe, inductively per the assignment for the operation symbols.

And, for sentences, the model assigns a truth value, inductively per the assignments for the operation symbols and the predicate symbols.

For open terms, the model plus an assignment for the variables inductively assigns a member of the universe.

For open formulas, the model plus an assignment for the variables inductively assigns a truth value.

A model M is a model of theory T if and only if every theorem of T is true per model M.

Now, per a given model, a mathematical object is a member of the universe of that model.

Each real number is a mathematical object.

Our theory says has the theorem:

{x | x is a real number} is uncountable

Now, for any model of our theory that is also a model that "correctly captures"*** the "intended meaning" of 'uncountable', the subset of the universe that is mapped to from the predicate symbol 'R' (for "is a real number") is indeed uncountable.

/

SOLUTION SETS

For example, the solution set for the equation

x+1 = 1+x
(where '+' is defined as an operation on the set of real numbers)

is

{x | x is a real number} = R

and we have the theorem:

R is uncountable

And, looking at it semantically, if we have model in mind that "correctly captures", then 'R' maps to the set of real numbers (or an isomorphic variant), thus the model maps R to an uncountable set.

In general, for a formula P with free variables x1....xn, the solution set is:

{<x1 ... xn> | P}

where n=1, we may drop the tuple notation and just say:

{x | P}

/

REPLACEMENT SET

As far as I can tell, that is a notion used in beginning informal high school algebra or instruction at that level. I don't know of an actual serious mathematical definition in this context. For the purpose of this discussion, I recommend just forgetting about "replacement sets". It is not needed for any explanatory purpose and only clutters an otherwise rigorous exposition of this topic.

/

* To be more accurate, only terms with no free variables are names, and only formulas with no free variables are statements.

** Throughout, I use some English in the formulations to facilitate exposition. In principle, these formulations would be just symbol sequences of the formal language.

*** To get avoid certain cases provided by Lowenheim-Skolem.

Every path (whether finite or infinite) leads to a different number. — keystone

Right. I was distracted by the dashed lines in the Wikipedia illustration. I recognize now that they're just for place keeping.

an ever distant mirage — keystone

For example, the square root of 2 does not remind me of a mirage. It is not problematic that it is the limit of a sequence of rationals but is not one of the entries in that sequence. But some people just can't grok the idea of the entries of a sequence getting arbitrarily close to a point but that point is not itself an entry in the sequence. But, alas, this brings us back again to the threads from a few months ago. We've been through it already.

arithmetic with real SB strings — keystone

I can only take your word for it that you've satisfactorily worked out that arithmetic. Don't forget that you have to manage not just finite sequences but infinite ones too.

it appears that each real number has a single path which can correspond to a sequence of rationals — keystone

If the details truly work out, then, yes, that is a nice feature.

'is a path on the left side of the SB tree' as a fourth competing definiens? It would be of 'is a real number between 0 and 1 inclusive'.
— TonesInDeepFreeze

Can you rephrase this? I'm not sure what you're asking. — keystone

Nevermind it; I was not on the right track there.

Potential infinity will suffice. — keystone

The tree itself is infinite. And every real is an infinite path.

It seems you're back to your old tricks again. If you don't want infinite sets, then state your axioms in which you derive mathematics without infinite sets. We went over all this 'potential infinity' business a while ago. To save my valuable time, rather than go full circle yet again with you, I'd do best to recommend that you or anyone can read those threads.

Of course, I agree that the computation is not the same as the result.

/

Also, back to an earlier juncture, it is decidedly not the case that I want to show off my knowledge. I don't claim to have very much knowledge about mathematics and mathematical logic. I have a real good firm grasp of some basics, but beyond those mere basics, I falter. I have forgotten so much of the mathematics (especially aside from mathematical logic and set theory) that I studied that I am not even competent when the subject gets very far. So the notion of me wanting to impress anyone is ridiculous. But I also am enthusiastic and like to share what I do know, and it bothers me when I see clearly incorrect or confused claims posted, so I do take some solace in posting corrections and explanations.

/

Anyway, I am interested in the idea of SB used for defining the reals, as another poster has proposed, but I'd like to see that notion developed beyond mere handwaving.

Thank you for that.

Different authors of textbooks in mathematical logic define these terms somewhat differently. I go with Enderton, and this is for first order logic in particular:

We have a countable set of symbols. Each symbol is of only one of these kinds:

quantifier
sentential connective
variable
n-place operation symbol, for some n >= 0
n-place predicate symbol, for some n >= 0

(also left and right parentheses, but I depart on that point, as officially I formulate with only Polish notation so that parentheses are not needed, though informally I use infix notation for some terms and formulas with parentheses.)

(I like that, in classical logic, a first order language needs only one quantifier (the other quantifier can be defined from the first) and only one sentential connective (either the Nicod dagger for "neither nor" or the Sheffer stroke for "not both", as from either one of those, all other connectives can be defined).)

the set of expressions is the set of finite sequences of symbols. (A bit odd to call them all 'expressions' since some of them are just gibberish and have no interpretations, but so it goes.)

the set of terms is defined inductively.

the set of closed terms is the set of terms that have no variables.

the set of well-formed formulas is defined inductively (and, commonly, we just say 'formula' with the same meaning).

the set of sentences is the set of formulas that have no free variables.

an equation is a formula of the form:

T = S, where 'T' and 'S' are terms.

/

it is not the case that all formulas have variables. of course there are formulas without variables, for example:

0 = 0

is a formula with no variables.

"1 + n = n + 1", but actually using the real numbers, not the variable? — Count Timothy von Icarus

If 'n' is replaced by a closed term, then the result is a sentence, in particular a sentence that happens to be an equation with no free variables.

If you tell a Turning Machine to add 2 to 2, that's different than subtracting 2 from 6, right? — Count Timothy von Icarus

Of course.

However, if all arithmetic expressions that = 4 are identical with it — Count Timothy von Icarus

Right there, you're committing the error of not distinguishing the name from the object. The expressions are not the number.

'4' is not 4 and '2+2' is not 2+2. But 4 is 4 and 2+2 is 2+2 and 4 is 2+2.

You are talking about two different subjects together: Mathematics and mathematical logic and, as I take your word for it, information theory.

I have no opinions about what you say about information theory. But I have corrected you on a number of mathematical points.

Here's one:

I think you are confusing the set of all computable functions with the set of all equations. — Count Timothy von Icarus

That came from your incorrect notion that the set of equations is uncountable. So I am not conflating the set of equations with the set of computable functions. The set of equations is a set of syntactic objects, viz. a set of certain kinds of sequences of symbols. The set of all computable functions is also countable, but I do not conflate those two sets. I'm talking about basic mathematical notions as formalized in mathematical logic. And in context of ordinary mathematical languages in which the set of symbols is countable.

I'd very much like to know whether you understand now that the set of all equations is not a countable set.

I'll look at that link.

/

I know what 'instantiate' means. I just don't know what you mean by "the abstraction". Which abstraction? And I don't know what you mean by "what does it mean"? Is there a particular implication from 2+2 = 4 that you are wondering about?

There are computations of arithmetic. And of course they can be conveyed as Turing machine computations. I don't know what makes problematic the theorem '2+2 = 4' or the interpretation of '=' as standing for the identity relation. I don't see why you think it's a problem that the steps are single and stepwise. Any routine, of course, is reducible to steps.

In the formalist interpretation of mathematics, where "an entity is what it does,"
— Count Timothy von Icarus

Where can I read that that is a formalist interpretation? — TonesInDeepFreeze

Still interested. I know what 'formalism' is in the philosophy of mathematics. But I don't know of formalism claiming "and entity is what it does".

"informational encoding"
"system"
"can recognize"
"uniquely specifies"
"without any symbolic manipulation having to be performed"

Would you please say where I can see a glossary of that terminology as you are using it? — TonesInDeepFreeze

I'm still interested. If you are earnest about communicating, then the least you could do is provide a resource for the definitions of your terminology.

If 4 shares an identity with 2+2, 3+1, 5+ -1, 8/2, etc. then the P≠NP problem doesn't make sense — Count Timothy von Icarus

But P vs NP does make sense, so from your conditional we would have to infer that 4 is not identical with 2+2. (By the way, I have no idea why you think that 4 being +2 implies that P vs NP does not make sense. I'm not asking though, since I'm not inviting you to spew yet more jumbles of undefined terminology and vague premises.)

I said as much at the outset. — Count Timothy von Icarus

You said the contrary at the outset and some time afterwards also.

If 4 + 4 = 8 and 10 - 2 = 8, what does that mean for the instantiation of the abstraction? — Count Timothy von Icarus

I don't know what you mean by "the instantiation of the abstraction".

In any case, 4+4 = 8 and 10-2 = 8. 4+4 is 8 and 10-2 is 8.

Having 4 $20 bills and being given 4 more is not the same thing as having 10 ($200) and giving away 2 ($40). Having 5 apples and picking two more isn't the same as having 9 and throwing two in the fire. — Count Timothy von Icarus

So what? Numbers are not bills or apples. Not only are you deluded and uninformed about mathematics, you're also jejune in your arguments about it.

That is, numbers exist as real abstract objects but computation is just a human language for describing their relationships. — Count Timothy von Icarus

I don't have a special opinion to state about that. Except that there is also an abstraction of computation in the theory of computability.

/

Meanwhile, a number of misconceptions by you that I explained in my first post:

https://thephilosophyforum.com/discussion/comment/803226

Let's flag this right away:

responds to a point about how some philosophers of mathematics don't think numbers exist outside formal systems, games we set up, with "they don't need to care about the philosophy of mathematics to know that 2+2 is 4," they aren't particularly interested in a discussion. — Count Timothy von Icarus

Here is what I posted:

I don't think most mathematicians particularly care that much about the philosophy of mathematics.
— Count Timothy von Icarus

They don't need to care about the philosophy of mathematics to know that 2+2 is 4. — TonesInDeepFreeze

YOU said that most mathematicians don't "particularly care that much about the philosophy of mathematics." And I agree with that. But I replied that they still know that 2+2 is 4. That does not imply that I don't care about the philosophy of mathematics or discussion about it.

I took the tone from posts starting with:

"All aboard the crazy train,"

"No, only as you are deluded. "

"Wrong." — Count Timothy von Icarus

That tone does not imply that my purpose is to show off my knowledge.

Generally in field with multiple subfields where the same term can refer to multiple things, it's common to ask if there might be a communication problem, not call someone an idiot. — Count Timothy von Icarus

(1) It's not a matter of terms having different meanings. We are not comparing definitions of 'equation', 'solution set', and 'model'. Rather, you made flat out incorrect claims about them.

(2) I didn't comment on your intelligence.

deluded or uniformed. — Count Timothy von Icarus

You are deluded and uniformed regarding the mathematics and mathematical logic claims that you made and that I mentioned.

when someone starts an interchange with calling you deluded — Count Timothy von Icarus

And I didn't start that way. You can see the first post to see.

you only responded to small fractions of each post — Count Timothy von Icarus

Your posts include a wide mess of undefined terminology thrown around. I replied specifically to parts where I could best offer exact corrections and explanations. Instead of recognizing those corrections now, you spread a lot of smokescreen as above.

/

"Correct, if there is a unique symbol for each real, then the set of symbols is uncountable."

Thank you. That's all I was saying. — Count Timothy von Icarus

No, that's not all you said:

For every real there is a 'theorem" in such a system of the form x = x. — Count Timothy von Icarus

And I explained why that is incorrect.

So your correct claim 'If there is a unique symbol for each real, then the language is uncountable' was used for a non sequitur that there is a theorem for each real.

As I wrote, we don't ordinarily work with languages with uncountably many symbols. It's not even clear what "use" would mean with a language of uncountably many symbols: If there are uncountably many symbols, then there is no decision procedure to decide what the symbols of the language even are, so it's not clear what it would mean to "use" symbols when we can't even know what is or is not a symbol for the language. Languages with uncountably many symbols are "theoretical" in the sense that they are not used for working mathematics but are instead used for theoretical investigations about languages.

Everything else you responded to was you jumping over yourself to demonstrate knowledge about terminology and irrelevant. — Count Timothy von Icarus

Your attempted mind reading is incorrect and presumptuous. You don't know that my purpose was to show off my knowledge and not instead to correct and explain your error. I have given you, gratis, information and ample explanations. Doing so does not warrant your snide and incorrect "jumping over yourself to demonstrate knowledge".

And what I wrote is exactly relevant in response to what you wrote. If you meant something different from what you actually wrote, then it is not my fault to have taken what you wrote as you actually wrote it.

you jumped to the formula x = x; — Count Timothy von Icarus

YOU wrote and discussed that formula, and I replied regarding it.

I'm talking about the informational encoding of any object such that a system can recognize that encoding X uniquely specifies Y without any symbolic manipulation having to be performed. As I mentioned originally, this is in the context of communications theory. — Count Timothy von Icarus

It's not my fault then that you also jumbled together whatever you mean by the above with continued and incorrect claims about equations and solution sets, then also to post an egregious falsehood about model theory, and a number of other misconceptions about mathematics.

"informational encoding"
"system"
"can recognize"
"uniquely specifies"
"without any symbolic manipulation having to be performed"

Would you please say where I can see a glossary of that terminology as you are using it?

Meanwhile, among other points, I hope that at least you understand that in ordinary mathematics '=' means identity, which is to say, for any terms 'T' and 'S',

T = S

means that 'T' and 'S' name the same object, which is to say that T and S are the same object.

An equation is a formula. An equation is a formula of the form:

T = S
where 'T' and 'S' are terms.

model theory works with uncountable symbols — Count Timothy von Icarus

You don't know what you're talking about. In ordinary model theory for ordinary first order languages, there are only countably many symbols in the language. That does not contradict that the universe of a model may be uncountable.

For every real there is the theorem of the form x = x. — Count Timothy von Icarus

Very wrong. You completely misunderstand this subject.

Every theorem is a sentence and every sentence is a formlula. There are only countably many theorems. There is not a different theorem for each real. Rather, the open formula 'x=x' is satisfied by uncountably many reals (i.e., the solution set of 'x=x' is uncountable). You are making the same mistake you started with, which is conflating solution sets with formulas. AGAIN:

There are only countably many formulas. But some formulas have uncountable solution sets.

A symbolic system with a unique symbol for every real cannot be smaller than the reals, no? — Count Timothy von Icarus

Correct, if there is a unique symbol for each real, then the set of symbols is uncountable.

But in ordinary first order languages, there is NOT a unique symbol for each real. We don't ordinarily work with uncountable sets of symbols, because doing so would defeat the intent that the set of symbols, the set of expressions, the set of formulas, the set of sentences, the set of axioms and the set of proofs are all decidable sets.

Get a good book on mathematical logic and study it step by step. Right now, you're very much ill-informed on the subject.

you can check on intuitionism versus Platonism versus formalism, etc. — Count Timothy von Icarus

I know about intuitionism, platonism and formalism. Meanwhile, you need to learn the most basic mathematics rather than throwing around a bunch of terminology that you don't understand. Clearly, you've never actually studied this subject step by step as the subject requires.

"And this has nothing to do with P vs. NP, which is a problem in mathematics that understands that 4 is the same object as 2+2."

I don't think most mathematicians particularly care that much about the philosophy of mathematics. — Count Timothy von Icarus

They don't need to care about the philosophy of mathematics to know that 2+2 is 4.

"Wrong. An equation is a certain kind of formula. In an ordinary mathematical theory (such as set theory, which is the ordinary theory for the subject of equinumerosity) there are only countably many formulas. But there are uncountably many real numbers. It's true that the set of equations is not 1-1 with the set of reals, but it's the set of reals that is the greater."

I think you are confusing the set of all computable functions with the set of all equations. — Count Timothy von Icarus

No, you are confusing what I said and meant with something you want to say or mean.

What I said is exactly correct: (1) There are only countably many formulas. (2) There are uncountably many reals. (3) Therefore, there are more reals than there are formulas.

We are talking about the set of all equations, which is the size of the set of all solution sets for all equations. — Count Timothy von Icarus

That is clearly incorrect. You know worse than nothing about this subject.

An equation is a formula. There are only countably many formulas. A solution set for an open formula with only one free variable is a subset of the set of reals. A formula may have uncountably many members in its solution set, but still there are only countably many formulas. One more time:

(4) The set of equations is a subset of the set of formulas. There are only countably many formulas, so there are only countably many equations.

(5) A solution set for a formula with only one free variable is a subset of the set of reals. A solution set for a formula may have uncountably many members.

So as you can see, there are infinitely more equations than reals — Count Timothy von Icarus

No, only as you are deluded. Your delusion is in conflating the cardinality of the set of equations with the cardinalities of solution sets for equations. Again: There are only countably many equations, but some equations have uncountably many solutions.

Here's what you need to provide for your SB proposal:

rigorous definition of 'is an SB_real' (then let SB_R = {x | x is an SB_real})
(and you'll have to provide for the negative real numbers too, though I guess that wouldn't be too hard)

rigorous definition of 'SB_<'

rigorous definition of 'SB_+'

rigorous definition of 'SB_*'

rigorous proof that <SB_R, SB_<, SB_+, SB_*> is a complete ordered field


We also don't yet have a rigorous (not just ostensive) definitions of the SB tree, 'R' and 'L'. But I don't doubt that there are ones, though complicated they probably are, so we could at least provisionally work with the ostensive definitions we know.

Also, you might want to consider taking reals not as paths but as sequences of nodes on paths. Perhaps it's easier to talk about sequences of nodes rather than sequences of edges, or at least it's more familiar.

Who are you trying to convince here? Philosophers who consider definitions optional?
— jgill

This is a chat forum, not a journal. We should be allowed to spitball here. — keystone

That's a strawman. He didn't say that all discourse has to be at the level of a mathematics journal.

I'm not talking about defining a particular real number. I'm talking about defining the PROPERTY 'is a real number'. Such a definition is of the form:

x is a real number <-> F(x)
where F(x) is a formula with no free variables other than 'x'.

So, three competing definitions:

x is a real number <-> x is a convergence equivalence class of Cauchy sequences of rational numbers

x is a real number <-> x is a Dedekind cut

x is a real number <-> Eid(x = <i d> & i is an integer & d is a denumerable decimal sequence)
[or something like that]

/

As for arithmetic on the Stern-Brocot tree — keystone

I'm not talking about that tree in that context. I was talking about the three competing definitions of 'is a real number' and how easy or difficult it is to define the operations for real numbers based on those definitions.

No, the paths are not real numbers. First, a path is a sequence of edges, not a sequence of nodes. Second, a sequence of nodes is not a real number. Rather the limit of the sequence is a real number.
— TonesInDeepFreeze

Feeding the aforementioned algorithm the string RL, it will treat it exactly as the golden ratio. If RL looks like the golden ratio and it behaves like the golden ratio, why do you not say that it is the golden ratio? — keystone

I take it that by 'RL', you mean the particular denumerable path. That is not a real number by any of the three competing definitions of 'is a real number'. (1) It is not an equivalence class of Cauchy sequences. (2) It's not a Dedekind cut. (3) It's not an integer and a denumerable decimal sequence.

Regarding (1), a real number is the limit of infinitely many Cauchy sequences of rationals, so if 'is a real number' would be defined as just one particular Cauchy sequence of rationals, then which of the infinitely many should it be? We don't have an answer to that question. So, instead, we take real numbers to be a whole equivalence class of the Cauchy sequences of rationals, where 'equivalence' is in the sense of 'mutually converging' (or whatever the actual technical term should be).

[EDIT: What I said in that paragraph is correct, but I was missing the point of an alternative where reals could be sequences from the S-B tree, which would eliminate the need to define as equivalence classes.]

all real numbers [...] are paths on the Stern-Brocot tree — keystone

'is a path on the left side of the SB tree' as a fourth competing definiens? It would be of 'is a real number between 0 and 1 inclusive'.

But, for example, what real number would be the edge {1/2 1/3}? And are you sure that every irrational number is one of the denumerable paths? And that the sequence of nodes of every denumerable path converges to an irrational number? Aren't there denumerable paths that stay constant on a single rational number?

we don't feel inclined to say that 2=1.9 — keystone

But 2=1.9. If your method entails that that is not the case, then I doubt that your method actually provides a complete ordered field.

/

PS. CORRECTION:

I initially misconstrued you. I was not reading carefully enough. I made the point that no irrational is a node on the tree. That is true, but not relevant, since your point (which I failed to read correctly) is that irrationals may be certain paths (not nodes).

In the formalist interpretation of mathematics, where "an entity is what it does,"
— Count Timothy von Icarus

Where can I read that that is a formalist interpretation? — TonesInDeepFreeze

I'm still interested. Where can I read a "formalist interpretation" that "an entity is what it does".

Choo choo! All aboard the crazy train!

If 4 shares an identity with 2+2, 3+1, 5+ -1, 8/2, etc. then the P≠NP problem — Count Timothy von Icarus

That's pure extemporization.

In ordinary mathematics, '=' is taken to have a fixed semantics such that:

4 = 2+2

means that

4 and 2+2 are the same object.

means that

'4' and '2+2' denote the same object.

That's the way it is in virtually all mathematics.

You don't get to declare otherwise.

You can have your own philosophy, but yours doesn't correspond to mathematics as understood by mathematicians.

And this has nothing to do with P vs. NP, which is a problem in mathematics that understands that 4 is the same object as 2+2.

If a unique description of an abstract objects, e.g., a number, is that number [...] then [unacceptable consequent] — Count Timothy von Icarus

Right, a description of a number and the number are not the same objects.

The set of all equations that are "equal" to any number X is infinitely larger than the set of reals as they cannot be set in 1:1 correspondence. — Count Timothy von Icarus

Wrong. An equation is a certain kind of formula. In an ordinary mathematical theory (such as set theory, which is the ordinary theory for the subject of equinumerosity) there are only countably many formulas. But there are uncountably many real numbers. It's true that the set of equations is not 1-1 with the set of reals, but it's the set of reals that is the greater.

look up some of Metaphysician Undercover's posts — jgill

Sorry, but that's an invitation to a crazy train.

common definitions of real numbers — keystone

equivalence classes of Cauchy sequences of rationals. Yes.

or

Dedekind cuts. Yes.

or

decimal representations. (Actually, I think two sequences. A finite sequence for the integer part and a denumerable sequence for the part after the decimal point). But, while, of course, every real has a decimal representation, it is not common to define 'is a real number' that way. First we have to have a rule for when a real has more than one representation. Also, I don't know how easy are the definitions of addition and multiplication compared with the definitions of those operations with equivalence classes of Cauchy sequences or Dedekind cuts. So I prefer to use equivalence classes of Cauchy sequences or Dedekind cuts, and usually I prefer Dedekind cuts since they are used in Enderton's 'Elements Of Set Theory', which is my go to reference.

They are equivalent in the sense that they each provide a complete ordered field, and all complete ordered fields are isometric with one another.

/

'destinations' and 'journeys'. I am not familiar with those as mathematical terminology.

Anyway, I don't say that a sequence is a real number. A real number is an equivalence class of Cauchy sequences, and every real number is a limit of various sequences of real numbers (and a limit of various sequences of rational numbers).

/

No, the paths are not real numbers. First, a path is a sequence of edges, not a sequence of nodes. Second, a sequence of nodes is not a real number. Rather the limit of the sequence is a real number.

[EDIT: When I wrote that, I was missing the point of an alternative where reals could be sequences from the S-B tree.]

/

A description is a linguistic object. A description is not a real number. However.

(1+sqrt(5))/2 is a real number.

'(1+sqrt(5))/2' is a description of that real number.

Also, there is the notion of a definite description. A definite description is an expression of this form:

'The x such that P(x)'. Where 'P(x)' is a formula with no free variables other than 'x' and such that it is a theorem that there is exactly one x such that P(x).

A definiens for a definition of a constant to stand for a real number can be reduced to a definite description. For example, defining the constant 'Phi':

Phi = the x such that x = (1+sqrt(5))/2
or reduced:
Phi = (1+sqrt(5))/2

What is your definition of 'completely described'?

Anyway:

Phi is explicitly defined:

Phi = (1+sqrt(5))/2

And:

If x not= y, then 2 = card({x y})

Why are these things even in question?

Meanwhile, going back to the original post, no irrational is a node of that tree, and if you want a tree that has irrationals as nodes, then you need to adduce a different tree, and it would not be by saying something like, "add another node to that tree as a limit node of other nodes". That's not what trees are.

[EDIT: When I wrote that, I was off-track by missing that the notion was to consider reals as paths, not, as I mistakenly thought, as nodes.]

On the other hand, if "RLRLRL..." converges to Phi, then, of course, it's fine (but redundant) to say that Phi is the limit of that sequence.

Do you think a Cauchy sequence of positive rationals can be used to describe the Golden Ratio? — keystone

I take it that by 'describe' you mean in the sense of 'represent' as ordinarily understand to be a denumerable decimal (or binary, whatever) expansion.

Meanwhile, GR [Phi], like any real number, is the limit of a Cauchy sequence of rationals.

might irrationals be all the infinite strings — keystone

Real numbers are not sequences. Real numbers are equivalence classes of Cauchy sequences of rationals. And a real number is the limit of a Cauchy sequence of rationals.

[EDIT: When I wrote that, I was missing the point of an alternative where reals could be sequences from the S-B tree.]

On the Stern-Brocot tree, might irrationals be all the infinite strings which do not — keystone

There are no irrationals that are nodes of that tree. Moreover, every node n of that tree is finitely many nodes away from any other node with which n shares a path in the tree.

So, you can't just magically add Phi as a node to this tree. If you want Phi to be node of a tree, then you need to adduce some other tree. And, of course, adding a node that is not connected to any other node results in something that is not a tree. So you can't just say, "Take the tree and append on to it such and such a node, even though the node is not connected to any other node of the tree". Period.

[EDIT: When I wrote that, I was off-track by missing that the notion was to consider reals as paths, not, as I mistakenly thought, as nodes.]

I don't know why you are particularly interested in this tree and Phi. We could simplify by taking the complete infinite binary tree. It starts by splitting into nodes 0 and 1 and then from every node splits again by adding a sequence with an added 0 or added 1. There's a formal definition too, but you get the picture. Now, every node of that tree is a finite sequence of 0s an 1s, and every node of that tree represents a rational number between 0 and 1. Then you might say, "But I'd like the tree also to represent an irrational number". But there is no irrational number represented by a node of that tree. Period.

/

Golden Ratio is irrational. As such, how are we going to describe it with the positive rationals? — Count Timothy von Icarus

I take it that by 'describe' you mean in the sense of 'represent' as ordinarily understand to be a denumerable decimal (or binary, whatever) expansion.

Meanwhile, there is no finite sequence of rationals such that Phi is the limit of that sequence. But there is a denumerable sequence of rationals such that Phi is the limit of that sequence.

[Phi] is not an abstract object, but rather a property of abstract objects — Count Timothy von Icarus

Phi is a real number. If one takes mathematical objects to be abstract objects, then Phi is an abstract object.

There are two different, but compatible notions:

(1) Phi is a certain real number. (viz. (1+sqrt(5))/2)

(2) Two real numbers, x and y, are such that x/y = Phi. (i.e., (x+y)/x = x/y.)

I don't think an infinite series can encode an irrational number while remaining itself rational — Count Timothy von Icarus

'sequence' ("string") and 'series' mean different things.

A series is a sum of a denumerable sequence of terms such that there is a limit to the sequence of the successive finite sums. [Note: That definition requires that the sequence has a limit, so a series and its limit are the same object, thus excluding the "undefined" sense of a series that does not converge. This is different from definitions that allow that there are nonconvergent series, which, formally, does not make sense (where we are not using the Fregean "scapegoat method) since the sequence is not the series and if the sequence does not have a limit then there is nothing that can be the value of such a "series" in the sense of a series being a certain kind of sum.]

Of course, if a series is irrational then it is not rational.

However, infinity is not a rational number because it is undefined as to its status as an integer — Count Timothy von Icarus

You were using 'infinite'. 'is infinite' is an adjective (a predicate); a set is infinite if and only if it is not finite. But 'infinity' is a noun. There are points of infinity - positive and negative - on the extended real line, but that's not what's in play here.

A ratio is necessarily a relationship — Count Timothy von Icarus

This is again conflating the two different but compatible notions. (1) A ratio of two real numbers is a real number. It is not a relationship between real numbers. (2) Two real numbers x and y such that x/y = z are in a relationship - a set of pairs of real numbers - in particular, {<r s> | r/s = z}.

Every real number is a ratio. For every real number x, there exist real numbers y and z such that x = y/z. Given certain real numbers, such as Phi, it is convenient to state equalities of those numbers as ratios, but the existence of a real number is contingent on the existence of an equivalence class of Cauchy sequences (or, with a different definition of 'real number', on the existence of a Dedekind cut), since that is what a real number is. Especially note that '/' is a defined symbol. So any definition of 'Phi' that uses '/' can be reformulated so that '/' is not used.

I call it "second-order abstract." — Count Timothy von Icarus

'second order' has certain meanings in mathematics. I guess your usage is in some philosophical but not mathematically defined sense.

In the formalist interpretation of mathematics, where "an entity is what it does," — Count Timothy von Icarus

Where can I read that that is a formalist interpretation?

What do you think a theory is?

What do you think set theory is?

What do you think an inconsistent theory is? (You claim ZFC+CH+~CH is not an inconsistent theory, so it's clear you don't know what an inconsistent theory is.)

What do you think a model is? (You think there's a model (or "world") of ZFC+CH+~CH, so it's clear you don't know what a model is.)

What do you think a model of set theory is?

where did you get that? — litewave

From several textbooks and articles on mathematical logic and set theory.

whether you call the multiverse a model or a collection of models — litewave

As far as I can tell (based on the page you mentioned, and the surrounding pages) the multiverse is not a model. It is a collection of models.

the fact remains that a set with cardinality between naturals and reals exists in this collection — litewave

Sets exist in universes (domains of discourses) for model. The collection is a collection of models. The only things that exist in that collection are models. The set exists in the universe of one of the members of the collection.

although it doesn't exist in all its subcollections (models), which is fine. And the same applies to the existence of any set that is defined by a consistent axiomatized set theory; that's why I said that all logically possible (consistent) collections exist. — litewave

And now we're exactly where we started. Read the conversation again, if you wish to understand. Or, better yet, get a book on the subject so that you can understand its basics. Or continue to ignorantly demand that you're right, no matter how carefully it is explained exactly why you are not.

Still, it is incorrect that "according to set theory, all logically possible (consistent) collections exist." It's your own personal notion, not found as a theorem of set theory and not even as an agreed upon, let alone well articulated, informal meta-principle regarding set theory.

Then the multiverse is a model of what? Or what is it? — litewave

I already told you. It's a collection of models (or "worlds" informally).

Are you interested in understanding anything about theories, models, set theory and models of set theory? Or are you just going to continue insisting that you're right about everything even though you know nothing about the subject?

"Clump"? Is that supposed to be another technical term? — litewave

Your snark is ludicrous in context.

You entirely skipped my specific and exact explanation, to instead try to gain a pathetic bit of snarky upper hand.

Actually 'clump' is not technical, because there IS NO technical notion of combining models of contradicting theories.

Then there are "CH worlds" and "~CH worlds". — litewave

There are many models of CH and many models of ~CH.

You are missing the point, which is that a set with cardinality between naturals and reals exists in the multiverse. — litewave

No, you are missing the point that such a set exists in some models in the multiverse and not in other models in the multiverse.

You utterly got Hamkins backwards. Not surprising, since you are an ignorant, intellectually dishonest and (now seen to be) petty crank.

No, he mentions that there are separate universes. That is the multiverse: The collection of separate individual universes. He doesn't combine universes all into one big clump. There is no such thing when there is not even what a model when the theories contradict one another. And there is no "the world of ZFC+CH" or "the world of ZFC+~CH". Rather, for each set theory, there are many non-isomorphic models.

Get it in your head: There is no model of an inconsistent theory. There is no model of ZFC+CH+~CH, let alone a model from combining two models.

And Hamkins doesn't say that he obtains a model made by combining models of ZFC+CH and of ZFC+~CH. That would be ludicrous.

And if Hamkins says that, aside from particular theories, in general and simpliciter, there is a set of cardinality between the naturals and the reals, then we would need to see the exact passage in which he says that.

What Hamkins mentions is that there are two approaches: (1) A universe view in which some particular model is taken as the one that determines all mathematical matters. (2) The multiverse view in which there is not one particular model that is taken as the one that determines all mathematical matters, but instead there is PLURALITY (his word, my emphasis) of models, and mathematical matters therefore are not settled simpliciter but rather with a plurality of answers, each depending on particular models. You completely mixed that up to think he's saying that the multiverse is a stew of all models thrown into a single pot to make another model itself. He says the OPPOSITE of that. And the question whether there exists a set of cardinality between the naturals and reals is not answered 'yes' according to some single multiverse, but rather answered 'yes' or 'no' according to different models that are in the collection of models that is the multiverse.

You don't know anything about what a theory or model is. Yet you make stubborn false claims about the subject, even misrepresenting Hamkins. That is intellectually shameful.

Rather than keep repeating your ignorance, you would do better to grab a book on the basics of the subject of theories and models and properly learn the concepts.

Meanwhile, https://thephilosophyforum.com/discussion/comment/766751 stands, and now we add:

* You are utterly confused on even the most basic notions of a theory and models of a theory.

It is as if you took these two statements:

(1) "This ball is red."

and

(2) "This ball is not red."

and concluded that these statements are contradictory. But you didn't notice that these statements are not about the same ball but about two different balls and thus there is no contradiction between them. — litewave

That's a variation of your abysmal ignorance of the basics of the subject of models of theories.

(1) and (2) are a contradiction. However, there is a model in which (1) is true and (2) is false, and a model in which (1) is false and (2) is true. And that may be the case on account of 'this ball' referring to different objects per different models (or, also, on account of whatever 'this ball' refers to being in the subset of the universe named by 'red', or not in the subset of the universe named by 'red' per given models).

Learn the basics of mathematical logic and model theory, toward the subject of models of set theory.

I suppose that you also think that a union of ZFC+CH and ZFC+~CH theories is an inconsistent theory. Yet according to Hamkins the worlds defined by these two theories are parts of a consistent multiverse. — litewave

Wow. You most clearly demonstrated your ignorance of the basics of this subject, and continue to carelessly misappropriate Hamkins.

Yes, ZFC+CH+~CH is inconsistent. That's clear on its face.

And worlds are models. Models are not consistent or inconsistent. Theories are what are consistent or inconsistent.

Yes, just as with Hamkins, different theories may have a different class of models from one another. Even certain kinds of theories by themselves have models that are not isomorphic with one another.

A (consistent) set theory has many models not isomorphic with one another. ZFC itself has models in which CH is true and models in which CH is false. But there does not exist any model of ZFC+CH+~CH, since inconsistent theories do not have models.

Hamkins points out that we are free to work separately in different models. He doesn't say that we combine a model of ZFC+CH with a model of ZFC+~CH.

You know virtually nothing about the subject of set theory and models of set theory.

As I mentioned, I doubt you even know what ZFC IS.

The union of all axiomatized set theories does. — litewave

That is an inconsistent theory. And even then it doesn't say what you say it does. You keep skipping the point that there is no apparent way to put your claim in the language of set theory.

And earlier you said ZFC is just an example. But now you premise on the union of all theories, of which ZFC is obviously one. But ZFC says no set is a member of itself, while you say that there are sets that are members of themselves, while you take existence from the union of all the theories. Now, granted, you might say that you are taking only from the individual theories the theorems that assert existences, not those that deny existences. It would take at least a bit of thinking to figure out how that would actually work, but at least it is overwhelmingly clear that it is nothing close to what set theory says.

Hamkins regards the world defined by ZFC+~CH as equally real as the world defined by ZFC+CH and that both worlds exist in the multiverse. So that's how the continuum hypothesis is settled. — litewave

That is the bare gist of it. And it doesn't say what you say it does. So I just repeat what you don't address:

And that quote is not at all tantamount to saying that we take as existing all the sets that are proven to exist according to different set theories. As it stands in and of itself, it could be mean the exact opposite - that there is no single universe that determines the totality of the sets. That what exists depends on each individual theory. He says there is are distinct concepts; yet your notion is that there is a unified concept that is arrived upon by collecting from all the distinct concepts, or from the union of what is proven among an uncountable number of theories. It remains to study Hamkins further to see exactly how his notion works, but at least from that passage, we cannot infer that it works as you claim it does. — TonesInDeepFreeze

I didn't put words in your mouth. I thought that when you used the word "naively" you referred to naive set theory. — litewave

Yet, I did not refer to naive set theory there. And I made clear previously that I was talking about set theory. And, if I now recall correctly, I had a least alluded to the fact that naive set theory is not definitely axiomatized (except usually it is understood to include the principle of comprehension). And in the second instance I made clear that I was talking about a naive, intuitive, informal way of thinking; and I did not in that regard say I was talking about naive set theory; moreover, since set theorists now do not work in naive set theory, it would make no sense to interpret me in the least charitable way possible - that I was talking about set theorists working in naive set theory.

So, please do not read my posts as carelessly as you read about set theory (I have no idea where you got your ersatz notions that you claim to represent "set theory", and you didn't even bother to read the Wikipedia article that you purported refuted one of my points, when actually it supported that point) and then purport that they say things that in fact they don't.

Back to the original point: You are incorrect that "according to set theory, all logically possible (consistent) collections exist". And you make it even worse with your handwaving about unions of theories and appropriating Hamkins (while you know not even the bare basics of the technical exposition). You are incapable of even conceding a single mistake, including the initial one.

Yes, I deleted that post, as I realized it failed to track with what you did say.

self-deleted. not needed.

Back to the very start: It is not correct that "according to set theory, all logically possible (consistent) collections exist". I've demonstrated that in several ways. Apparently you are not willing to admit to even such a basic mistake.

And by now you've been corrected on these points:

* What set theory is. (I doubt you even know what the term 'set theory' actually refers to.)

* What naive set theory is (as you didn't even read the very article about it that you cited).

* That your own personal glib hand waving is not a sufficiently definite notion - mathematically or philosophically.

* That you conflate your own personal glib hand waving with what set theory and naive set theory actually are.

* That your own personal glib hand waving doesn't work out the way you think it does.

* What a definition of a set is as opposed to a definition of a property of sets.

* That you have not shown a determining passage from Hamkins regarding your own notions vis-a-vis his.

* That you have not supported that Hamikins says what you claim he does about CH.

* That you falsely put words in my mouth about regularity, and without subsequent retraction, and evaded my point about CH by misconstruing it. And you present no even remotely definite sense of what you mean by 'exist' - whether by syntactical definition, or by membership in a universe for a model, or otherwise.

* You show no recognition of the distinction between defining a particular set versus a claim that there exist sets having a certain property - which is a distinction crucial to allowing this subject to be discussed intelligibly.