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.

We're getting our wires crossed. You are talking about all "well formed formulas," which I didn't use as a term for a reason. These are countable under standard set theory because it is assumed that a formula is a finite string, by definition. But that's begging the question on the topic of equations being equivalent to identity. If you want to say most of mathematics generally assumes this, I would agree, I only brought up this tangent because I'm not sure if it should accept this given. Semantics vs syntax.

For example, model theory works with uncountable symbols. For every real there is a 'theorem" in such a system of the form x = x. This is only true where x is a symbol that uniquely identifies each real. If the reals are uncountable then so to are the symbols required to show that x = x for all reals or x + 1 = 1 + x. Set theories formulas are countable, but they are so by definition. A symbolic system with a unique symbol for every real cannot be smaller than the reals, the set of symbols must be, by definition, in one to one correspondence with the reals. And since these symbols can also be combined in an encoding, there are more combinations (infinitely more) than there are reals.

Probably my fault for the language. This is why I started with "encodings," but accidentally slipped into "equations."

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.

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. Now will you allow that, given an alphabet where every real has a unique symbol, those symbols could be used in arithmetic, such that, for example, the symbol for 1 added to itself is equivalent to 2?

Everything else you responded to was you jumping over yourself to demonstrate knowledge about terminology and irrelevant. I mentioned all possible encodings of the type "any real = itself,"etc. (i.e. one such string for every last real), which I thought was apparent given the context. This set has nothing to do with well formed formulas re: standard set theory. You jumped to the formula x = x; that's not what I was referring to. If I meant "x + 1 = 1 + x" in terms of the variables I would not have bothered listing out the variations using integers or mentioned solution sets as an analogy to illustrate the point. "x +1 = 1 + x," is not one such an encoding because x is a variable, not a real. Since the context was encodings' equivalence with the thing they encode, in an abstract sense, I thought this was obvious.

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. My point is that, even if we imagine an infinite computer with an infinite alphabet, it still must use step wise transformations to relate symbols to each other.

Which yes, is a point far adjacent to mathematics proper, but this isn't a math forum lol.

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.

Yes, because the statement "model theory can be used to examine infinite symbolic alphabets," is equivalent to my saying "ordinary model theory for ordinary first order languages uses infinite alphabets." Notice how you had to throw in a bunch of specifiers into that sentence so you could show how wrong I am and how smart you are? Have you heard of the principal of charity?

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.

I took the tone from posts starting with:

"All aboard the crazy train,"

"No, only as you are deluded. "

"Wrong."

Generally in field with multiple subfields where the same term can refer to multiple concepts, it's common to ask if there might be a communication problem, not call someone an idiot. And I'll readily admit I misused the term sequence earlier, which was pointed out to me in a helpful way.

Second, that assumption also came from the fact that I tried expressing the point at length and you only responded to small fractions of each post, only where there appeared to be an in for calling me deluded or uniformed.

Anyhow, thanks anyways, as it's always good to see what the least charitable take would be on an idea so as to better polish up the description. Although, even if I know "less than nothing," about mathematics, I think I know enough about conversation to know that when someone starts an interchange with calling you deluded, or 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.

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.

If we continue down the tree with this alternating pattern RLRLRLRLRLRL... we approach the Golden Ratio.

Is there anything wrong with completing this tree and saying that the infinite digit RL is the Golden Ratio? — keystone

Constructively speaking, there's nothing wrong with your identification of real numbers with "infinite" paths, i.e. the non-wellfounded sets known as "streams", provided such paths are finitely describable. For a computable real is equivalent to a circularly defined equation that can be lazily evaluated for any desired number of iterations to yield a finite prefix. In your case, that would be an impredicatively defined binary stream such as S, defined as the fixed point condition

S = 1 x ~S

where _x_ is the cartesian product and ~ is logical NOT (i.e. S is the liar sentence).

To faclitate the identification of streams with cauchy convergent sequences, S can be considered equivalent to other streams for which it shares a bisimulation with respect to some filter for deciding how streams should be compared. The stern-brocot tree can also be interpreted as a game-tree, such that a computable real number is identified with a "winning strategy" for converging towards an opponent's position who attempts to diverge from the player's path to some epsilon quantity.

Surreal Numbers also share a similar binary- tree construction, and their fabled ability to embed the real-numbers might be recalled. But this rests upon the assumption that transfinite induction is valid, which isn't constructively permissible due to it's reliance on the axiom of choice. Your indicated idea of using fixed-points to define real numbers, although not original is more promising.

I believe the non-standard identification of real-numbers with streams and more generally co-algebras, was originally due to Peter Aczel in the eighties, who became famous for inventing/popularising non-wellfounded set theory. For an alternative approach to non-standard analysis that is constructive and sticks to well-founded sets by merely augmenting them with additional axioms to denote terms at the fixed points, see Martin Lof's notes under "The Mathematics of infinity"

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.

I said as much at the outset.

Here is the problem, if computation is not reversible. If 4 + 4 = 8 and 10 - 2 = 8, what does that mean for the instantiation of the abstraction?

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.

This entails that computation is not instantiated in the world at all, that adding two apples to seven is the same thing as taking two from nine, a tough argument to make, or claiming that the relationship described in 4 + 4 is acomputational, that it exists without reference to computation. That is, numbers exist as real abstract objects but computation is just a human language for describing their relationships.

That's a fine way of looking at it, maybe one of the more popular. My problem is that it seems hard to explain why we can recognize identity sometimes but not others, and formal systems for describing information do not account for this. So there appears to be a problem with the formalizations or the ontology in this respect.

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

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.)

"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.

You could look at the link I shared about it from SEP.

Instantiation =

The word "instantiate" is related to "instance". If someone says, "Name some things that are red." you could answer, "For instance, roses are red, apples are red, blood is red." In other words, roses, apples and blood are instances of the property red. In other words, roses, apples, and blood instantiate the property red.

That's all "instantiate" means. An object x instantiates a property p if p(x). That is, x instantiates p if x has the property p, if x exhibits p, if x is an instance of p. All are ways of saying the same thing (with possibly some subtle metaphysical distinctions).

So, yes a property can be instantiated by another property. The property "is a color property" is instantiated by the property red.

Five apples is an instantiation of the number five, etc. e.g., Plato's Theory of Forms.

Informational encoding: the signal in Shannon-Weaver information if you're familiar with that.

System: from physics, either physical or in an abstract toy universe. By this I just meant any computer we can envisage. I didn't want to get into a specific definition because I want to consider all possible computers.

Can recognize: not a signal sent in ambiguous code, one signal cannot refer to two+ different outcomes for a random variable.

Without symbolic manipulation having to be performed: I should have said "without multiple computational steps (quintuples)," to be more precise. One step would be "see symbol x, print symbol y on that section of the tape (doesn't matter which way to move the tape after)." As opposed to how computation generally has to be performed, utilizing multiple spaces on the tape in a stepwise fashion.

The only way to avoid having to use multiple spaces on a tape for at least some computations, even with an infinite symbol system for every real number, would be to have a unique symbol for every arithmetic combination of those symbols.

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".

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.

Ah, my mistake. This is supposed to say "If 4 + 4 = 8 and 10 - 2 = 8, what does that mean for the instantiation of the computation?
"

I.e., how is computation instantiated in the world? (E.g., https://plato.stanford.edu/entries/computation-physicalsystems/).

If you tell a Turning Machine to add 2 to 2, that's different than subtracting 2 from 6, right? But they have the same answer, 4. However, if all arithmetic expressions that = 4 are identical with it, why does this seem so unintuitive and why do we think dividing 16 by 4 is different from adding 2 and 2 when it comes to computation. We tend to think of computation as being the thing a Turing Machine does to produce outputs, not the outputs themselves.



Yeah, that was very unclear with the wrong word in there.

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.

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.

Yeah, I thought I understood the miscommunication there and I did not. You're correct re: well formed formulas. I had always thought formula = well formed formulas and equation could be defined more broadly as "any two equivalent expressions," such that an equation would allow for things that a formula would not, e.g. having infinite length or an infinite symbolic alphabet. Also that formulas have variables whereas an expression needs none. What would be the term for all statements following the form "1 + n = n + 1", but actually using the real numbers, not the variable?

Basically, if the reals are the solution set of all the values that can be put into 1 + n = n + 1, what do we call the set of all the "things" (I said equations before) that the values are solutions to?

The set that has 1 + π = π + 1, etc. as its members?

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.

Thanks for that.

I apologize for this whole digression anyhow because I had the realization that the thought I had that kicked this off is irrelevant to the thing I'm actually interested in, the ways in which the "process" by which 2 and 2 are added together, the computation, is different from just the output of the process. Total blind alley.

It's shockingly hard to find a discussion of computation that isn't just "computation is the processes that lambda calculus, Turing Machines, etc. can define." You can find a lot of articles on "what are numbers," or "what is entailment," but I've had trouble saiting my interests on this front. You'd think that all the interest in physics re: pancomputationalism would have sparked more philosophical interest in the topic? IDK, maybe I'm just looking with the wrong terms.

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.

He may have devised the continued fraction expansion of the equation — jgill

That's certainly a beautiful definition of the golden ratio.

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. — TonesInDeepFreeze

Accepted. I suppose I changed the topic a little, hoping to show how arithmetic with real decimals (which troublesomely starts from the right of the string) is less manageable than arithmetic with real SB strings (which conveniently starts from the left of the string).

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? — TonesInDeepFreeze

The SB tree might offer something here since it appears that each real number has a single path which can correspond to a sequence of rationals [as you hinted].

'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.

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? — TonesInDeepFreeze

My intuition says yes, but I wouldn't know how to prove it. I think it's clear that every decimal number can be captured by a SB string (of L's and R's) but that is no proof.

Aren't there denumerable paths that stay constant on a single rational number? — TonesInDeepFreeze

Every path (whether finite or infinite) leads to a different number. Finite paths lead to rational numbers. Infinite paths lead to irrational numbers. Or I think you'd be more comfortable saying that infinite paths that don't end in R or L lead to irrational numbers. I think of the limit of the tree as the real number line as depicted here:

https://imgur.com/vWBO6U9

But as we agreed, there is no bottom of the tree. This limit exists somewhat like an ever distant mirage.

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. — TonesInDeepFreeze

I agree that RLR (1.9) converges to 2 (R). So in the conventional sense of equality they are equal. However there is no row of the tree where the two paths intersect. If equality corresponds to intersecting paths, then in this stricter sense of equality I argue that they are not equal. I believe there is value in both uses of equality. The former is useful for practical purposes (e.g. working with calculus) while the later is useful for philosophical purposes (e.g. calculus, which is inseparably tied to real numbers, is the mathematics of the journeys (paths), not of the destinations (nodes)). If we move our focus away from arriving at a destination (which is a mirage after all) then we lose our attachment to actual infinity. Potential infinity will suffice.

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

Fair enough. I think I got defensive because I interpreted it as 'get educated and then talk' when his comment may have simply been something like 'if you want to take this to the next level you need to formalize your ideas'.

Here's what you need to provide for your SB proposal: rigorous definition...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. — TonesInDeepFreeze

I agree that for this to be taken seriously I must present a rigorous definition. However, at this time I'm not equipped to do it. I like the idea of paths/journeys because it corresponds to a process. But yes, I like the idea of describing a path/journey by the intermediate stops along the way (i.e. sequence of nodes).

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. — TonesInDeepFreeze

Cool. Yeah, I would too.

EDIT: When I mentioned a rational path intersecting an irrational path, I meant that rational dashed line never intersects with an irrational solid line, similar to image below.

https://upload.wikimedia.org/wikipedia/commons/3/37/SternBrocotTree.svg

It's clear you're providing useful comments but you've mentioned multiple topics/terms which I'm not familiar with. I've got some homework to do! Thanks.

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.

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.

I did come across the term I meant to use if reference to solution sets. It's the replacement set, so for a formula like 1 + x = x + 1 this would be the variables replaced with each real number.

Now if the solution set is the real numbers, does that mean the replacement set is the same size? And if so, what do we call the members of the replacement set if not equations? Expressions? Or, as I thought might be the case from your post, can we say that the replacement set is actually smaller than the same formula's solution set?

Here is why I thought the replacement set might smaller, but tell me if I'm wrong:

The things in the replacement set seem to be equations, 1+2 = 2+1 is an equation, at least as defined as two expressions with a equals sign (which maybe is a definition lacking rigour?). However, if equations are necessarily finite, how could we have one for every real number? We would need an uncountable number of such equations, one for each real, which seems to violate the logic you described.

You'd either need an infinite string for the equation to put the real number in, since you can't do it with a finite number of digits, or a unique symbol for each real, something like π. But then you would need an uncountable number of symbols, which also violates the logic. This would mean that there is not a well formed equation for every solution of 1 + x = x + 1, the old "some truths aren't expressible in a system."

I looked around for answers but it's hard to find something specific and then some online sources aren't vetted and conflict. I figured the answer has to be either that the members of the replacement set have a different name than "equation" or that the replacement set is counterintuitively smaller than the corresponding solution set in cases where the solution set it the real numbers.

But could you have a set of "equations" from a system that does allow an infinite alphabet? These wouldn't be valid equations under set theory, but they would be a set in the way we can have a set of mathematical models, or a set of library books. I guess "a set of statements from a language in an infinite symbol language."

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. — TonesInDeepFreeze

Aside from never being able to complete the computation involving infinite strings, there is a scenario where the algorithm may appear to hang. This is where it would continue to absorb more input digits without generating more output digits. A decimal analogy might be when subtracting 0.9 from 1.0 where (depending on your approach) you might get stuck in the step of continually looking ahead for a non-zero digit to borrow from.

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. — TonesInDeepFreeze

From the Stern-Brocot perspective, what is a point if not a node on the tree? Sqrt(2) does not converge towards any node on the tree. However, it appears to converge to a node that exists at 'row infinity'. Of course, there is no 'row infinity' which is why I relate it to a mirage. There is an inconsistency in claiming that both (1) the real number line exists and (2) 'row infinity' does not exist. You can't have it both ways. The real number line is an incredibly useful mirage.

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.