Unless I've overlooked something, it seems to me that it's easy to prove that it is not the case that any two strict linear orderings with the least upper bound property are isomorphic:

Proof: Take any two ordinals with different cardinalities. The standard ordering on an ordinal is a strict linear ordering with the least upper bound property.

Even ridiculously trivial: The membership relation e_0 on 0 is a strict linear ordering with the least upper bound property. The membership relation e_1 on 1 is a strict linear ordering with the least upper bound property. But <0 e_0> and <1 e_1> are not isomorphic since there is no 1-1 function from 0 onto 1.

So, unless I've overlooked something, I think we need to mention that there are fields involved. (Or maybe it's tacitly understood that there are fields involved.)

This bears on another thread lately.

Two definitions of 'the continuum'"

the continuum = R

the continuum = <R L> where L is the standard less than relation on R

Definition of 'is a continuum':

A topological space S is a continuum if and only if S is connected, compact and Hausdorff

Let z be the standard ordinal ordering on 2^aleph_0.

z is a strict linear ordering on 2^aleph_0 and with the least upper bound property.

But it is not the case that <R L> and <2^aleph_0 z> are isomorphic, even though R and 2^aleph_0 are equinumerous.

So, it seems to me that the claim that R is unique to within isomorphism, or that <R L> is unique to within isomorphism, i.e. that the continuum is unique within isomorphism, should be regarded as false or it needs to be taken as tacit that what are isomorphic are the fields (fields with strict linear ordering with the least upper bound property). What are isomorphic are certain systems (S p t h) with (R + * <).

And that serendipitously ties in with the topic of "systems", as a reminder that another sense of 'system' is that of a tuple with a carrier set and operations and relations on that carrier set.

(Another sense I'd mention is 'logistic system'.)

To each his own, but I don't feel a much difficulty in adjusting to contexts, so that in some contexts I pay attention to the formal implications of the definitions but other contexts I go with a looser flow.

The formal context I mention here is formal Z set theory and its Z based variants.

Is the natural number 3 identical to the real number 3? — fishfry

Formally, no. As you mention, they are different kinds of sets but there is an embedding of the naturals in the reals.

Informally, yes.

Consider the x-axis [and the] the y-axis. Are they the same? Clearly not, they're different lines in the plane with only one point of intersection. Yet they are both "copies," whatever that means, of the real numbers. But in set theory there's only one copy of any given set. Where'd we get another copy? Are the x-axis and the y-axis the same? — fishfry

"whatever that means" indeed. I'd have to hear someone's definition of "copy".

Formally, the x-axis is {<p 0> p in R} and the y-axis is {<0 p> | p in R}. So the domain of the x-axis = R = the range of y-axis. And we can define less than relations: for all p and q in R , <p 0> less than <q 0> iff p < q, and <0 p> less than <0 q> iff p<q.

Informally, pretty much, nothing at odds with the formal notion, as far as I can think of.

"the unique Dedekind-complete linearly ordered set" uniquely characterizes the real numbers. — fishfry

Do you mean just a "strict linear ordering with the least upper bound property" or do you mean a "complete ordered field"?

Of course, we know that any two complete ordered fields are isomorphic, but that involves more than a strict linear ordering with the least upper bound property. It also involves having + and * and the needed statements about them. So, from "any two complete ordered fields are isomorphic" can we infer that "any two strict linear orderings with the least upper bound are isomorphic"? (Or maybe it's tacitly understood that there are fields involved.) [EDIT: See next post.]

Would you say that 4 and 2 + 2 are two "instances" of the same number? — fishfry

I don't know what 'instances' means, but in context of the standard interpretation of the symbols '4', '2' and '+, it is the case that 4 and 2+2 are the same. '4' and '2+2' are not the same, but they name the same thing. 4 = 2+2. 4 is 2+2.

One sense of 'instance' or 'occurrence' I do understand is that of occurrences of a symbol or string of symbols.

In '4 = 2+2' there are two occurrences of '2'. The first occurrence of '2' is as the third entry in the string and the second occurrence of '2' is the fifth entry in the string.

One formal approach is to take such strings as finite sequences, which are functions whose domain is a natural number. The string is displayed just as one character after another, but formally, it is this sequence:

{<0 '4'> <1 '='> <2 '2'> <3 '+'> <4 '2'>}

Let f = {<0 '4'> <1 '='> <2 '2'> <3 '+'> <4 '2'>}

And we know 5 = {0 1 2 3 4}.

So the domain of f is 5. The range of f is {'4', '=',' 2', '+'}. And:

f(0) is '4'
f(1) is '='
f(2) is '2'
f(3) is '+'
f(4) is '2'

So '2' is both the third entry and fifth entry in the sequence.

Or we can use tuples instead of functions. So '2+2 = 4' would be <'2', '+', '2', '=', '4'>.

But it seems to me that for additional handling, such as concatenation, functions are easier to work with mathematically even if lengthier to display.

And, for example, the string '= 2+' occurs as a substring twice in '4 = 2+2 -> (4 = 2+2 & 5 = 2+2+1)'.

Etc.

So expressions themselves are mathematical objects. But, wait, not so fast, because the symbols are not themselves mathematical objects. Ah, but we can take even the symbols to be mathematical objects. For example, we can take them to be natural numbers.* And not just mapped to natural numbers as with Godelization, but taken to actually be natural numbers.

So, for example, we regard any display of the typographic shape, that is two lines crossing as with

+

to not actually be the symbol but rather to be a visual aid for marking an occurrence of the symbol which is a natural number. For example, when we specify the symbols of a language, '+' could be, say, the natural number 16. Then, by the way, we when we Godelize a language, we don't have to map the symbols to numbers; they already are numbers.

* For an uncountable language (which is not a formal language but is a language, and is used for proving the existence of a model that provides for nonstandard analysis) we could take the symbols to be real numbers. Or we could take the symbols to be members of an uncountable ordinal.)

Please stay tuned. — keystone

The show I'm interested in seeing is a hoped for episode in which you account for all the configurations to determine which are a continuum and which are not. If that goes well, then the studio might consider extending the series into another season.

Your strawman arose because you don't take seriously (though you make it a point to say that you do) the posts to which you reply.

You suggested that topology might be a distraction and that my definition might be unnecessarily complicated

I replied:

I wanted to provide a mathematical definition of 'is a continuum'. I find it in topology. People have been knocking around the term 'a continuum' in a math context. So what is a mathematical definition? I provided one. That's the opposite of distraction.

And topology is a study that informs analysis and put analysis in a broad context. Understanding the real numbers and the continuum in context of topology is definitely not a distraction. And why would topology be a distraction but your half-baked verbiage not be a distraction? — TonesInDeepFreeze

and

I provided a quite streamlined definition. If an equivalent definition could be simpler then you're welcome to state it. Less formally:

A topological space C is a continuum if and only if C is compact, connected and Hausdorff.

So I provided definitions of 'compact', 'connected' and 'Hausdorff'. Those definitions depend on the definition of 'a topology' so I provided that definition, which I couched with only these non-logical notions: 'element' (primitive), 'subset', 'power set', 'union', 'pair' and 'binary intersection', which are quite basic notions of mathematics (except for 'element', they also can be defined back to the sole non-logical primitive 'element'). — TonesInDeepFreeze

You replied by saying again that topology is not necessary, as if I had said it is necessary, as you were replying to a post in which I said that there's nothing in my definition that requires knowing anything more than a few basic set theory concepts. So, clearly, you ignored what I said and made it appear that I had been saying that you'd have to study topology to track my definition - the opposite of what I said. That is bad faith, whether intentional or a product of not bothering to read.

After a lot of that kind of thing with you, and much worse instances, in this thread and previous ones, I eventually get to the point of feeling that what is most interesting about you is not the particulars of your confusions about mathematics but what goes on in the head of person who is so wrapped up in himself that he wants other people to take their time and labor to understand his own personal, malformed musings while he pretty much just blows off the substantive and informative information and explanations he gets in reply.

The most salient instance lately is your ridiculous claim that you don't object to set theory but only to philosophy about it, yet you do object to the axiom of infinity, while evading that to do so is indeed to object to set theory, not just philosophy about set theory, and then you so sophistically evade that point too by endorsing not the axiom of infinity but a principle very different from the axiom of infinity! Sheesh!

And you claim not to object to set theory but to "misinterpretations" of it, yet your examples of supposed misinterpretations are not misinterpretations! Sheesh and double sheesh!

I felt hurt and turned off by your tone but for some reason in this thread I'm actually quite appreciative of our interactions. — keystone

You're a saint.

I do pla[n] to respond — keystone

If you wish to engage me with this, then know that first I need for you to determine what are all the possible configurations and then to say exactly which are a continuum and which are not, as I mentioned.

What misinterpretations of the meanings of foundational mathematics? What writings by mathematicians or philosophers are you referring to?
— TonesInDeepFreeze

Let me restate the examples I mentioned:

Naïve infinite set theory is thought to be about actually infinite sets when I think it is really about potentially infinite algorithms for constructing the infinite sets. (I want to stay clear of axiomatic set theory since I haven't read the required material.)

Cauchy sequences are thought to be sequences of actually infinite terms when I think they are really about potentially infinite algorithms for constructing the infinite sequences.

When we draw a cartesian plot it is thought that there exist actually infinite points in the plot when I think there really are only finitely many continua, each having infinite potential for partitioning. — keystone

What is your trip, man?

You said, "my only qualm with set theory is the philosophy underlying it (centered around actual infinities)".

So that would be taken to mean that you have no objection to set theory itself but only to certain philosophy about it. But set theory itself is the set of theorems derived from the axioms. So you would have no objection to the axioms. But you do object to the axioms (and not just philosophy about them). For example, you do object to the axiom that there exists a successor inductive set while instead you approve the very different principle "there are potentially infinite algorithms for constructing infinite sets".

News flash: "there are potentially infinite algorithms for constructing infinite sets" is NOT the axiom of infinity. Saying you approve "there are potentially infinite algorithms for constructing infinite sets" but that you don't approve "there exists a successor inductive set" is NOT saying you don't object to the axiom of infinity but only object to philosophy regarding it; rather it is saying you DO object to the the axiom of infinity but approve a very different principle. Again, you take me for stupid if you think I don't see that. It is a childish game you play.

Same for Cauchy sequences and the Cartesian plane.

First off, why do textbooks for courses in U.S. Civics not mention John Locke, William Blackstone, the Federalist Papers, John Marshall or Plessy v Ferguson?
— TonesInDeepFreeze

...not an American. — keystone

You're as bad at history as you are at honest, logical thinking. The Federalist Papers, John Marshall and Plessy v Ferguson are American. John Locke and William Blackstone are English but at the deepest foundation of American principles of government and law.

It seems you missed the analogy. You asked why topology is not very much featured in calculus and analysis books you've read. Putting aside that topology is often part of analysis books, the answer is that a calculus book is about setting forth the most basic mathematics that is put to use in different fields of study; it's not about very much understanding the foundations of that mathematics or its broader mathematical context. An analysis book is more foundational than a calculus book but still mainly about diving right into the real and complex numbers as opposed to getting deep into the details of the set theoretic foundations and the topological generalizations.

By analogy, a book for a course in U.S. Civics is mainly concerned with describing the federal government, its organizations and functions, relationship with the states and with its residents and citizens, with some of the philosophical and legal background but not deep into those.

I don't think you're sincere in wanting to communicate. If you were, you would give people the consideration of clearly articulated concepts.
— TonesInDeepFreeze

Mathematicians hold a high bar for clarity. Might it simply be that I'm not a mathematician? — keystone

I think it's more likely that you're a self-infatuated poseur.

I asked you already: Who do you think it's suitable for? Especially if not for a mathematician, then who?
— TonesInDeepFreeze

At this point, a mathematician who can piece together informal ideas. At a later point (once I've read more), a mathematician. — keystone

In other words, a mathematician of a certain temperament and then later a mathematician. Huh?

Anyway, I speculate that the reason you won't read the substantive material in my posts is psychological. You divert to the false claim that the definition I gave is too specialized....You are so busy espousing that you don't read that to which you respond.
— TonesInDeepFreeze

I admit that sometimes when it gets too heavy I glaze over the details. But have I really not adequately responded to many of your points in this thread? — keystone

Your idea of a "detail" is a one-liner axiom. You would glaze over just about anything. With all the glazing over you do, you should be in the business of hams or donuts.

How many examples do you need to appreciate that that bot flat out lies
— TonesInDeepFreeze

It works well sometimes though. I see it moreso as a handy tool to use with caution. — keystone

A quart of raw milk with listeria works well to quench your thirst. A handy beverage to drink with caution.

that's not the axiom of infinity! It is nonsense to say that you don't object to set theory by recourse to agreeing not with the axiom of infinity but with something very very different! How stupid do you take people to be? How stupid do you think people are not to see the sophistry you just pulled? You're insulting.
— TonesInDeepFreeze

I haven't studied axiomatic set theory but I have taken axiom of infinity to mean that there exists an inductive set. Is that not it? — keystone

That is it.

What I want to reinterpret this as is 'there exists an algorithm to construct an inductive set'. — keystone

Make up whatever principles you want. But "there exists an algorithm to construct an inductive set" is not the axiom of infinity and it's not a "reinterpretation" of the axiom of infinity. It's a decidedly different idea.

Again, when you say you don't object to set theory but you do object to one of its axioms while preferring to plunk down for a different principle, you make no sense! Would you please stop it? Have at espousing whatever alternative principles you like, but it makes no sense to say you don't object to set theory when explicitly you do!

/

And you skipped recognizing that you strawmanned when you said you'd have to study topology to understand my definition.

For reference, here are definitions of 'is a graph', 'is an undirected graph' and 'is a loopless graph':

G is a graph
<->
EVDf(G = <V D f> &
V not= 0 &
V/\D = 0 &
f is a function &
dom(f) = D &
Ab(beD -> Epq(f(b) = {p q} v f(b) = <p q>)))

In other words: a graph is a non-empty set of vertices, a set of edges disjoint from the set of vertices, and a function that assigns to each edge either an unordered or ordered pair of vertices. (An unordered pair of vertices has no direction; an ordered pair of vertices has a direction.)

G is an undirected graph
<->
EVDf(G = <V D f> &
V not= 0 &
V/\D = 0 &
f is a function &
dom(f) = D &
Ab(beD -> Epq(f(b) = {p q})))

In other words: an undirected graph is a graph such that there are edges with direction.

G is a loopless graph
<->
EVDf(G = <V D f> &
V not= 0 &
V/\D = 0 &
f is a function &
dom(f) = D &
Ab(beD -> Epq(p not= q & (f(b) = {p q} v f(b) = <p q>))))

In other words: a loopless graph is a graph such that there is no edge between any vertex and itself.

Once you satisfactorily cover https://thephilosophyforum.com/discussion/comment/935856 I'll see about rigorously defining 'is a k-continuum' as a graph. But I see trouble ahead anyway with your personal notion of partitioning. I pessimistically predict that ultimately you won't be able to get past handwaving with it.

What logic do you use? Classical? Intuitionistic? Other? If you don't state your logic then I will take it to be classical.

What set theory axioms do allow? Perhaps Z set theory without the axiom of infinity and the axiom of regularity (though, I predict you will soon enough be stumped without the axiom of infinity). Note: Just saying "finite processes, no infinity" is hand-waving unless you mathematically define 'is a process' without recourse to infinite sets. If you mean Turing machines or equivalents, then let's see how you actually couch your definitions and inferences with them.

For reference, here are the axioms of Z set theory without the axiom of infinity and the axiom of regularity (a symbol guide for the text symbols I use is in my forum About panel):

(1) extensionality:

Axy(Az(zex <-> zey) -> x = y)

For any sets x and y, if they have the same members, then x = y.


(2) separation schema:

If F is a formula in which x does not occur free, then all closures of
ASExAy(yex <-> (yeS & F))
are axioms

If F is formula, then for any set S, there is the subset {x | xeS & F}. Even more informally, for any formalizable property R and set S, there is the set of elements of S that have the property R.


(3) union:

ASExAy(yex <-> Ez(zeS & yez))

For any set S, there is the set of all and only the members of members of S.

Notated as US = {y | Ez(zeS & yez)}.


(4) pairing:

ApqExAy(yex <-> (y = p v y = q))

For any sets p and q, there is the set such that both p and q are in the set and nothing else is in the set.

Notated as {p q}.


(5) power set:

ASExAy(yex <-> y is a subset of S)

For any set S, there is the set whose members are and only the subsets of S. Notated as PS = {y | y is a subset of S}


Summarizing:

If x and y have the same members, then x is y.

For any formalized property and for any set, there is the subset whose members have that property.

For any set S, there is the union of S.

For any sets x and y, there is the set {x y}.

For any set S, there is the set of subsets of S.

(If we add the axiom of infinity and the axiom of countable choice, then we can derive classical analysis. cf. 'Notes On Set Theory' - Moschovakis pg. 116)

I'm going to put 'k-' in front of words to be clear it's your terminology not to be conflated with the usual mathematical usage.

So far, you haven't defined 'is a k-point' and 'is a k-curve', so, at least for now, I'll take them as primitive, with the axiom: No k-point is a k-curve.

It seems that a k-continuum is a certain kind of finite(?), undirected(?), loopless(?) graph whose k-vertices are either k-points or k-curves. And (I surmise) no k-vertex is connected to itself.

You should state definitively what kinds of configurations are k-continua and are not k-continua. Here's what I have so far:


These are k-continua:

one k-curve

one k-curve connected to one k-point

one k-point connected to a k-curve connected to another k-point


These are not k-continua:

a graph in which occurs a k-point not connected

a graph in which occurs a k-point connected to another k-point

a graph in which occur two or more k-curves but at least one of them is not connected to another k-curve (I surmise)

a graph in which occurs a k-curve connected to more than two k-points


Is there a natural number n>1 such that there is no k-continuum such that there occurs n number of k-curves connected to one another? (In other words do you disallow that a k-continuum may have arbitrarily finitely many connections of curves from one to another?)


Are these disallowed from being k-continua?:

a graph in which occurs a k-curve connected to a k-point connected to another k-curve

a graph in which occurs a k-curve connected to another k-curve connected to a k-point


I'm not sure that exhausts all possible configurations. You should figure it out to define 'is a k-continuum'.

That should be: I'm not just a non-mathematician.

Then halfway through all of it suddenly made sense. — jgill

Same for me. But it was in 7th grade in sex ed.

It's not as if i recoil in horror before matter — Gregory

I'm not a big fan of matter. How nice it would be to exist without being subject to the vicissitudes of objects - massive, medium size and subatomic - clashing and banging all around you, wantonly careening at you, and roiling inside you without regard for the effect it all has on you. Matter doesn't care at all about me, so why should I respect it? Well, I do respect some of it - nice people, a lovely beach, a perfect avocado, and some jazz records and math books. But most of the rest of it, phooey! One thing for sure, no one ever involved in a head on automobile accident ever said, "Thank the universe for the laws of physics".

I am working on putting some of your illustrated explanations into actual mathematics. Might take me some time to assemble into a post, hopefully I will finish and post.

But I've been down this road already with you in another thread. I took a lot of time and effort to turn your gibberish into communicative mathematics. Then, all along the way, you revised your idea, so I revised in response, which is fair. But eventually, your proposal came to an impasse of illogic, yet you wouldn't budge and merely insisted on your notions though they had been shown inconsistent. A dead end with you. But maybe this time it could be different. Hope springs eternal.

What misinterpretations of the meanings of foundational mathematics? What writings by mathematicians or philosophers are you referring to?
— TonesInDeepFreeze

For example:
That set theory is about infinite sets.
That Cauchy sequences are infinite sequences.
That reals are numbers in the same sense that rationals are numbers.
That the Cartesian (and related) coordinate systems lie at the heart of basic calculus. — keystone

You said you have no objection to set theory itself but that you object to misinterpretations of it.

The first in your list there is just an observation that the main areas of interest for prominent set theorists concern infinite sets. That's not a misinterpretation of set theory.

The second is just a theorem of set theory. A theorem of set theory is not a misinterpretation of set theory. So, if you approve set theory itself, then you approve its theorems, including the one you just mentioned.

The third is not something I recall ever reading.

The fourth is just an observation that is seen to be true by opening a textbook. It's not a misinterpretation of set theory.

The mathematical definition is given in topology. How could the actual mathematical definition not be at the very heart of comparing the mathematical definition with alternative definitions? It seems to me that you're rationalizing your unwillingness to inform yourself on the subject.
— TonesInDeepFreeze

Why is it that the intro to calculus/analysis textbooks I’ve read never mention topology? Is it because these texts don’t need a general definition of continua since they only work on the continuum, whereas topology is needed for a defining continua? — keystone

First off, why do textbooks for courses in U.S. Civics not mention John Locke, William Blackstone, the Federalist Papers, John Marshall or Plessy v Ferguson?

Next, my guess is that, yes, analysis textbooks don't usually define 'continua' for the reason you just mentioned. However, as to topology in general, many analysis books do delve into some topology too. Analysis and topology go hand in hand. It's arguable whether it's better to learn topology first for its generality and elegance that applies to analysis (though that lacks analysis as a motivation and source of examples) or analysis first for its motivation and source of examples (though that lacks the generality and elegance of topology). While, of course, both are a gas to study in and of themselves.

Keep in mind, you’ve already given me a reading list that I’m just a few pages into. Adding topology isn’t a problem—I’d even prioritize it if it made sense. But I think it’s fair for me to question whether expanding my reading list is really necessary. — keystone

Ha! Blatant strawman! I didn't say you have to study topology to understand the definition of 'continua'. It's the opposite since I gave you a definition that is self-contained and needs only understanding of a handful of first chapter set theory concepts. That was the very point, as I mentioned it: to give a definition that needs nothing other than that handful of chapter one set theory concepts.

What? You didn't immediately apprehend that was a spoof?
— TonesInDeepFreeze

Ha! No, I didn’t. But I was being honest with my response. There was so much technical jargon that I had no idea what you were talking about, so I asked ChatGPT. That only made me more confused, so I stopped. — keystone

Next time I'll make it even more outlandish for you so that it is inescapable. I'll include Madame Chiang Kai-shek, Krusty The Clown, a wig factory in Duluth, Minnesota and a toothpick on the ground as examples, respectively, of a homeomorphism, isometry, generalized convergence and a product space.

And I told you that. The same goes for topology—I stopped because I’m not informed on the subject, and I told you that too. — keystone

And I told you that you don't need to know topology to understand the definition. You don't need to have a prior understanding of one single sentence of topology to understand the definition I gave. And you would see that even just by reading the definition itself. But, of course, prophets such as yourself need not bother with the writings of others.

but you haven't the slightest inclination to even glance over a mathematical definition given to you by a person who has, at extreme length and in extreme detail (in at least two other threads) engaged your notions. Why is that? Could it be in your personal characteristics? (Some variation of being so overly infatuated with your own mind that there's little intellectual juice left in you to bother learning much about the mathematics that other people have given lives of intellectual labor to?)
— TonesInDeepFreeze

I think the main reason we're not fully connecting is that I’m not presenting my points in a way that’s suitable for a mathematician, — keystone

I asked you already: Who do you think it's suitable for? Especially if not for a mathematician, then who? Who do you think would read your stuff and understand it at any level that qualifies as grasping and absorbing a reasonable version of what you have in your own mind? I don't think you're sincere in wanting to communicate. If you were, you would give people the consideration of clearly articulated concepts. Anyway, I speculate that the reason you won't read the substantive material in my posts is psychological. You divert to the false claim that the definition I gave is too specialized. I'd rather stay on the subject of you. Whatever psychology is going on in you that makes you your own special variation of a crank is more interesting than your malformed, ersatz math musing.

I’m not fully understanding some of your points because they’re not framed in a way that’s accessible to a non-mathematician. — keystone

You're talking about mathematics. Of course discussion then is going to include mathematics.

And, again, the definition I gave requires only familiarity with a few first-chapter, easy concepts in set theory.

You are so busy espousing that you don't read that to which you respond.

I’ve been using ChatGPT as a tool to help me grasp the more complex ideas, but you discourage that. — keystone

How many examples do you need to appreciate that that bot flat out lies, pretends to explain when it is terribly botching the subject, and is incapable of, or unwilling to, check its misinformation, confusion and illogic. It is really really pathetic that you would so seriously louse up the integrity of your study by reference to a source so inimical to truth, understanding and clarity.

And as if it's not enough that the Internet is inundated with cranks, now we have to contend with bots that are cranks!

I only have a limited amount of intellectual juice to dedicate to this. — keystone

You don't ration wisely.

I am passionate about my idea but I don't think that's the main factor here. — keystone

Contrary to evidence.

But in this discussion, we see people refer to both 'the continuum' and 'continua', so we should be careful not to conflate those terms.
— TonesInDeepFreeze

Yes I get that. I've been consistent with this. — keystone

That deserves a fact check, which I'm too lazy now to research.

So you have no objection to the axiom of infinity itself, only with philosophizing that there exist "actual" infinite sets? And what do you mean by "actual"? If one views mathematical sets to be mathematically actual but one does not opine as to whether there are physically actual sets, is that okay with you? If one holds that abstractly there are infinite sets but one does not opine that physically there are infinite sets, is that okay with you?
— TonesInDeepFreeze

My view is that the Axiom of Infinity represents an inductive algorithm for constructing the inductive set, which is said to have a cardinality of aleph-0. — keystone

That's not the axiom of infinity! It is nonsense to say that you don't object to set theory by recourse to agreeing not with the axiom of infinity but with something very very different! How stupid do you take people to be? How stupid do you think people are not to see the sophistry you just pulled? You're insulting.

It is to be vigilantly mistrusted.
— TonesInDeepFreeze

But should it be mistrusted in 2-5 years? — keystone

I don't know. It could be magnitudes worse - fully Orwellian and dystopian (if it continues on the path it's now on) or it could be profoundly better - paradisiacal, or anywhere in a continuum ('continuum', nice word that) between. But I'm pessimistic.

But so what? We're not dealing with what it will be. We're dealing with the toxic, pernicious thing it is now.

Who do you think they are comprehensible to, other than yourself?
— TonesInDeepFreeze

A mathematician with plenty of patience and an open mind — keystone

So a hypothetical being. Not any actual mathematician you've ever met? Not even evidence that there is such a mathematician. And not evidence that even the most patient and open-minded Bodhisattva of a mathematician wouldn't tell you, "Get back to me when you've worked out some math".

And you think a mathematician would not reply in terms that presuppose, at the least, basic undergraduate mathematics?

so far it's only been ChatGPT... — keystone

I can't handle the pathos of that now.

I'll look at this later, if my time, patience and supply of snacks is adequate.
— TonesInDeepFreeze

The next bag of Sweet Chili heat Doritos is on me. :P — keystone

Leave it to you to offer the worst junk food.

I'll look at this later, if my time, patience and supply of snacks is adequate.

My ideas are half-baked and I'm not great at communicating them in a way digestible to mathematicians. — keystone

Who do you think they are comprehensible to, other than yourself?

using ChatGPT as an academic source is grounds for a warning. — fdrake

Should be Wikipedia too.*

* Actually, I don't favor censoring references to the output of AI bots or publicly edited encyclopedias. And Wikipedia is magnitudes better than AI bots (which is not saying much for Wikipedia). But that doesn't mean I shouldn't say how dangerously unreliable they are.

Are you mixed up? You said ChatGPT struggled to make sense of my actual definition, not my parodic definition. When I put my actual definition to ChatGPT, it replied as I posted. Meanwhile, it seems that the point of my parody went past you, even though I said what the point is. And ChatGPT listed possible purposes of that whacky definition but overlooked parody or put-on as one of them.

AI is remarkable. The ability, on the fly, to compose text, to code programs, to create images, to make spreadsheets, etc. is amazing. But it's not intelligence. It can't discern truth from falsehood even in a million different simple cases. It fails often to prove even simple mathematical theorems. Etc. And it can't explain its own errors. It's a source of gross misinformation and misunderstanding. It is to be vigilantly mistrusted. Humankind doesn't deserve to degrade itself by blind acceptance of every inferior and dangerous stage of technology. And AI doesn't have a sense of humor at a level greater than a five-year-old.

misinterpretation of their meaning — keystone

What misinterpretations of the meanings of foundational mathematics? What writings by mathematicians or philosophers are you referring to?

I think it makes sense to focus on the topics that are absolutely essential for understanding continua — keystone

The mathematical definition is given in topology. How could the actual mathematical definition not be at the very heart of comparing the mathematical definition with alternative definitions? It seems to me that you're rationalizing your unwillingness to inform yourself on the subject.

I suspect I'd have to read at least one textbook to truly understand what you mean by "A continuum is a decomposition of a hyperspace module with sidewise-partitionable step-wise radii." — keystone

What? You didn't immediately apprehend that was a spoof? But a spoof with a point; a point that you missed even though I mentioned it explicitly.

Thus you went past my point, which is: If you don't understand the actual definition I gave, then you can ask me. It's curious that you wish to engage people with your notions, to provide them with, at best, vague definitions that they should take time and energy to understand or persist asking for explanations to then also take time and energy to understand, but you haven't the slightest inclination to even glance over a mathematical definition given to you by a person who has, at extreme length and in extreme detail (in at least two other threads) engaged your notions. Why is that? Could it be in your personal characteristics? (Some variation of being so overly infatuated with your own mind that there's little intellectual juice left in you to bother learning much about the mathematics that other people have given lives of intellectual labor to?) But I should hush myself lest one of the Pecksniffian pearl clutchers here accuses me of "ad hominen".

you stated that 'is a continuum' is an adjective, which it isn’t. — keystone

What? You started your post by agreeing that it is an adjective. It is an adjective, a predicate in this case.

You said you agreed but then you disagree. So I doubt you got the point at all:

'the continuum' is a noun phrase that designates a particular mathematical object, viz. the set of real numbers.

'is a continuum', formally, is an adjective phrase, a predicate, that designates a property of certain mathematical objects, viz. the property of being a compact, connected, Hausdorff topology (or, put another way, being a compact, connected, Hausdorff set relative to a topology on that set).

'is a continuum', informally, is an adjective phrase, a predicate, that designates a property of certain things, e.g. a rainbow.

But in this discussion, we see people refer to both 'the continuum' and 'continua', so we should be careful not to conflate those terms.

my only qualm with set theory is the philosophy underlying it (centered around actual infinities). — keystone

So you have no objection to the axiom of infinity itself, only with philosophizing that there exist "actual" infinite sets? And what do you mean by "actual"? If one views mathematical sets to be mathematically actual but one does not opine as to whether there are physically actual sets, is that okay with you? If one holds that abstractly there are infinite sets but one does not opine that physically there are infinite sets, is that okay with you?

But wait, out of curiosity, I put my definition to ChatGPT verbatim, and ChatGPT replied:

This text outlines definitions related to topological spaces and various properties in the context of set theory and topology. Here’s a breakdown of the key components:

Topology: The first definition establishes the criteria for a topology T on a set
X. It requires that:

T is a subset of the power set of X (denoted PX).

The whole set X and the empty set must be in T.

Any union of sets in T must also be in T.

Any finite intersection of sets in T must also be in T.

Compactness: The second definition describes compactness in terms of a topology T on a set X. A set C is compact if every cover of X by open sets in T has a finite subcover. This emphasizes the idea that even if you have a potentially infinite collection of open sets that covers X, you can always find a finite number of these sets that still cover X.

Connectedness: The third definition characterizes connectedness. A set C is connected if it cannot be partitioned into two non-empty, disjoint open sets. This property ensures that C is "in one piece" in a topological sense.

Hausdorff Condition: The fourth definition introduces the Hausdorff property, which states that for any two distinct points p and q in X, there exist neighborhoods (open sets) around each point that do not overlap. This separation property is crucial for many results in topology.

Continuum: Finally, the last definition states that a continuum is a set that is compact, connected, and Hausdorff. This is a significant concept in topology, as it combines these important properties into a single structure.

Comment
These definitions are foundational in understanding topological spaces and their properties. They help formalize concepts that are essential in analysis and topology. By establishing clear criteria for these properties, they pave the way for deeper exploration into how different topological spaces behave and interact. If you have a specific area you'd like to explore further or questions about these definitions, feel free to ask!" — ChatGPT

So, I don't know why ChatGPT "struggled to make sense of it" for you.

But ChatGPT got these wrong:

"A set C is compact if every cover of X by open sets in T has a finite subcover."

No, not a set C. Rather, it should be (in the manner of ChatGPT):

A set X is compact per a topology T on X, if every cover of X by open sets in T has a finite subcover.

"A set C is connected if it cannot be partitioned into two non-empty, disjoint open sets."

No, not a set C. Rather it should be (in the manner of ChatGPT):

A set X is connected per a topology T on X if it cannot be partitioned into two non-empty, disjoint open sets in T.

an unpartitioned continuum — keystone

You've not defined 'a continuum' (a bunch of itself undefined verbiage is not a definition). So I don't know what you're talking about in mathematics when you use the word 'continuum'.

And every set has a partition. Many partitions for larger and larger sets.

you've encountered far more actual continua — keystone

As I understand, you reject using infinite sets. But you say that we encounter continua. So continua are finite?

I can tell you've taken care to read my words closely. — keystone

Not close enough. If I had, I'd have more bad things to say about them.

Is your stuff supposed to be mathematics, or mathematics infused with philosophy, or philosophy, or philosophy infused with mathematics, or something else?

"fundamental objects"

"by its very nature"

"multiple vertices representing the same node" Vertices are nodes.

"broken down into finer and finer elements"

"change the continuity"

"acts"

"algorithm describes"

"The cardinality of an algorithm is determined by comparing it to other algorithms whose cardinalities are already known" Circular.

"continua one partition at a time" You were supposed to be defining 'continua'.

"connected" (in your sense)

/

Maybe more simply, why don't you get back to me when you've figured out your definitions from a starting point rather than backwards from undefined to undefined?

Meanwhile though, still my question: Is your stuff supposed to be mathematics, or mathematics infused with philosophy, or philosophy, or philosophy infused with mathematics, or something else?

And when you cite ChapGPT for mathematics, I figure that you are as lacking in credibility as it is.

You asked me to define continua, which I assume was prompted by my earlier claim that "the continua" [...] — keystone

Again, best to keep things straight:

(1) 'the continuum' is noun that names the set of real numbers.

(2) 'is a continuum' is an adjective that we talking about defining.

Also, the discussion had been, for a while, primarily (pretty much solely) mathematical. Of course, one should be allowed to discuss perspectives other than mathematics alone, but I suggest that one should be clear at each point as to what they are talking about - mathematics, physics, philosophy, or some combination that is explained.

I haven’t yet reached the point of formalizing my ideas into a logical system. — keystone

Let alone that you don't offer primitives, axioms, definitions and proofs, I wonder whether you grasp the idea. Making definitions that rely on yet more undefined terms needs to finally arrive at the base set of primitives, which are undefined but inferences mentioning them are governed by axioms. Or, even if not that formal, at least to state starting notions that are so basic that they at least they can be comprehended intuitively at a basic level. Such things as "increasingly refined composite object made up of arbitrarily many fundamental elements" are so detached from a clear meaning that they are not at all comprehended at a basic intuitive level.

So, even if not formal mathematics, but rather as informal exposition of intuitions, ideas or philosophical views, I can't make heads or tails of whatever it is you're trying to say.

You mentioned providing examples. Ostensive indications and understandings not from explicit definition but rather from gleaning in context are fine and useful, mainly at the stage of basic intuitions. Indeed, I doubt that I could define such terms as 'is', 'in' or 'at' but rather I presuppose that people understand them however they contextually got to that understanding. On the other hand, your notions are not of that basic kind but rather are intricate enough that they need more than ostensive definitions.

However, my primary concern is the continua used in basic calculus — keystone

Then that is mathematics. Or do you mean the study of physical phenomena that calculus is used for?

In any case, as far as I've ever seen, basic calculus doesn't mention the adjective 'is a continuum'. Rather, if at all, basic calculus mentions the continuum = the set of real numbers.

I wonder if delving into topology might be more of a distraction. — keystone

I wanted to provide a mathematical definition of 'is a continuum'. I find it in topology. People have been knocking around the term 'a continuum' in a math context. So what is a mathematical definition? I provided one. That's the opposite of distraction.

And topology is a study that informs analysis and put analysis in a broad context. Understanding the real numbers and the continuum in context of topology is definitely not a distraction. And why would topology be a distraction but your half-baked verbiage not be a distraction?

I also question whether your definition of a continuum is unnecessarily complicated. — keystone

I provided a quite streamlined definition. If an equivalent definition could be simpler then you're welcome to state it. Less formally:

A topological space C is a continuum if and only if C is compact, connected and Hausdorff.

So I provided definitions of 'compact', 'connected' and 'Hausdorff'. Those definitions depend on the definition of 'a topology' so I provided that definition, which I couched with only these non-logical notions: 'element' (primitive), 'subset', 'power set', 'union', 'pair' and 'binary intersection', which are quite basic notions of mathematics (except for 'element', they also can be defined back to the sole non-logical primitive 'element').

For reference, even ChatGPT struggled to make sense of it — keystone

What folly and how insulting!

(1) ChatGPT is so often stupid and dishonest about mathematics. When you posted a purported "proof" by ChatGPT, I even exactly showed you that ChatGPT doesn't know what it is doing*. And that is only one example and not even the worst instance of ChatGPT's stupidity and dishonesty. I could give you as many examples as requested (I'm not going to though since I waste my time only when I feel like wasting it).

* You didn't acknowledge that.

(2) Of course ChatGPT wouldn't be able to understand purely text-symbol rendering of mathematical symbols (for example, if you don't know that 'E' is to stand for the existential quantifier then 'Exy' would be gibberish to you).

My definitions are very straightforward use of just the basic logical symbols and a few basic symbols of set theory.

And if you don't understand the notation (though I specified most of it explicitly in a post only a few posts prior), then you can ask.

And, by the way, the definition I gave is especially efficient as you don't have to read all the chapters in the topology book that lead up to the definition; but instead my definition is extracted from that context while being standalone.

(3) But wait, out of curiosity, I put my definition to ChatGPT verbatim, and ChatGPT replied:

This text outlines definitions related to topological spaces and various properties in the context of set theory and topology. Here’s a breakdown of the key components:

Topology: The first definition establishes the criteria for a topology T on a set
X. It requires that:

T is a subset of the power set of X (denoted PX).

The whole set X and the empty set must be in T.

Any union of sets in T must also be in T.

Any finite intersection of sets in T must also be in T.

Compactness: The second definition describes compactness in terms of a topology T on a set X. A set C is compact if every cover of X by open sets in T has a finite subcover. This emphasizes the idea that even if you have a potentially infinite collection of open sets that covers X, you can always find a finite number of these sets that still cover X.

Connectedness: The third definition characterizes connectedness. A set C is connected if it cannot be partitioned into two non-empty, disjoint open sets. This property ensures that C is "in one piece" in a topological sense.

Hausdorff Condition: The fourth definition introduces the Hausdorff property, which states that for any two distinct points p and q in X, there exist neighborhoods (open sets) around each point that do not overlap. This separation property is crucial for many results in topology.

Continuum: Finally, the last definition states that a continuum is a set that is compact, connected, and Hausdorff. This is a significant concept in topology, as it combines these important properties into a single structure.

Comment
These definitions are foundational in understanding topological spaces and their properties. They help formalize concepts that are essential in analysis and topology. By establishing clear criteria for these properties, they pave the way for deeper exploration into how different topological spaces behave and interact. If you have a specific area you'd like to explore further or questions about these definitions, feel free to ask!" — ChatGPT

that math is rooted in actual infinities. I just don’t see how applied mathematics has any need for or use of actual infinities. I see an alternative. — keystone

For Pete's sake again! The original post in this thread purported to prove that the continuum does not exist. The argument wasn't that it doesn't exist in nature but that it doesn't exist mathematically. The argument was bunk. Then there was discussion of the mathematics, so it was needed to give a mathematical definition of 'is a continuum'. And the definition itself makes no presupposition that there exists an infinite set. But I would bet that we can prove that if <X T> is a continuum then X and T are infinite. If you can provide an finitistic mathematics in which there are finite continua or in which there are no continua, have at it; but that is a task not completed by posturing with a bunch of undefined terminology.

In any case, I’ll provide the requested definitions in my next post. — keystone

Then I would be right to ask you to defined the undefined terminology in those definitions, and again until (1) You finally fail to reach primitives or (2) You end up in a circle or (3) You do reach primitives.

Here's a definition:

A continuum is a decomposition of a hyperspace module with sidewise-partitionable step-wise radii

decomposition: limitless regarding of tangents in non-Euclidean operatives

hyperspace module: arbitrary [1 3] transcendental intersections modulo mono-component recursions

sidewise-partitionable: bounded reaching of proximal antitheses in the quadrant independent ordinal numbers

step-wise radii: graph-based ultra-empty full-product numeric manifestations concentrically regarded

There, now I've really set the stage.

The following is not a formal, finalized definition, but I hope sets the stage for the discussion — keystone

Is what you wrote supposed to be informal unfinalized mathematics, or informal unfinalized mathematics infused with philosophy, or informal unfinalized philosophy, or informal unfinalized philosophy infused with mathematics, or other?

Stage for what discussion? Discussion of your stuff. Meanwhile, the stage for the overall discussion has included some actual mathematics.

If you're interested in more than just your own stuff, I suggest you read the mathematics I gave. Then you could compare your own stuff with the product of mathematicians deeply dedicated to the subject. That's a great place to start. Then, if someone such as you wants to advance on the beachhead of an alternative, then at least we can compare that alternative to where we stand on ground won by dint of the combination of profound mathematical imagination and heroic intellectual discipline.

Finite object: Finite in the sense that its complete set of attributes can be fully described without invoking infinite processes. — keystone

Define:

"complete set of attributes"

"can be fully described"

"invoking infinite processes"

Continuous object: In 1D, the proposed fundamental objects are of two types: (1) open-ended curves, which are inherently continuous, and (2) points. A composite 1D object is the union of these fundamental objects and is continuous if, when duplicates are removed, the following conditions are met:
Points are connected to 0–2 curves (but not to other points).
Curves are connected to 0–2 points (but not to other curves).
No objects are disconnected from the composite structure. — keystone

"open-ended curves" means "open curves"?

Define:

"inherently continuous"

"duplicates removed"

"connected to"

"0-2 curves"

"0-2 points"

Potential for arbitrarily fine partitioning: The continuum can be subdivided into an increasingly refined composite object made up of arbitrarily many fundamental elements, maintaining its continuity. — keystone

Define:

"refined composite object"

"increasingly refined"

"fundamental elements"

"maintaining continuity"

Characterized by the cardinality 2^aleph_0: The partitioning process can be described algorithmically, such that no algorithm can be devised allowing for further division. Although this algorithm would not halt if executed, the structure of the algorithm itself reveals that the potential for infinite subdivision aligns with the cardinality 2^aleph_0. — keystone

Define:

"partitioning process"

"algorithm devised allowing further division"

"algorithm reveals"

"aligns with"

I have refrained from providing examples or illustrations for the sake of brevity, though they could help clarify my position. — keystone

To clarify your stuff, the best thing to do would be to define your terms. If your stuff is supposed to be mathematical, then proposing to "define" by adducing yet more undefined, impenetrable verbiage that never resolves to primitives is a mug's game.

Do really think there are people other than yourself who can make sense of your verbiage? Who in the world is supposed to know what you mean by things like "refined composite object", "fundamental elements", "align with" in context of this subject?

If I've asked for definitions that exist already in the literature as clear and rigorous mathematics, then I would like to know where to look them up. Can you direct me to writings in which such verbiage is defined in the sense you use it? Can you tell me what the prerequisite readings are? Or is your stuff couched ultimately and merely as you wish to personally use words, with no way for people to inform themselves of the usage to follow along with you, either to accept your stuff or to show errors in it?

As I told you of books that would be a starting point for mathematics, what books would you direct me to that are starting points for keystonematics? If there are no books to recommend, then your best bet would be to tell me your primitives, your axioms and your train of definitions and proofs.

The axiomatic method is the opposite of exclusionary. Anyone may look at the primitives and axioms, which are stated right up front, and opt in at least to see what they lead to, or, on the other hand to reject them from the start. And when one does follow along to see where they lead, the process of seeing whether a purported proof is indeed a proof is objective; no one, not even the author, may rule by decree in that regard. Checking the purported proof is open to anyone. On the other hand, your style is quite exclusionary. You are the only one who knows (or thinks he knows) what in the world you mean by all that verbiage. You can fashion any arguments you want out of it and no one can verify that you're correct or demonstrate that you're incorrect, because the meanings are ultimately free floating in your own mind, unmoored to a clear (let alone rigorous) system of definitions.

Anyway, like I said, you would serve yourself well by understanding the topological meaning of 'is a continuum'. Then at least you could see how your stuff, from the left field of your ruminating, relates or not to mathematics.

https://thephilosophyforum.com/discussion/comment/934861

There has been unclarity in this discussion. Two concepts have not been held distinctly:

(1) the continuum (a noun)

(2) is a continuum (an adjective)

The use of those terms in mathematics is confusing to those who haven't read their definitions.

(1) I've given the definition of 'the continuum', though even that term itself is used in different ways:

(1a) the continuum = R

(1b) the continuum = <R L> where L is the standard ordering of R

(2) I mentioned that 'is a continuum' may mean different things in different contexts. But, at least in topology (which is central to this discussion), there is a clear definition. I'll give it here (hopefully, there are not many typos or mistakes):

Df.
T is a topology on X
<->
(T subset_of PX &
XeT &
AS(S subset_of T -> US e T) &
Akm({k m} subset_of T -> k/\m e T))

Df.
C is compact per X and T
<->
(T is a topology on X &
C = <X T> &
AS((S subset_of T & X = US) -> EF(F is finite & F subset_of S & X = UF)))

Df.
C is connected per X and T
<->
(T is a topology on X &
C = <X T> &
~EBC(BeT & CeT & B not= 0 & C not= 0 & B/\C=0 & X=BuC))

C is Hausdorff per X and T
<->
(T is a topology on X &
C = <X T> &
Apq(({p q} subset_of X & p not= q) -> EUV({U V} subset_of T & p e U & q e V and U/\V = 0)))

C is a continuum per X and T
<->
(C is compact per X and T &
C is connected per X and T &
& C is Hausdorff per X and T)

/

Th.
{S | S is an open subset of Q} is a topology on Q

Th.
{B | B is an open interval of Q} is a base for {S | S is an open subset of Q}

Th.
{S | S is an open subset of R} is a topology on R

Th.
{B | B is an open interval of R} is a base for {S | S is an open subset of R}

Show: ~ <Q {S | S is an open subset of Q}> is a continuum per Q and {S | S is an open subset of Q}

Show: <R {S | S is an open subset of R}> is a continuum per R and {S | S is an open subset of R}

/

Df.
(T is a topology on X & W is a topology on Y)
->
(f is an isomorphism between <X T> and <W Y>
<->
(f is 1-1 from X to W &
Aj(jeT <-> f(j) e W)))

I correctly mentioned that the system of real numbers is a complete ordered field and that famously it is a theorem that all complete ordered fields are isomorphic (applying the usual definition of 'isomorphic' for systems). But are all continuums isomorphic (applying the definition of 'isomorphic' above)?

/

This mathematics does not preclude other senses of 'is a continuum' that people may wish to specify in philosophy and science.

As I understand from this conversation:

Let:

Q for the set of rationals
Q_o for the set of open subsets of Q
Q_i for the set of open intervals of Q

R for the set of reals
R_o for the set of open subsets of R
R_i for the set of open intervals of R

Then:

Th. Q_o is a topology on Q
Th. Q_i is a base for Q_o
Th. Q is disconnected per Q_o

Th. R_o is a topology on R
Th. R_i is a base for R_o
Th. R is connected per R_o

The takeaway: R is connected per R_o but Q is disconnected per Q_o

/

For reference, definitions:

U for the 1-ary union operation. P for the power set operation. E for the existential quantifier. A for the universal quantifier. e for member_of. 0 for the empty set.

Df.
T is a topology on X
<->
(T subset_of PX &
XeT &
AS(S subset_of T -> US e T) &
Akm({k m} subset_of T -> k/\m e T))

Th.
T is a topology on X -> 0 e T

Df.
EX T is a topology on X ->
(B is a base for T
<->
(B subset_of T &
As(seT -> EJ(J subset_of B & s=UJ))))

Df.
B is a base for a topology on X
<->
ET(T is a topology on X &
B is a base for T)

Df.
S is an open subset of Q
<->
(S subset_of Q &
Ax(xeS -> Ey(yeQ & y>0 & Aj((jeQ & |x-j|<y) -> jeS))))

Df.
S is an open interval of Q
<->
(S is an open subset of Q &
Axyz((xeS & yeS & x<z<y) -> zeS))

Df.
S is an open subset of R
<->
(S subset_of R &
Ax(xeS -> Ey(yeR & y>0 & Aj((jeR & |x-j|<y) -> jeS))))

Df.
S is an open interval of R
<->
(S is an open subset of R &
Axyz((xeS & yeS & x<z<y) -> zeS))

Df.
T is a topology on X ->
(X is disconnected per T
<->
EBC(BeT &
CeT &
B not= 0 &
C not= 0 &
B/\C=0 &
X=BuC))

Df.
T is a topology on X ->
(X is connected per T
<->
~ X is disconnected per T)

Can you give me an example of an open set of rationals? — fishfry

If I'm not mistaken, no non-empty set of rationals is open in the reals.

I don't know what you mean by "isn't already contained in R". R is the complement of L in the rationals. No irrational number is in R...period. (By the way, I wish the letter 'R' weren't being used for the complement of L, since I'd like to use the letter 'R' for the set of real numbers.)

My definition is more streamlined than taking a Dedekind cut to be a partition; with my definition, a Dedekind cut is simply the lower part.

There are two different things: (1) Do you want to provide a definition of 'open' and 'closed' in the rationals? (2) Consider open and closed in the reals.

Regarding (2), the definitions are ('R' stands for the set of reals, and x, y, and j range over reals):

S is open in R
<->
S is a subset of R and
Ax(x in S -> Ey(y>0 & Aj(|x-j|<y -> j in S)))

S is closed in R
<->
c(S) is open in R

Then, if I did the math right (?), no nonempty set of rationals is open in R.

Without comment on the rest of your post, the very first claim is incorrect :

when open sets L and R are used to define a Dedekind cut L|R for an irrational number r, the generated closed set [r] is disjoint from both L and R, and yet their union equated with the continuum. — sime

With your letters:

The interval [r] is just {r}.

And L and R are sets of rationals only; no irrationals are members of L or R. The only irrational in Lu{r}uR is r. Lu{r}uR is not the set of real numbers.

You are flat out incorrect in claiming that Lu{r}uR is the set of real numbers.

With my letters ('Q' for the set of rationals, 'c' for the complement operation on sets of rationals, '<' for the less-than relation on rationals):

Definitions of 'Dedekind cut' differ in particulars, but one concise definition:

D is a Dedekind cut
if and only if
D is a non-empty proper subset of Q and
For all t and v, if t in D and v < t, then v in D and
D has no greatest member

Then the definition: x is a real number if and only if x is a Dedekind cut

So your irrational r is itself a Dedekind cut, call it 'D'. And your R is c(D).

So D u {r} u c(D) = D u {D} u c(D). The only irrational in d u {r} u c(D) = D u {D} u c(D) is r = D.

In plain words, you are flat out incorrect that the union of the lower cut with the one member interval and with the complement of the lower cut is the set of reals. Neither the lower cut nor its complement have any irrational members.

You conflated the cut (which is a set of rationals) with a real interval.

.

Your picture of all of this is much too woozy
— TonesInDeepFreeze

I am sorry if that is true — Gregory

Then you are sorry.

If Godel is widely misunderstood, the blame falls on those who explain it because i've seen many contradictory explanations of it — Gregory

The blame falls on those who presume to explain it but don't know what they're talking about. It's not Godel's fault nor the fault of those who do understand mathematical logic that there are ignorant, confused bozos that spout ill-premised misrepresentations of the mathematics. Reading well-written textbooks by people who understand the subject won't expose you to contradictory explanations.

Godel might have proven something about human conceptual thinking — Gregory

The incompleteness theorem itself is a mathematical theorem. It is not itself a philosophical take on the subject of human conceptual thinking.

The argument that @MoK gave involved the real numbers and their ordering, and real intervals, and his own confused notion of infinitesimals. He gave a definition of 'continuum' that sputtered. And he argued that the reals are not a continuum. His arguments were a morass. And given his personal definition of 'a continuum', he was refuted that the reals are not one.

Define 'continua'. Preferably a mathematical definition. And most preferably not free-floating, hand-waving verbiage.

There is no consistent formal theory that proves all the arithmetic truths. But it's not the case that there is an arithmetic truth such that there is no consistent formal theory that proves that arithmetic truth.

It would seem that incompleteness entails that there is no consistent formal axiomatization whose axioms are all logical truths and that proves all the arithmetic truths. Though there may be differences as to what 'logicism' means.

But incompleteness entails even more. Incompleteness entails that there is no consistent formal axiomatization that proves all the arithmetic truths.

Moreover, David Hilbert hoped that a finitistic proof of the consistency of infinitistic mathematics would be found (Hilbert's program). That hope was "shattered" by the second incompleteness theorem. Interestingly, Godel was trying to find such a finitistic proof, but he saw that he could prove the incompleteness theorem thus the second incompleteness theorem too. He ended up proving the opposite of what he had started set out to prove. Later, Bernays and Hilbert (mainly Bernays?) provided the details Godel left out in proving the second incompleteness theorem. When Godel saw Alan Turing's formulation of the notion of computability, Godel recognized that the second incompleteness theorem does indeed preclude Hilbert's program. Moreover, the second incompleteness proves that the consistency of PA (and even Q, i.e. Robinson arithmetic) does not have a finitistic proof. However, Gentzen did prove the consistency of PA with (for lack of a better term) "quasi-finitistic" means.

Those are great logicians, great intellectual achievements. And a lot more (not necessarily in chronological order): Predecessors: Boole, De Morgan, Peirce, Cantor, Peano, Dedekind, Frege. Then Lowenheim, Skolem, Whitehead & Russell, Zermelo, Fraenkel, Church, Tarski, Lukasiewicz, Lesniewski, Post, von Neumann, Rosser, Kleene, Herbrand, Henkin, Hintikka, A. Robinson, Montague. And the constructivists: Brouwer, Heyting, Markov, Kolmogorov, Curry. Then Kripke models. And Shelah, Friedman, Woodin, Silver, Martin ... so many ... Rich intellectual history.

unprovable assumptions — Gregory

Your picture of all of this is much too woozy.

You're referring there to MoK. He argued that the continuum does not exist. I don't recall that he mentioned paradox (maybe he did?).
— TonesInDeepFreeze

I should not have used the word 'paradoxical' but rather logically impossible. — keystone

I don't recall the notion of logical impossibility being mentioned (maybe it was?). However, of course, if from certain premises we derive that the continuum does not exist, then that contradicts the claim that the continuum does exist. But the point of the argument by @MoK was to first simply show that the continuum does not exist. That argument by him was shown to be ill-premised and confused.

point-based — keystone

Define 'point based'.

In greatest generality, a point is a member of a set.

Df. the continuum = <R L> where R is the set of real numbers and L is the standard ordering on the set of real numbers.*

So, of course, there are points involved.

* Perhaps a more common definition is:

Df. the continuum = R where R is the set real numbers

But, along with some authors, I prefer to explicitly mention the ordering, especially as usually when we talk about the continuum, we have not just the set but also the ordering in mind.

I'm not sure who Mok meant by other poster but I assumed it was you. For example you wrote the following:

An ordinary mathematical notion is that the continuum is the set of real numbers along with the standard ordering of the real numbers; then a continuum is any set and ordering on that set that is isomorphic with the continuum. — keystone

In other posts, I emphasized definitions of 'the continuum' and 'continuous function'. But lately I overlooked that I also defined 'a continuum' as above. I have qualms about that definition of 'a continuum'. It might be correct - equivalent with other definitions around - but I realize now that I'm not completely sure. Definitions of 'a continuum' vary, and it seems, based on context.

So, I'll remove to the safer ground of my definitions of 'the continuum' and 'continuous function' and leave 'a continuum' alone.