[2] Continuum hypothesis. There is an insurmountable gap between aleph-0 and aleph-1. There are no infinite cardinalities in between. — alcontali

This is true by the definition of the Aleph numbers. It's not a statement of the continuum hypothesis. I hope we can clarify this if we're going to talk about CH.

There is no transfinite cardinal between $\aleph_0$ and $\aleph_1$, period. It's a matter of definitions within ZF, Zermelo-Fraenkel set theory without the axiom of choice.

$\aleph_0$ is by definition the cardinality of the set of natural numbers.

$\aleph_1$, by definition, is the cardinality of the set of all distinct well-order types of countable sets. $\aleph_2$ is the cardinality of the set of all distinct well-order types of sets of cardinality $\aleph_1$, and so forth. If you don't know what that means that's ok, I don't want to get sidetracked by technical detail. All that's important at the moment is that each Aleph number has a specific technical definition such that the they form a sequence of discrete cardinalities, one after the other. There can be no Alephs strictly between two Alephs simply by virtue of their definitional construction. That's the key point. They're defined such that there are no "in-between" Alephs.

Now consider the real numbers. We can code each real number (between 0 and 1 for simplicity and without loss of generality) as an infinite binary sequenc; a bitstring with an implied binary point in front. It's clear that a real number is essentially a function from the set of natural numbers to a set with two elements (zero and one). Such a set (again omitting technical details) has cardinality $2^{\aleph_0}$, which is defined as the cardinality of the set of all functions from the natural numbers to a two-element set.

So $2^{\aleph_0}$ is SOME cardinal; and if we assume that every cardinal is an aleph (this actually depends on the axiom of choice) then there must be some ordinal number $x$ such that $2^{\aleph_0} = \aleph_x$. Which $x$ could that be?

The continuum hypothesis is the claim that $2^{\aleph_0} = \aleph_1$,

The generalized continuum hypothesis, GCH, is the claim that $2^{\aleph_\alpha} = \aleph_{\alpha + 1}$ for every ordinal $\alpha$.

[If you've never seen the ordinals they're the coolest thing ever. They show up in logic and sometimes computer science, especially in the theory of hypercomputation].

As far as anyone knows, $2^{\aleph_0}$ could be huge -- it could be $\aleph_{100000}$ or worse. Gödel thought that it was $\aleph_2$. Cohen thought it could be much larger. Or ... and there's a lot of support for this position ... the question could be entirely meaningless.

Your idea isn't entirely without merit. Cohen wrote that intuitively, exponentiation is a much more powerful operation than taking successors. For example going from $2^4$ to $2^5$ is a much larger jump than going from $2^4$ to $2^4 + 1$. (My example, not Cohen's). By that analogy, CH would be false. That's essentially your argument I think. You're analogizing with the permissible cardinalities of finite fields. It's not a bad idea as far as it goes if you find it helpful.

Some of what you wrote is a little off the mark IMO but I wanted to nail down the definition and meaning of CH first. What you call uncanny I call "a stretch" not justified by the math. I accept your analogy only insofar as it's helpful to you personally.

ps ... this also really jumped out at me.

[1] A Reinhardt cardinal, [2] An n-huge cardinal, [3] A huge cardinal, [4] An extendible cardinal, [5] A supercompact cardinal, [6] A superstrong cardinal, [7] A Woodin cardinal, [8] A measurable cardinal, [9] A (strongly) inaccessible cardinal.

The complexity proposed is substantially beyond what is done in existing, established theories. Therefore, what he is doing, looks quite ... ambitious. — alcontali

Those technical terms are the meat and potatoes of 20th century set theory, especially of the last thirty or forty years. They're not "beyond existing theories." They're just beyond what you've heard about. Measurable cardinals for example were studied in 1930 by Stefan Banach. An inaccessible cardinal was implicitly used by Wiles in his proof of Fermat's last theorem, though specialists believe his proof could be done without it. Advanced set theory doesn't make the TED talks and math blogs but it's out there. And it's really "out there."

That's a good point but are your hands moving relative to your keyboard when you type something. I think we can begin there too. — TheMadFool

This is an interesting comment. I could say, suppose the light switch is in the next room and an evil demon electrician is flipping it? There's motion even though we can't observe it. In fact if I understand your point, you are saying that if there is change there must be motion. Is that true? Suppose instead of a light switch and an electrician, there's an electronic timing circuit? There's no motion, unless you count the vibrating electrons. But electrons vibrate even when there's no change! So I don't know if this idea is true. There can be change without mechanical motion, that's what the electronic revolution is all about. What do you think?

Finite fields (in which arithmetic is permitted) must have a prime-power size. Therefore, there tend to be gaps between permissible calculation field sizes. E.g. a size of 13 is allowed, but 14,15 are not; 16=2⁴ is a prime power and is again allowed, and so on. So, in a way, we could wonder why it would be any different -- the fact that there are gaps -- in between infinite field sizes in which arithmetic is permitted? Of course, this kind of pattern is not a proof, but it seems to point in a particular direction. — alcontali

The Alephs aren't fields. Finite fields have absolutely nothing at all to do with this. They're apples and rutabagas.

But then again, since CH is provably not provable/disprovable from ZFC, proving CH from any alternative set of axioms, that is not just itself, may not be particularly "simple". It would require a meaningful distance, i.e. a meaningful number of non-trivial derivation steps from these axioms to CH, while these axioms would also have to be provably independent of ZFC. I do not see how else anybody could make progress in CH? — alcontali

One can Google around. A lot of work has been done by Woodin, Hamkins, and other contemporary set theorists. Work on CH has been ongoing for decades. It just doesn't make the mainstream news.

One big new idea is Woodin's Ultimate-L. It's so new and so technical it doesn't have a Wikipedia entry. This MathOverflow thread has some references. Nothing in this topic is comprehensible to laymen, just mentioning it since it's the state of the CH art.

https://mathoverflow.net/questions/269780/the-ultimate-l-in-a-nutshell-on-descriptive-articles

Another idea is Hamkins's set-theoretic multiverse.

http://jdh.hamkins.org/themultiverse/

Here's an accessible article that surveys the modern developments.

https://www.ias.edu/ideas/2011/kennedy-continuum-hypothesis

According to Cantor's theory, the infinities (Beth numbers) (or Aleph numbers) are a series of successive numbers, with the smallest one being countable infinite, the next one uncountable infinite, and each successive infinity, inf[n] = 2^inf[n-1], the cardinality of the power set of the previous one. Such sequence of infinities is not particularly compatible with the idea of one God, which would correspond to one infinity. — alcontali

Cantor thought that the limit of that process, essentially the union of all the infinite cardinals, was God. God is the absolute infinity. Cantor's theology is forgotten today but it was important to him. This would be relevant if you're trying to use set theory to make a theological argument.

In any event, there are many uncountable cardinals. All the cardinals other than the countable one Aleph-0 are uncountable. The Beth numbers are the Alephs in the presence of the continuum hypothesis; they're distinct in its absence.

We know for certainty that we, A, are in motion relative to the sun, B because we have night and day, seasons. — TheMadFool

I'm not sure this is sound. If you turn a light in your living room on and off, you have "day and night," but that's not proof of relative motion between you and the lamp. Likewise if you turn the thermostat higher and lower you have "seasons." Night and day and seasons are not proof of relative motion absent other facts, such as ... well, such as the relative motion of the earth with respect to the sun. Besides, night and day don't require relative motion between the earth and the sun, it only requires the earth to rotate on an axis. Or, for someone to be messing around with the light switch.

When the population overreaches its capacity, there’s going to be much more deaths. — Purple Pond

Malthus said that 200 years ago and Erlich said it in the 1960's. Both turned out to be wrong. Why should I believe you today?

Of course from a numeric perspective, there are more deaths every day since the population is increasing. But that's not an argument, it's just an observation that everyone dies and the more people there are, the more people die in absolute numbers. Did you have a more nuanced point to make?

Tell that to the people who died in the pandemic, the famine, the tidal wave, the war, the sinking boat, the earthquake... — Bitter Crank

Sinking boats and earthquakes support the idea of ecological collapse? What are you talking about? People do get hit by city buses, I'll grant you that. Totally did not follow your point.

It requires monumental stupidity for a species to paint itself into such a corner that it depends on some future technology that might never materialize to stave off an existential threat. — RogueAI

We should have stayed in caves? I don't follow your point. I've got 200,000 years of human progress on the side of my argument. You've got 200 years of failed doom and gloom predictions going back to Malthus and spectacularly exemplified by Erlich.

Again, for the third time I spell it out: I only contest your ability to specifically predict they will be here in 20 years — god must be atheist

Slow day around here?

You did NOT talk about fusion. You mentioned cold fusion, a technology that's never been conclusively demonstrated to work at all. Of course you're right, I did not mean 20 years literally to the day, just indicating a likely or possible general time frame based on the high interest of governments and private startups, and the status of currently successful pilot projects.

Therefore as far as I can see, the 20-year prediction is yours.

How do you do it? — god must be atheist

Other sources. I've been reading up on fusion power lately. You're acting like those people in 1995 who said the Internet was a fad. Keeping up with technology news is not like picking lottery numbers.

what's the winning combination of Lotto 6/49 in five weeks? And who won the World Series in 2032? — god must be atheist

That's silly. Fusion is a technology currently getting a lot of government and private investment. Experimental fusion reactors are already generating power. If I said that AI and robotics will be important in the future would you make that same disingenuous remark? Read the article I linked.

Well, if someone creates cold fusion reactors, we've got it made. For another 100,000 years,then the same problems will rear their ugly rears. — god must be atheist

Regular old fusion is on the 20 year horizon. It runs on seawater. No risk of meltdown or runaway chain reaction.

https://www.cbsnews.com/news/ten-serious-nuclear-fusion-projects-making-progress-around-the-world/

All of Erlich's predictions were wrong. He lost all his resource price bets. I'd say the same will happen in the future. Human ingenuity will defeat doom and gloom as it has for thousands of years.

I don't mean for this to sound glib or cute. I am trying to understand your use of the term meaningful. Is discussing the upcoming American pro football season meaningful? Is eating lunch meaningful? Is being born, living a life, and dying meaningful? Is the universe meaningful?

After I understand those things, I can attempt to ponder whether discussing metaphysics is meaningful. Ultimately it's something you can do to pass the time "between the forceps and the stone," as Joni Mitchell once sang. And discussing metaphysics is certainly less harmful than, say, shooting up a Walmart. But is either act meaningful? All those people were going to die anyway, as is everyone else who was at the mall that day, and all of us. What is meaningful?

When he talks about religion, however, his views are just a rehash of Christian-Lebanese political tactics. — alcontali

Your perspective is interesting. I enjoy his books a lot and also his irreverent and "I'm smarter than everyone else" style. On learning that he has some religious beliefs that blend in with some politics that someone has an opinion on, my instinct is to not care. I allow people to have their personal beliefs independently of my enjoyment of their professional work. Not being particularly religious myself, I'm never surprised at the kind of things religious people believe. This could be take to extremes, of course. I wouldn't be an admirer of Hitler's watercolors. But I can and do enjoy the work of many people who have views I don't subscribe to and don't feel like investing the energy to learn about. Taleb's a math guy. A lot of math people have unpopular or odd political opinions. The mathematician and prolific textbook author Serge Lange was an HIV denier of some sort. The great physicist Werner Heisenberg worked on German's atomic bomb project. Should we be troubled by this whenever we think of his famous uncertainty principle? It's a good question. These days some people believe that the Betsy Ross flag (just to pick one example that comes to mind) is racist and should be #cancelled as they say. It's in the air.

But if you don't assume the idea that everything is made of parts, you don't have to assume the unintuitive :wink: idea that a line is made of points: you can think of lines and points as two different types, and an intersection of two lines as a function from pairs of lines to points. — Mephist

The revenge of Russell. Type theory triumphant.

I'm all typed out, no pun intended, nothing else to say at the moment.

I completely agree. — Mephist

I should quit now!

I just wanted to say that in the cold light of day I no longer believe what I wrote last night. If you start with classical math and remove LEM you get some form of constructivism. If you add LEM back in, you get back classical math. So there's much less than meets the eye when you say you can add in axioms to recover classical math.

Secondly, you said you can import a library into Coq to recover standard math. But if that were true, standard math would be computerizable, and then that would remove the biggest benefit of constructivism.

I confess to being just as confused as ever, but at a somewhat higher level.

It's too hot to type here today so I'll save the rest for later.

ps:

But proofs of existence are algorithms, in any formal logic. — Mephist

Ok let me just try to get some clarification. If I use the axiom of choice to whip up the Vitali set, the set of representatives of the equivalence classes of $\mathbb R / \mathbb Q$, that set is not "constructive" by any conceivable definition of the word.

But if you code up the axioms of formal ZFC into a computer, the proof of the existence of the Vitali set can be cranked out by a program. Even by an undergrad! So in that sense, proofs are constructions even if they are constructing things that are in no way constructive.

Have I got that right? What I mean is, am I confused about something sensible?

I don't remember which one is the SEP article. Could you send me a link? — Mephist

https://plato.stanford.edu/entries/mathematics-constructive/

They echoed many of the things you've been saying, such as that there are several (four in fact) variations of constructivism, and that there's a constructivist logic ... well basic stuff, but stuff I've been reading without comprehension before and can now understand. Also I've learned that a lot of this goes back to the nineteenth century. Poincaré for example was troubled by nonconstructive thinking.

That's why you should never be frustrated about not being able to get through to me. You are getting through by osmosis. I've never bothered to try to understand constructivism at a technical level before and this is an education. The article talked about putting explicit bounds on the rates of convergence of Cauchy sequences, and that related back to the Italian paper, so it's starting to feel familiar. One learns by repetition.

I will say one thing though. I still just don't get constructivism as a philosophy. I DO understand that certain formulations of constructive math lend themselves beautifully to computerized proof systems. Nobody would deny the digital deluge that's flooding the world right now, why should math be spared?

But I don't get the metaphysics. "A thing only exists when we have an algorithm to instantiate it." But the world is full of things that have no algorithm. The world itself is not an algorithm.

And here I think is the source of my discomfort with constructivism. Many these days believe that the world IS an algorithm. I disagree with that proposition. To the extent that some -- not all, ok, but some -- aspects of constructivism are in support of the belief that everything that exists must be an algorithm; I oppose that aspect. I instinctively believe that the constructivists are missing a lot: namely, everything that is important but that can't be explicitly constructed.

Yes, the AC can't be construed as a computation, and it's not part of constructivist logic. What I am saying is that AC can be added to a constructivist logic framework such as Coq if you want to use standard logic: standard logic can be obtained as an extension of constructivist logic (in a similar way as metric geometry can be seen as an extension of projective geometry): if you want to use classical logic in Coq (and that's how it's used most of the time) you just have to import the Classical_Prop library: — Mephist

You've said this to me probably several times and I finally understand it. If I've got this right, you can add various axioms to constructive math and recapture standard math. So we get the benefit of computerization and we can still do the things standard math cares about. Which is pretty interesting, if true.

Ah, "Classical_Prop." Coq has a library you can import that makes it act like standard math. You've connected adding axioms in a formal system, with with importing a library in a program. Wow. Thanks. Great insight. I got it.

I don't know much about the guy. I read today that he once said of Hillary Clinton:

She’s got dyed blonde hair and pouty lips, and a steely blue stare, like a sadistic nurse in a mental hospital.

https://www.theguardian.com/politics/2015/feb/11/boris-johnson-sings-hillary-clinton-praises

Works for me!

In this sense the axiom is an oracle: it allows you to compute f — Mephist

As I think of it, perhaps a new axiom is an oracle. I understand that point of view.

So I'm willing to be agreeable on this point, but still no less confused on the issues I've raised earlier.

Also ... no form of the AC could possibly be construed as a computation. That, I just do not see at all. It's a pure statement of existence. A particular set exists with no means to possibly compute or construct or identify any of its elements. Whether this is worse or the same as the powerset axiom I can't say. Perhaps it's no worse. But a lot of people historically regarded it as worse. I can't put any of that in perspective.

I've re-read the SEP article on constructive math with a much deeper level of comprehension than I've had in the past. Your efforts have not been in vain.

I have never done anything "useful". No discovery of mine has made, or is likely to make, directly or indirectly, for good or ill, the least difference to the amenity of the world. (Hardy) — alcontali

Hardy was a number theorist. At the time he wrote those words, number theory was regarded as beautiful but useless. Today it's the mathematical foundation of public key cryptography, underlying all Internet security and cryptocurrencies. I wonder what he would say if he came back and discovered that his belovedly useless number theory was intensely studied by the spies at the NSA.

For a discussion of the pragmatics of the axioms of set theory, see Penelope Maddy's Believing the Axioms parts I and II.

https://www.cs.umd.edu/~gasarch/BLOGPAPERS/belaxioms1.pdf

https://www.cs.umd.edu/~gasarch/BLOGPAPERS/belaxioms2.pdf

I guess I got confused because you guys were talking about numbers as points, where I was thinking about zero-dimension points on a two-dimensional line. This stuff to me right now is very esoteric, as I don’t remember the terminology for the different kinds of numbers. I was always better at calculations like an engineer than I was so much interested in or ever had any exposure to theory. — Noah Te Stroete

A line is only one dimensional. A plane is two dimensional. Points are zero-dimensional. How a bunch of zero dimensional points make a line is a bit of an ancient mystery. Newton thought of a point as tracing out a curve as the point moves through space. But that only pushes the mystery to space itself.

There's really only one standard real number line that most people care about, the one from high school math used by physicists and engineers. The other stuff is esoteric. Constructivism is the idea that every mathematical object must be able to be explicitly constructed. That leads to a different kind of notion of the real line. But I really can't speak for constructivist philosophy, since I don't know much about it. You can see the difficulties @Mephist is having in explaining it to me!

↪fishfry Can’t a real number go on and on for infinity? Or is that an irrational number...? — Noah Te Stroete

You mean the decimal representation of a real number? Yes, they're infinitely long. For example 1/3 = .333... and sqrt(2) = 1.4142... and so forth. Even terminating decimals like 1/2 = .5 = .4999... have infinitely long representations.

But the representation isn't the real number. The real number is the abstract thing pointed to by the representation, just like 2 and 1 + 1 are two different representations of the same abstract number.

Well, thank you for trying to explain it to me. I guess I’m really not interested enough to do further studying. It seems like a meta question or concern whose answer may have little consequence... but it may just seem that way to me because of my ignorance.

I apologize for the intrusion. — Noah Te Stroete

No intrusion at all, I love to talk about the standard real numbers! I'm not sure which part of your question I didn't clarify, I'd be glad to try again if you have more questions. As I understand it you asked how the real numbers can be made of points. You take all the real numbers and collect them into a set. That's the set-theoretic viewpoint. So the real numbers are made of points just as the set of natural numbers {0, 1, 2, 3, ...} is made up of the individual numbers 0, 1, 2, 3, ...

There are no gaps in the real numbers, but i'm not sure which part of this you are asking about.

A line is made of points with no gaps? Now I don’t know any theory, but what I intuitively grasp from taking calculus is that a point has zero dimensions. A line is not made of points. A point just divides a line into two segments. — Noah Te Stroete

In the standard set-theoretic account of the real numbers, the real line is made of points. That is, the line is the union of all the sets that each contain one point, if you think of it that way.

That doesn't mean there's a point and then a "next" point. They're not lined up like bowling balls. Between any two real numbers there's another. In fact between any two real numbers, the points between can be stretched into a new copy of the entire real line. So it's like maple syrup, not bowling balls if that helps.

There are no gaps. "Every Cauchy sequence converges." That's the technical condition that ensures there are no gaps. It means that every sequence that should converge, does converge.

As a helpful example, consider the rational numbers. Between every two rational number there is another. But there are still gaps! For example the sequence of rational numbers 3, 3.1, 3.14, ... that converges to pi, doesn't converge in the rationals. There are gaps in the rationals at every irrational.

But the reals have no gaps. That's their defining property. All the gaps in the rationals are filled in.

I can't speak to constructivist or intuitionist conceptions of the real line. And there's also the hyperreal line, in which every real number is surrounded by a cloud of infinitesimals.

Forgive my elementary level knowledge on the subject. — Noah Te Stroete

It's a deep question, the nature of the mathematical continuum. People have been arguing about it forever.

It's not about verifying proofs. In every formal logic proofs can be verified mechanically, otherwise it wouldn't be called "formal" logic. It's about the complexity of rigorously written proofs. In ZFC complete formal proofs become so long that nobody uses them in practice. In Coq the situation is much better. In HOTT is even better.
You don't have to abandon standard math to use Coq or HOTT. You can use HOTT to do perfectly standard math with a much simpler formalism for proofs: that's why it's proposed a new foundation of ALL mathematics. — Mephist

Yes yes this is the part I understand. Coq is a tool for helping to avoid errors in published math. That was Voevodsky's idea.

It's the mystical mathematical metaphysics of Brouwer I don't understand. The claim that every mathematical object should come with a recipe for building it. I've never been excited by that idea. It seems unnecessarily restrictive. After all there is no reason to believe the world is a computer, no matter how many TED talkers assert the proposition.

I
I think I'll give up with this discussion because I see that is leading nowhere! — Mephist

On the contrary. It's been very valuable to me. It's just that I have no more points to make. And you've tried to explain things to me that I haven't understood. The fault is mine.

I
The intuition of a line being made of points and having no gaps is VERY unintuitive, and it's NOT used in standard mathematics: integrals and derivatives are defined as limits of sequences: no sets of points at all. But I am sure I can't convince you about this and I don't see the point of going in circles repeating the same things... — Mephist

You're right. You could never convince me of that since you're wrong. What if there were a sequence that morally should converge, but there was no limit there? That's the point of Cauchy completeness. You need it to ensure the existence of the limits you need for your integrals and derivatives.

But I sense that you are unhappy or frustrated. At my end I've found this very enlightening and helpful. I appreciate the conversation. If we're done that's good. I may peruse your links from time to time.

You can use exactly the same definition of Cauchy-complete totally ordered field in constructivist logic. Even rational numbers are locations on the real line. The problem is with continuity: points are not "attached" to each-other, right? — Mephist

Right. The real numbers are a continuum. The rational numbers aren't because they're not Cauchy complete. The computable reals are not because they are not Cauchy complete. (I proved this earlier). And the constructive reals ... well I can't say, because I still don't understand the difference between the computable reals and the constructive reals. That's another point of mystery for me.

But even in the standard reals the points are not attached to each other like bowling balls. It's more like infinitely stretchy taffy. You can take any two points, no matter how close together, and stretch the line between them to any length you like.

If you read my posts I have always said the same thing: constructivist logic DOES NOT MEAN assuming that only computable functions exist! — Mephist

Ok. I will stipulate to being thoroughly confused on this point. But that's ok.

If somebody that is reading this knows about constructivist logic and thinks that this is not true, PLEASE reply to this post and say that it's not true! — Mephist

I'll stipulate you are correct on this point. I just don't understand the mechanism of construction. Perhaps that's what I'm missing.

P.S. Now I know what computable reals are, but I still don't have a definition of non computable reals: I imagine that you mean functions from naturals to naturals that are not computable — Mephist

A computable real is a real number that is computable in the sense of Turing 1936. A noncomputable real is a real that isn't. This is perfectly sensible, isn't it?

And of course you can identify noncomputable reals with noncomputable maps from the naturals to themselves, just as you can identify all the reals with that set.

P.S. Maybe I'll try to explain the difference between assumptions and definitions (I know that it's an elementary concept, but maybe it's not so clear).
In constructive logic I can write "Let FF be a function from natural numbers to natural numbers." This is an assumption.
I know nothing about FF: it can even be non computable.
Then I can write "Definition: GG is the function that takes a number xx and returns the number x2+F(x)x2+F(x)".
Now, if FF is not computable, even GG will be not computable, but I don't need to know how FF is defined in order to define GG. I can compute 2∗G(x)−F(x)2∗G(x)−F(x) and obtain 2x2+F(x)2x2+F(x).
F(x)F(x) will never be expanded or computed, because is treated as a "black box" that I assume to exist but can't be used to calculate actual natural numbers. — Mephist

Is a black box like an oracle, a device that can solve a noncomputable problem?

So, the point is that I can assume the existence of non computable functions, but I must use computable functions in my definitions (in this case the square and the addition functions). — Mephist

Ok, you can refer to noncomputable functions and then computable expressions involving them. That makes sense.

In standard logic this is not true, because I can use the axiom of choice even in my definitions. — Mephist

You haven't related anything to the axiom of choice, I totally didn't get this reference here. It doesn't seem to apply to anything you said. You don't need the axiom of choice to have noncomputable functions.

In constructivist logic instead you shoud add the axiom of choice as an assumption (or axiom), and then you can use it as an "oracle function" ( a black box, such as F(x) ) inside your definitions. — Mephist

I think you are using AC in a funny way. I tried to correct a couple of instances of this earlier. Nothing we're talking about has anything to do with the axiom of choice.

I do not believe that an oracle in computer science is in any way analogous to the axiom of choice. If you know of a connection perhaps you could explain it. An oracle allows you to compute something that was formerly noncomputable. The axiom of choice doesn't let you compute anything. In fact the elements of the sets given by the axiom of choice have no properties at all and could never be computed by any stretch of the word.

At the end, using constructivist logic you can even do exactly the same proofs that you do in standard logic, but you have to add the appropriate axioms explicitly as assumptions, because they are not built into the logic itself. — Mephist

I believe you but there's a kernel I'm missing. If the constructive reals let you prove all the same theorems, what is the point? What's wrong with the original proofs? And by "built into the logic," you have a lot of meaning for that phrase in your mind but I don't know what you mean.

Perhaps you need to make your point more simply because I'm not seeing it at all.

NO! A constructive real DOES NOT REQUIRE a computable Cauchy sequence!
ALL Cauchy sequences of rational numbers (computable AND INCOMPUTABLE) are PERFECTLY VALID real numbers in constructive logic. — Mephist

Ok that said. This is different from what I understood from the Italian paper, which is that a real number is characterized by (or identified with) a funny-Cauchy sequence defined by a particular rate of convergence that allows us to show that the modulus of the sequence is computable. If there's anything technical I got from this thread it was exactly that.

But I understand as you've explained to me that there are many different approaches to constructivism; and that in some of them, all Cauchy sequences of rationals represent real numbers, not just the computable ones.

I believe you if you tell me this. But the collection of all [equivalence classes of] Cauchy sequence of rationals is exactly the standard real numbers.

So I believe you if you tell me, but actually, I don't believe you. Because you've just constructed the standard real numbers. So I'm confused again.

NO! A constructive real DOES NOT REQUIRE a computable Cauchy sequence!
ALL Cauchy sequences of rational numbers (computable AND INCOMPUTABLE) are PERFECTLY VALID real numbers in constructive logic. — Mephist

Ok, I see that I said something that induced you to write in caps. Before responding to the interesting technical points in your following posts, let me just go meta for a moment.

I'm trying to wind down my involvement in this thread.

* I have not said anything new for quite some time. *

* My next move, if I were interested enough to make it a priority, would be to dive into the wonderful references you've been giving me all along.

So if it happens that I'm totally wrong about constructivism, totally wrong about everything for that matter, it's ok with me. I've learned more than I ever knew about constructivism and it still baffles me. I understand Coq and the desire to mechanize the checking of published proofs so as to avoid error. That was Voevodsky's original motivation.

I DON'T necessary understand the mathematical metaphysics that seems to accompany neo-intuitionism. I've looked into Brouwer a little bit over the years and I confess I just don't get it. Why tie your hands with constructibility? Much less computability, which is even more restrictive. Computability is trendy right now. After all we mastered the technology only in the past 50 years, and it wasn't tell ten years ago that everyone started walking around with a supercomputer in their pocket named after a fruit. So I'll forgive a certain amount of computational metaphysics. "We're a simulation," "We can upload our minds to a computer," etc. It's nonsense but it's perfectly understandable. Just as everyone thought the world was a machine in the century after Newton. It will pass. Turing showed that there are naturally expressed problems that can't be solved by computer. This lesson hasn't been internalized in our zeal to reinterpret reality in terms of the computable.

Hence neo-intuitionism. The revenge of Brouwer. I'm all for it.

But I just haven't got any level of advocacy or passion in this thread at the moment, since I'm perfectly well out of things to say. So I'm startled to see that anything I said induced a response involving a lot of upper case typing. I assure you that eliciting such a response was not my intention. I'm not dogmatic about anything, I've pretty much said my piece and learned some things.

I'll try to comment on your posts but I'm most definitely not wanting to engage in any disagreements that involve a lot of capitalization.

OK, so I have a question: what is this morally correct model of real numbers? The set of all infinitely long decimal representations? — Mephist

I can live with that. In fact I've read something along those lines in some of the constructivist references. The decimal representations (modulo dual-representations, of which there are only countably many) are a perfectly good model of the reals. They have the least upper bound property so they are in fact the standard reals.

I wanted to comment on the concept of moral correctness. In 1940 ‎Gödel developed the constructible universe, denoted $L$, which is a model of ZF in which both the continuum hypothesis and the axiom of choice are true. In so doing, he showed that these two propositions are at the very least consistent with ZF. [In 1963 Cohen proved that their negations are consistent as well, showing their full independence].

[Note: The constructible universe is a proper class and not a set, therefore it is not actually a model of anything. I've always been confused on this point].

Now everyone could just go, well, we have a nice model, it's a nice universe of sets that decides the truth of these two propositions. Why don't we just assume that $L$ is the true universe of sets and be done with it?

That is, if we denote by $V$ the von Neumann universe, why don't we just adopt the axiom $V = L$, the axiom of constructibility?

The fact of the matter is that we don't do that. Mathematicians feel that $L$ is not the "morally correct" universe of sets. For one thing it doesn't have enough sets.

This is a good example of the concept of moral correctness: That although mathematicians may act like formalists when they're writing proofs (or computerized proof verification systems), deep down inside they are all raging Platonists. Nobody does math because they truly believe it's a meaningless formal game like chess. Mathematicians believe that their work is about something. They have intuitions about what these things are; and just because there is a funny model with some property, they do NOT stop there. If the model doesn't match the intuition, the model becomes a handy formal device but the search for mathematical truth continues unabated.

I had an idea to solve the question about Cauchy completeness, that I should have had a long time ago: just look at the book that proposes constructive mathematics as the "new foundations"! — Mephist

Ok second my replies to your two longer posts of 6 days ago.

Here's the book: https://homotopytypetheory.org/book/ — Mephist

Ok true confession I haven't had a chance to look at that yet but it's on my reading list.

Chapter 11.3 Cauchy reals:

"""
There are three
common ways out of the conundrum in constructive mathematics:
(i) Pretend that the reals are a setoid (C, ≈), i.e., the type of Cauchy sequences C with a coincidence relation attached to it by administrative decree. A sequence of reals then simply is
a sequence of Cauchy sequences representing them. — Mephist

Setoid. Buzzword of the day. From Wiki:

"In mathematics, a setoid (X, ~) is a set (or type) X equipped with an equivalence relation ~."

Well that tells me nothing at all. Given a set there are lots of equivalence relations on it. I assume they must mean some particular equiv rel of interest to someone. Type theorsist or constructivists or such. But what you quoted didn't explain much I'm afraid, it seems to be missing details. What's a coincidence relation? There's a key detail missing. But ok, onward.

(ii) Give in to temptation and accept the axiom of countable choice. After all, the axiom is valid
in most models of constructive mathematics based on a computational viewpoint, such as
realizability models. — Mephist

Ok this relates to something I ran across, that Cauchy and Dedekind completeness are equivalent if you accept countable choice. I haven't understood yet why that is but I gather it's not too difficult. And if you accept countable choice, which a lot of constructivists do, then things are not so bad. Countable choice (or sometimes dependent choice) is often enough for the theorems we care about.

(iii) Declare the Cauchy reals unworthy and construct the Dedekind reals instead. — Mephist

Unworthy! LOL. "We're not worthy!!"

Such a verdict is perfectly valid in certain contexts, such as in sheaf-theoretic models of constructive
mathematics. However, as we saw in §11.2, the constructive Dedekind reals have their own
problems. — Mephist

As far as I can see, the constructivists acknowledge the problems I'm mentioning and are trying to work their way out, but without complete success. Literally: Without "complete" success!

No version of constructively-defined real numbers can possibly be Cauchy complete unless you change the definition.

As far as I can see, I'm not wrong about that even though I'm ignorant of the mighty constructivist struggles to finesse their way out of the problem.

Using higher inductive types, however, there is a fourth solution, which we believe to be
preferable to any of the above, and interesting even to a classical mathematician. The idea is
that the Cauchy real numbers should be the free complete metric space generated by Q.
""" — Mephist

Oh what a cool buzzphrase.

I know what a free group is, and a free Abelian group, and a free module over a commutive ring. I can believe that one can define a free complete metric space over Q and I would love to see that definition, since I would probably understand it. Perhaps that's the answer, or at least an answer, to this dilemma.

Well, I see that the problem is much more complex than I thought... — Mephist

We have a meeting of the minds on this. It's certainly more complex than I thought.

However, solution (i) is the one used in Coq standard library (the one that I knew). And, as I said, in that case Cauchy completeness is an axiom, so.. no problem! :smile:
However, I don't know how to prove that for each ZFC real there exists only one constructive real that verifies all possible propositions expressible in ZFC, and maybe you are right that this cannot be done. — Mephist

Don't know. Haven't ever looked at Coq. It seems to me that the desire to be able to mechanically verify proofs (Voevodsky's original motivation) is not exaclty the same as Brouwer's desire to constrain mathematical objects to the realm of the constructible.

In other words Coq is all well and good, but if it's just to verify proofs, must we abandon standard math to use it? Perhaps I'm missing the subtleties here.

He proposes a fourth solution, based on the hierarchy of infinite sets typical of Homotopy Type Theory, and in 11.3.4 he proves Cauchy completeness. But I don't know HOTT enough to prove anything in it.. — Mephist

I find it fascinating that constructivists claim to be able to prove the Cauchy completeness of a set, namely the real numbers, whose elements manifestly can not all be computed. But at least I've learned that people are making the effort.

So, well, in the end I don't have a definitive answer :sad: (but maybe somebody on https://mathoverflow.net/ could have it) — Mephist

It's worth a shot on math.stackexchange since Andrej Bauer hangs out there I believe. Maybe I'll ask one of these days.

P.S. I found an answer about countability of constructivist real numbers here: https://stackoverflow.com/questions/51933107/why-are-the-real-numbers-axiomatized-in-coq — Mephist

I read that but I don't know Coq so had to skip the details. But the checked answer invoked Lowenheim-Skolem and adding extra axioms; and also made the point you did (which I completely agree with) that even in ZFC we can only define or compute countably many subsets of the naturals. I'll take away that these points are regarded as valid by constructivists and that if I knew more about all these things I'd be enlightened. But then I'd lose my apparenly naive faith in the standard real numbers. I'd hate to lose that!!

"""
As for your second question, it is true that there can be only countably many Coq terms of type R -> Prop. However, the same is true of ZFC: there are only countably many formulas for defining subsets of the real numbers. This is connected to the Löwenheim-Skolem paradox, which implies that if ZFC is consistent it has a countable model -- which, in particular, would have only countably many real numbers. Both in ZFC and in Coq, however, it is impossible to define a function that enumerates all real numbers: they are countable from our own external perspective on the theory, but uncountable from the theory's point of view.
""" — Mephist

Yes yes, perfectly well agreed. Nevertheless the noncomputable (or nondefinable, which is a slightly different notion) reals have a purpose, which is to plug the holes. But clearly my argument hasn't gotten any more sophisticated than that, and in the end I'm probably just not appreciating constructivism.

Yes... well, half of it: the "proofs-as-programs" interpretation is valid even in the standard first order natural deduction logic, if you don't use excluded middle (the intuitionistic part). Here is a summary of all the rules of first order intuitionistic logic with the associated expressions in lambda calculus: https://en.wikipedia.org/wiki/Natural_deduction#/media/File:First_order_natural_deduction.png — Mephist

Ok, first of two responses to your lengthy posts of page 9.

I didn't understand this para but you said your point was half of Curry-Howard so I'll trust you.

The only rule that doesn't fit with this interpretation is excluded middle. You can take a look at the paragraph "Classical and modal logics" in https://en.wikipedia.org/wiki/Natural_deduction for an explanation of why this happens. — Mephist

I'll have to stipulate that I didn't follow any of this. I can't detour into modal logic right now.

OK, I see your point now! — Mephist

Ok!! This was regarding the fact that there aren't enough computable numbers to plug the holes in the leaky boat of the constructive real line.

But consider this (taken from https://en.wikipedia.org/wiki/Real_number under "Advanced properties"):
"It is not possible to characterize the reals with first-order logic alone: the Löwenheim–Skolem theorem implies that there exists a countable dense subset of the real numbers satisfying exactly the same sentences in first-order logic as the real numbers themselves."

Even ZFC can't distinguish between countable and uncountable reals! — Mephist

First, that's not true. Cantor's theorem is a theorem of ZF and it shows that the powerset of the naturals is uncountable. And it's easy to prove the powerset of the naturals is bijectively equivalent to the set of all infinite binary sequences, which can be coded as the reals. So your point appears wrong.

Second, but really first, all Lowenheim-Skolem objections are suspect, because nobody seems to care. So why do constructivists suddenly care? Skolem as I understand it used his theorem to claim that sets are not sufficiently clear from the axioms. Perhaps so. But nobody in a hundred years has ever said, "Oh noes, set theory is destroyed, we have to start over!" Except the constructivists I guess. But I don't see why L-S is suddenly relevant.

Third, the standard reals are (as I understand it but I confess confusion on this point) a second-order theory because of the least upper bound property.

The second order reals are (again, as I understand it, and I am not an expert on these logical matters) categorical, meaning that any two models are isomorphic.

You often used this argument:
The constructive (ie computable) reals are too small, there are only countably many of them. The hyperreals are too big, they're not Archimedean. Only the standard reals are Cauchy complete. — Mephist

Yes and no. Yes, I do make that point. But I am not making a detailed technical argument. Just stating a curiosity that the standard reals have the Goldilocks property with respect to Cauchy completeness.

I mentioned that as a curiosity, not as an argument. I never imagined anyone would disagree.

Still, what I say is true, and you have not convinced me that the constructive reals are Cauchy complete unless you change the definition of a Cauchy sequence in such a way as to change its essential meaning. This is my working thesis and you have not convinced me otherwise.

Of course there are funny models but funny models are not arguments, they're just funny models.

And my answer is: you can assume without contradiction that constructive reals are uncountable, exactly as you do in ZFC! The fact that you can build only a countable set of real numbers only means that you can only consider a countable set of real numbers in the propositions of logic (the same as in ZFC), not that the logic does not allow the existence of an uncountable set of real numbers. — Mephist

If a constructive real requires a computable Cauchy sequence, there can't be enough of them to make up an uncountable set except in the "funny model" sense. I'm rejecting funny models in favor of the standard models.

Now, let's consider "computable reals". Computable reals are functions from natural numbers to natural numbers (you know better than me: take as input the position of the digit and produce as output the digit). Well, not all of these computable functions are valid terms in Martin-Lof type theory. — Mephist

Martin-Lof is a buzzword I hear a lot in these discussions but it doesn't mean anything to me. But of course there are far more functions on the naturals than there are computable numbers. You agree with that.

The set of functions from Nat to Nat that are allowed as valid expressions depends on the details of the rules, and in Coq sometimes it even changes from one version to the other of the of the software. The reason is that the rules of the language must allow to build only total recursive functions (functions that always terminate), otherwise the logic becomes inconsistent.
So, the functions that you can actually build using the rules of the logic correspond only to the Turing machines that are provably guaranteed to terminate (by using the meta-logic of the language, not the language itself), and that of course is a strict subset of all the Turing machines. But this IS NOT the entire set of functions from Nat to Nat that is assumed to exist, at the same way as the set of real numbers of which you can speak about in ZFC is not the set of real numbers that is assumed to exist. — Mephist

Well you are claiming that constructivists now regard noncomputable reals as existing. That's certainly contrary to my understanding. But if you allow noncomputable numbers in your ontology you have the standard real numbers and not something else.

That is: If you now claim that the constructive reals are the computable reals plus the noncomputable reals, you've completely conceded my point; and, I imagine, horrified all the constructivists who don't believe that at all.

About the Cauchy completeness problem, I don't know how to address it, — Mephist

I can see that!! (lol)

because in constructivist logic there are different axiomatizations of real numbers that are not equivalent between each other, and even not equivalent to ZFC reals. You can consider it a defect of the logic, but you can even consider it an advantage, because the thing that all these the axiomatizations have in common (well, not sure if all, but at least the two or three that I have seen..) is the set of theorems that is sufficient for doing analysis. So, the degree of freedom that is left out and makes the difference between the various constructivist axiomatizations corresponds to the aspects of ZFC reals that are not so "intuitive", such as for example the difference between point-based or pointless topology. This, in my opinion, means that there is not only one intuitively correct definition of what are real numbers. — Mephist

That's an awful lot of handwaving IMO. But there IS an intuitively correct definition of the real numbers: A Cauchy-complete totally ordered field. That's a second-order, categorical axiomitization. It was familiar to Newton and to every high school student.

I can believe that constructivists prefer a different model or models. I can NOT believe that anyone trained in math claims to not have an intuition of the standard reals. That is, I can imagine a constructivist saying, "The standard reals can't be right because you can't explicitly construct most of them." Of course that's true. But I can't believe anyone saying they have no mental picture of the standard reals as locations on the real line.

I believe the main source of confusion here is the concept of a model. If you take ZFC and remove some axioms (the axiom of choice and the logical axiom of excluded middle) the set of theorems that you can prove will be smaller, but the set of models of the theory will be bigger.
All models that were valid in ZFC will be still valid in the "reduced" ZFC, because all valid proofs in the reduced ZFC are still valid proofs of ZFC: you can't prove propositions that are not true in the model if you couldn't do it with the full ZFC. — Mephist

Ok I read the next two of your posts, and they contain a lot of meat. I want to respond to them but not tonight, it's late here. I will just say ahead of time that I reject all Lowenheim-Skolem arguments for the following meta-reason. The theorem dates from the 1920's or whatever and can always be used to trump any argument. "Oh yeah, well for all you know the reals are countable because L-S."

Now, here is the thing. For a hundred years, mainstream math hasn't cared. L-S is regarded as a curiosity and of course it's technically true, but nobody thinks that way.

So you now have to be making the claim that the constructivists are making the following argument: "For a hundred years we didn't take L-S all that seriously; but now we do." So I'm not convinced.

Also looking ahead to my more detailed comments to come, I believe that there IS a canonical intuitively correct model of the real numbers, and it's the standard reals. The second order standard reals. With the least upper bound axiom, the axioms for the reals become second order (I've always been a little bit shaky on this point so correct me if I'm wrong) and therefore are not subject to L-S.

The standard reals are the model of time in the Schrödinger equation. Everyone thinks of them the same way. The high school analytic geometry reals. Those are the morally correct reals. All other models are not. I believe that even a constructivist must be able to conceive of the standard reals (and is thinking of the same structure as the rest of us) and must recognize it at some level as the "real" reals. The only way to avoid this is to fall back on formalism. This is my belief. It may well be that I'm nothing but a product of my education, but I have a very strong intuition of the standard reals and I refuse to believe that others don't.

Anyway those are two points I wanted to make after a quick read of your two lengthy posts, but there is a lot of stuff in there and I'll get to it tomorrow.

Everything that is needed is a set W, some Qi, that can be "anything", a function Qi from the Qi to real numbers, and a function "complement" on the Qi. — Mephist

At the moment I haven't bandwidth to engage with the probability topic. So to roll this back, I made the following claim:

Claim: Without the axiom of choice you can't do modern probability theory.

I'll now retreat to saying that this is my belief, but if you have refuted my point then so be it. However skimming your argument I see that you refer to pointless topology and locales. These are modern concepts and somewhat specialized. So at worst, my Claim is operable up to a certain level of math, which is still relatively high up the chain in the scheme of things.

In the end, math is not an opinion, right? — Mephist

But yes, it is. It's opinion. What math is, how mathematicians think about and practice math, is under constant change.

In a trivial sense a proposition, once proved, is proved forever. But which proofs and which proof frameworks are considered meaningful, changes over time.

Techniques of proof vary over the years, as do standards of rigor. Gauss was the first person credited, in his doctoral thesis of 1799, with a fully correct proof of the fundamental theorem of algebra (FTA), which says that a polynomial in one variable with complex coefficients has a root in the complex plane; and as a corollary, that every n-degree polynomial has n roots, counted to multiplicity. (ie $x^2$ has one root but it's counted twice).

But by today's standards, Gauss's proof wouldn't be acceptable from an undergrad. It is no longer seen as rigorous. In fact the final subtlety in this theorem was nailed down as recently as 1920.

Likewise the categorical revolution started by MacLane in the 1940's has already caused set theory to lose its foundational status in many parts of math. Brouwer's intuitionistic dreams failed to become mainstream in the 1930's, but may end up doing so in the near future. I think of HOTT and constructivism as "Brouwer's revenge."

Likewise type theory, which Russell proposed in order to solve Russell's paradox; but instead, the axiom schema of specification won the 20th century (that's the axiom that says that you can form a set from a predicate only if you have an existing set to quantify the predicate over. And now type theory's a big deal again.

Math is not about the theorems. It's about how mathematicians think about mathematical objects. And in that respect math is constantly changing. Math is a historically contingent human endeavor. It's social and it can be political, just like anything else.

I had an idea to solve the question about Cauchy completeness, that I should have had a long time ago: just look at the book that proposes constructive mathematics as the "new foundations"! — Mephist

Hi @Mephist

I've been percolating on all this a few days. The insight I'm taking away is that constructivists care about Cauchy completeness. I hadn't formerly realized that. And I have a bridge from what I know, the detailed bits and bytes of Cauchy sequences (ie "epsilonics") and the corresponding notions in constructive math.

I've learned about as much as I'm going to about constructivism at the moment; and much more than I ever knew in the past.

I still don't see how constructivists can claim (if they do) that they have a model of constructive reals that are Cauchy complete. If that is true, to me it seems that it cannot also be "morally" Cauchy complete. I'm wearing my Platonist hat now, also called realism. And in particular, how constructivists deal with the holes that I claim must be present in the real line absent noncomputable reals. I will most likely need to defer learning more about this till a future time.

But you did point me at a reference that you say sheds some light, so I'll read that at least.

I will now dive into the thread since I was last here and see what there is to see. My intention is to answer any questions that I can, but not make too many assertions that I would have to defend.

Real numbers are a concrete unique object, because there is only one model of real numbers. — Mephist

I said no such thing and of course believe no such thing.

Basically, the point is that building proofs in intuitionistic logic is equivalent (it's really the same thing) to building computable functions: there is a one-to one correspondence between intuitionistic proofs and terms of lambda calculus (sorry, lambda calculus again...). This is really a very simple and practical thing, but I don't know why all explanations that I am able to find on the web are extremely abstract and complicated! — Mephist

Is this the Curry-Howard correspondence?

Ok I believe I've caught up to your series of posts replying to my most recent of the other day. I still owe you a response to the second long post you replied to me a few days ago.

But wait let's stop here for a moment. If you didn't know what I meant by a computable real, then let's go back to the basic argument. The computable reals can't be Cauchy complete because there are far too few of them to plug all the holes with only computably Cauchy sequences as defined in the Italian paper.

And again, I only mentioned it to kind of wrap up our original discussion. Saying in effect that, "Ok constructivism is all the rage and there's a ton of stuff I'll never know, but the standard reals are the only Cauchy complete model of the reals." That was my original point. If you haven't known what I meant by a computable real all along, this has been a vacuous discussion with an empty intersection.

Also of course there are many (many!) exotic models of ZFC, each with their own version of the reals. That doesn't bear on my simple and basic point.

Consider this question: why modules don't have a base? Is it because they can't see all those inverse functions of multiplication that vector spaces see? Well, yes! But does it mean that modules are vector spaces when you close your eyes to not the inverse of multiplication? — Mephist

Pretty funny. The statement "Every vector space has a basis" is logically equivalent to the full axiom of choice. So every time I give you a vector space and you say, "Well of course it has a basis," you are asserting AC.

To answer your question directly, I suppose that if someone asked me, why don't modules always have a basis, I suppose I'd say, "Some modules aren't free." Which doesn't answer the question, since a free module is just a module that has a basis. But even a module with a basis is not necessarily a vector space, so I think your analogy is a little off. The integers are a free module over the integers, with a basis, but are not a vector space.

But why do all vector spaces have a basis? It's because of the axiom of choice! Without choice, there's a vector space that has no basis. Consider how strange such a thing must be, before you casually abandon choice.

In my opinion ZFC, or ZF, makes sense only in first order logic (of course!), and intuitionistic logic makes the most sense in the context of type theory. But of course you can choose whatever random combination of axioms and rules, and find their implications, but I doubt that in this way you can discover something meaningful about what the "real" real numbers are... — Mephist

I can't comment on this. But if you haven't understood all along what a computable real is, and that there are only countably many of them, and that the standard reals are not countable, then you have not understood, let alone challenged, my argument.

Which is: The constructive reals can not legitimately be called Cauchy complete. There aren't enough of them to do the job.