But we have to distinguish between the extensional concept of a number of hotel rooms that can be built, visited, observed, realized etc, versus the intensional concept of a countably infinite set of rooms. — sime

There's no consideration of intensionality in the illustration.

a piece of syntax representing an inductive definition of the natural numbers. — sime

I have no idea what you think you're saying, and, after a number such exchanges with you, my bet is that you have no idea what you think you're saying.

Edit: On second thought, I do see your point: The uncountable co-idemponality topos decountably isomporphizes with the muti-hyperprimes to induce a Booleantype semi-constructive syntax field.

Yes, how could I have missed that?

There's no consideration of intensionality in the illustration. — TonesInDeepFreeze

You do recognise that Dedekind-infinite sets aren't extensionally meaningful, right?

So if one writes down an inductive definition of the natural numbers

1 + N <--> N

where <--> is defined to be an isomorphism, then to say N is "Dedekind-Infinite" means nothing more than to restate that definition.

Inappropriate extensional analogies for understanding dedekind-infinite sets , such as unimaginable and unobservable completed infinite sets of hotel rooms are going to appear paradoxical .

extensionally meaningful — sime

Definition of 'extensionally meaningful'?

And, if you respond with word salad, as you are wont to do, then I can't help you.

So if one writes down an inductive definition of the natural numbers

1 + N <--> N

where <--> is defined to be an isomorphism — sime

That's not a definition of anything, let alone the set of natural numbers.

then to say N is "Dedekind-Infinite" means nothing more than to restate that definition. — sime

'N is Dedekind infinite' means that there is a 1-1 correspondence between N and a proper subset of N. There's no need to drag isomorphism into it. The function {<j j+1> | j in N} is provably a 1-1 correspondence between N and a proper subset of N, so it proves that N is Dedekind infinite (and notice, contrary to your incorrect claim, choice is not involved). But that proof is not a definition of anything, let alone of the set of natural numbers.

And intensionality has nothing to do with this.

Your messages each time are as good as scribbled postcards from the Rabbit Hole.

Definition of 'extensionally meaningful? — TonesInDeepFreeze

The extensional meaning of a set are the items it refers to, in contrast to the definition of the set in terms of a formula, that is to say it's intentional meaning. Countably infinite sets cannot be given an extensional definition for obvious reasons, which is why finitists object to the reality of such objects, even if conceding that such 'sets' have instrumental use for generating numbers.

That's not a definition of anything, let alone the set of natural numbers. — TonesInDeepFreeze

Its just short-hand for the inductive definition of the Naturals in terms of an F-algebra with respect to an arbitrary category in which 1 --> N represents '0' and N --> N represents the successor function that corresponds to the Dedekind infiniteness property.

'N is Dedekind infinite' means that there is a 1-1 correspondence between N and a proper subset of N. There's no need to drag isomorphism into it. — TonesInDeepFreeze

That's a fair enough remark, given that only the right arrow is involved.

'

The function {<j j+1> | j in N} is provably a 1-1 correspondence between N and a proper subset of N, so it proves that N is Dedekind infinite (and notice, contrary to your incorrect claim, choice is not involved). But that proof is not a definition of anything, let alone of the set of natural numbers. — TonesInDeepFreeze

I never said that Choice was involved in the definition of dedekind infinity, i said that the presence of Choice causes all infinite sets in ZF to become Dedekind infinite by default, which is a major failing of ZFC in ruling out the only sort of "infinite" sets that have any pretence of physical realisability in the sense of extension.

You seem to be either (1) Using your own made up ways of talking about this or (2) Speaking in a different context from me.

If we don't agree on how we speak about the subject or our context for the way we speak about the subject, then we can't communicate.

My context is ordinary mathematical logic, which is the context of Hilbert himself. Though his Hotel predates the fuller context of mathematical logic, at least we're in the ballpark.

If you have a different context in mind, then one needs to know where to can reference it.

So, from my context, here goes:

extensional meaning — sime

I don't look for sets to have a meaning. Word, symbols, sentences have meanings. Where do authors speak of sets having meaning?

the items it refers to — sime

Sets don't refer to items. Sets have members. Where do authors speak of sets referring to items?

definition of the set in terms of a formula — sime

Granted, authors often speak of 'defining sets'. But to be accurate, there are two senses of defintion:

(1) Syntactical. Defining a symbol.

Example

'N' is the symbol.

The definition of 'N' is

N = x <-> (x is an inductive successor set & x is a subset of every inductive successor set)

Sometimes, we have definitions such as this:

B = {0 5 8}

I take that is an example of what you mean by an 'extensional definition'.

(2) Semantical. For a given interpretation of a language, showing a formula using n free variables that is satisfied by certain n-tuples of members of the domain of the interpretation. (That's putting it roughly, though.)

Example (with first order PA)

With the standard model

x = x

defines the set of natural numbers.

category — sime

I didn't know you're using category theory.

the presence of Choice causes all infinite sets in ZF to become Dedekind infinite by default — sime

Of course, though I wouldn't state that way.

which is a major failing of ZFC in ruling out the only sort of "infinite" sets that have any pretence of physical realisability. — sime

It rules out that there are infinite sets that are not Dedekind infinite. But I don't know your definition of 'physical realizability'. Sets are not physical anyway (though, outside of set theory, we could have sets whose members are physical objects). Also, even without choice, we can't prove that there do exist infinite sets that are not Dedekind infinite.

There's an interesting question: In ZF we cannot prove 'There does not exist an infinite set that is not Dedekind infinite'. But do we ever have this situation?:

A definition of 'Q' and a proof of 'Q is infinite' but such that there is not also a proof of 'Q is Dedekind infinite'.

In other words, it's conceivable that even though we can't prove that every infinite set is Dedekind infinite, for any infinite set we define, we can prove that it is Dedekind infinite. Hmm, not sure, but I think that might be the case, since a definable set will be constructible (?) and the axiom of choice holds for constructible sets. (?)

Now that you've explained some of your terminology (I had no way of knowing that, out of the blue, you would be using category theory), I'm going to revise some of my earlier comments:

(1) Your bit about choice didn't make sense to me earlier, but I might have been incorrect to mention it in connection with proving that N is Dedekind infinite.

(2) I said your remarks about N were not a definition. Perhaps they are in a context of category theory.

(3) Ordinarily I take 'intensionality' to be about intensional contexts such a modal, epistemic, etc. operators. Now that you've clarified your sense of 'intensionality' and 'extensionality', I don't dispute that, in that sense, the ordinary definitions regarding infinite sets are not extensional. But I don't like that terminology because it jams against the ordinary sense of 'extensionality' we use in set theory.

(4) Perhaps you're not down the Rabbit Hole. But I'm cautious not to fall into it.

/

On the other hand, only by extraordinary shifting of context would you not be incorrect in your description of the hotel as finite but always expanding. Hilbert meant an infinite hotel, not a finite one always expanding. If you want to set up a different story from Hilbert's then of course that is fine. But we shouldn't conflate them.

EDIT NOTE: I meant to include 'not' before 'down' in the comment about the Rabbit Hole. It's corrected now.

Undefinable real numbers have no place in my view.
— keystone

You reject vagueness then. That is certainly the usual thing to do. — apokrisis

I believe that cuts made to a continuum are perfectly precise since I can draw it with no vagueness. For example, consider this drawing of y=0 and y=x^2-2:



There's no blurriness to my drawing. However, when I start to measure it (usually through calculus), my measurements may be imprecise. For example, if I try to measure the decimal expansion of the coordinates where they intersect I will generate a rational approximation which does not correspond to the exact point (however I can get arbitrarily close as long as my calculator has got the time and capacity).

Hopefully this explanation makes more clear my view of physics where the planck constant applies to a limitation of measurement, not of space/time itself. This is how I think space/time can be continuous but measurement discrete. And I like to think of our reality as something like a simulation by a computer which has finite time and capacity. The planck length and other limits of resolution are direct consequences of limitations of that finite computer.

And how do you know there is a rule unless you have ever seen some exception? — apokrisis

I'm confused with where we've landed with this and I think I we need to step back. You said "I've already raised this point, asking if sqrt(2) is the exception or the rule." I don't see sqrt(2) as a rule. I see it as a description, perhaps of an infinite set. As I've discussed with TonesInDeepFreeze, I see no problem in there being a description of an infinite set, even if the infinite set doesn't actually exist.

That is a better analogy. I prefer my own still – the static on the TV screen which is both every show you could ever see, but all at once ... or else just meaningless noise. — apokrisis

I like my stereogram analogy more because it highlights that the observer plays an active role in actualizing the rabbit all the while doing no actual manipulation of the page (not actual cutting of the continuum). But I agree that it's not a perfect analogy because the creater of the stereogram encoded the rabbit in the page, whereas in mathematics it's a true blank slate.

If you are experienced and trained in this area and would be up for helping me out through paid mentoring, please let me know. — keystone

There's an Australian mathematician, Norman Wildberger, on YouTube who doesn't accept infinities.
Here's a link to one of his videos.
Difficulties with real numbers as infinite decimals
https://www.youtube.com/watch?v=tXhtYsljEvY
You might try contacting him.

Must an ordered field necessarily be a field of numbers?
— keystone

No. But all complete ordered fields are isomorphic with one another. So all complete ordered fields are isomorphic with the system of reals. — TonesInDeepFreeze

Ok, if we've only proved that the reals are an ordered field, then is it possible that we haven't proved that sqrt(2) is a number?

the axioms of set theory are not in concordance with the intuitive notions of 'finite sets'
— keystone

All the axioms are in that concordance, except one. — TonesInDeepFreeze

Not an axiom, but at the heart of set theory is the definition of equinumerosity of infinite sets which is not in concordance with MY notion based on finite sets. Let me try to explain using finite sets A and B and hopefully my use of set theory terminology is correct.

When I think of A being equinumerous to B, I think that there exists a bijection AND no injection between A and B.
When I think of A being more numerous than B, I think that there exists an injection from B to A AND none from A to B.
When I think of A being less numerous than B, I think that there exists an injections from A to B AND none from B to A.

If I apply this intuition to infinite sets A and B, then A is neither equi/more/less numerous to B. The only way that this is possible is if A and B are both the null set. In other words, my intuition based on finite sets leads me to believe that infinite sets are all empty.

I want to highlight again that I'm not saying that set theory is wrong, I'm just proposing that set theory might not be about actually infinite sets, but instead the potentially infinite algorithms that describe the infinite sets.

Hilbert's idea was that we can work in infinitistic mathematics if we have a finitistic proof of the consistency of infinitistic mathematics. Famously, we found out that there is no finitistic proof of the kinds of systems we'd like to use, not only not of set theory but even of PRA, the system itself that we may take as exemplifying finitistic reasoning at its "safest". Yet, if I understand correctly, Hilbert's condition was a sufficient condition not a necessary one. — TonesInDeepFreeze

Interesting!

That's interesting. But, if that is to be a statement in the system, we'd need to see "described" couched mathematically. I have a hunch that your notion is pretty much the same as 'there exist potentially infinite sets', and as I've said, I don't know a system that says it. — TonesInDeepFreeze

Yes, I use the word 'described' without explaining exactly what I mean but essentially I'm referring to the descriptions already used in set theory today to describe the contents of an infinite set. I'm not proposing that we reframe set theory to be based on potentially infinite sets as that would not be satisfactory. For example we can't describe a circle using an endlessly increasing list of points. That list will never suffice (proof Cantor). Instead, we describe a circle using a potentially infinite 'algorithm' for generating the actually infinite list. For example, the equation x^2+y^2=1 describes a circle in completion.

And yet Universal Turing Computation is a mathematical object – conceived in Platonia. This is the kind of "paradox" we are meant to be figuring out here, not simply saying one is the other as if the differences were moot. — apokrisis

Maybe we both agree that our universe is like a simulation by a finite computer and as such our universe has some limitations in measurement, e.g. as it relates to Planck length. What I want to highlight is that our universe's computation is performed by one computer (I actually think it's more like a cluster of computers but let's leave that aside). Conversely with mathematics we're not constrained to a single computer. The math that I do on my pocket calculator might offer a different precision than the math that can be done on a supercomputer. We need our understanding of how computers participate in mathematics to be general and apply to all computers. As such, we cannot say that mathematics has limits of resolution, but instead that the computer does. And to keep things connected with reality, let's stick with finite computers having arbitrarily fine (but in all cases finite) precision.

Mathematics, in many branches, is brimming with sets. Analysis, topology, abstract algebra, probability, game theory... Can't even talk about them, can't get past page 10 in a textbook, without sets.

But of course, one can use the theorems of mathematics for engineering without tracing the proof of those theorems back to axioms, in particular the set theoretic axioms. That's not at issue. — TonesInDeepFreeze

Okay, I'll concede this point and agree that set theory is used to formalize many branches of mathematics...and I don't think it's going away. But as I said before, we never actually manipulate the infinite sets directly. We 'describe' infinite sets and work with the 'descriptions'.

A sequence is a set. And it has a domain, which is a set, and a range, which is a set. An infinite sequence is an infinite set with an infinite domain.

Of course, one can leave that unconsidered, not in mind, when working in certain parts of calculus. That is not at issue. But when we trace the proofs of the theorems of analysis back to axioms, then, in ordinary treatments, those are the axioms of set theory. — TonesInDeepFreeze

I concede that there may be no better way to formalize calculus than set theory. I don't want to challenge the efficacy/validity of set theory. I only want to challenge the philosophy/interpretation of it. We don't actually work with the infinite sequence itself, we work with descriptions of the infinite sequence.

A circle is a certain kind of set of points. I don't know what you mean. — TonesInDeepFreeze

Yes, the conventional way to think of a circle is as an (actually) infinite set of points. What I'm proposing is that we think of the circle as the description (a finite string of characters) of those points. For example, the equation defining the circle. For example, consider this unorthodox drawing of a circle:



In this drawing there are no points. You certainly cannot identify a point and provide its coordinates. However, this drawing captures everything 'circular' about the green curve because it's associated with an equation. This equation is the description I've been referring to in my other posts. It describes, not the points the curve is made of but instead the potentially infinite points that would emerge IF I add more curves to the system. For example, let's add y=0:



In this case, two points emerge with this 'cut'. And because we know the equations for these two curves we can 'measure' the coordinates of these points.

With this approach we start with a continuum and progressively cut it. At any stage the objects are finite in number and the potential for additional cuts is never exhausted.

Conversely, if we start with points to build a continuum we're going to need infinity of them...and even then, I think such an approach is problematic and paradox filled.

I believe that calculus is more closely aligned with this parts-from-whole approach than it is with the conventional whole-from-parts approach.

I am saying much the same thing. But the question is not where the numbers need to be represented or stored. It is how many decimal places do you really need for the task in hand? — apokrisis

If you're going to describe a continuum with numbers you will need infinite decimal places. But of course, it doesn't have to be decimal. Binary is more pure. But that doesn't reduce the number of digits required. You can't do much math with one bit.

I believe that calculus is more closely aligned with this parts-from-whole approach than it is with the conventional whole-from-parts approach. — keystone

How would you catalogue all continuous curves? That would be a starting "point". In order to have derivatives and integrals you would need some kind of function derivable from a catalogued example. Sorry, but the whole approach sounds absurd.

(I had no way of knowing that, out of the blue, you would be using category theory) — TonesInDeepFreeze

Nowadays it's an extremely popular rabbit hole, complete with patios, restaurants and theaters. Doesn't do anything for me. Both Category theory and K-theory arose in the mid 20th century. I've mentioned before that one of my instructors from a half century ago was at a talk about the latter when the lecturer started entertaining comments and questions from the audience, became so annoyed with the flack he was receiving, he practically yelled, "Just believe me and I can prove anything!!"

Well, not in mathematical analysis.

But one can't fairly criticize the road of set theory if one is not addressing it as an axiomatization. And even if not criticizing set theory but instead just saying mathematics can be done with unformalized "potential infinity" instead, then it's not a fair comparison since one is an axiomatization and the other is not. — TonesInDeepFreeze

Ultimately I want a mathematics that is formally supported by axioms and my preference is to keep what we've got (ZFC) because I'm guessing that so much is based on ZFC because it works so well. Also, I don't have the skill to craft new axioms, ha!

But here's my beef.

With geometry (specifically drawings) we can do things that cannot be capture with algebra. For example, I can draw a perfect finite picture showing the intersection of y=x^2-2 and y=0 even though I couldn't do the algebra to tell you the infinite decimal coordinates of the intersection.

With 'algebra' (specifically the manipulation of equations through calculus) we can do things that cannot be captured with geometry. For example, I can perform a finite set of operations (derivative) to easily calculate the slope at a point but I could never actually draw the infinite steps as described by the limit definition.

However, we want there to be an equivalence between algebra and geometry and so infinity leaks in. If instead we see algebra and geometry to be complementary (instead of equivalent), then we don't need infinity. In geometry we would limit ourselves to what is finitely possible. In algebra we would do the same. And sometimes we would need to switch back and forth between the two, similar to how in physics we switch back and forth between complementary wave and particle descriptions of light.

But to accomplish the above, we need to rethink how geometry (specifically how we draw things) and set theory (specifically whether set theory is about points or algorithms). But as I've said before, I don't think this changes much math...I think we've been doing it right all along. I just think what we say doesn't correspond to what we do.

We say that there are infinite numbers of points on a graph when we actually only explicitly draw a finite number. We say that sets are about points when we actually only work with 'algorithms'/descriptions.

But when I started grad school at another university in 1962 one of the first required courses was an introduction to foundations using Halmos' Naive Set Theory and the Peano Axioms. It was quite illuminating. — jgill

As this conversation progresses, I've warmed up to set theory (although I currently hold an unorthodox view that set theory might not actually be about sets).

For me, the problem is not so much that there is anything counter-intuitive about this, but rather that it's rude and bad business practice to keep waking guests up in the middle of the night and make them pack and move to another room, especially an infinite number of times. Not only that, but the poster keystone has added lamps that keep turning off and on, which is extremely annoying when people are trying to get a good night's rest for the next day when everybody is going out to see Zeno's 10K Charity Run where Achilles will have to run through an infinite number of distances and suffer the ignominy of getting beat by a turtle. — TonesInDeepFreeze

LoL.

the very fact that anytime you have probability with infinitely many "contestants", whether it's dense space or whatever, you will necessarily either give the "contestants" a probability of 0 or be faced with adding up over 100% (since reiteratively summing any non-zero quantity indefinitely will approach over 100% at some point).

Your "solution" isn't a solution in that it doesn't talk about what the problem talks about. The "problem" is referring to continuity in dense contexts: it's not at all a "problem" in nondense contexts, this is equivalent to solving the Liar paradox by just saying "what if the guy doesn't lie?" — Kuro

What I'm proposing is that there is no "contest" involving infinitely many "contestants". For example, I'm proposing that to do calculus we don't need to assume that a continuum is built from the assembly of infinitely many points. Can you provide the simplest possible example in calculus where we need to assume that there are infinitely many points?

I believe that cuts made to a continuum are perfectly precise since I can draw it with no vagueness. For example, consider this drawing of y=0 and y=x^2-2:

There's no blurriness to my drawing. However, when I start to measure it (usually through calculus), my measurements may be imprecise. — keystone

This is Russell's argument. This photograph of Keystone's face might be blurred, but Keystone's face itself is not. Therefore vagueness is merely epistemic and not ontic. It exists in our representation of reality and not reality itself.

But the reverse argument also applies. The representation can be sharper than what it represents. The right facial recognition algorithm could separate a dim CCTV image of Keystone in a hoody from all the other faces stored in a police data bank. Signal processing can extract structural information that stands behind any amount of confusing surface detail.

So sure, one can eliminate vagueness by using well-defined algorithms like your equations. They are linear and not non-linear after all. Errors in measuring initial conditions don't matter as the uncertainty only grows in polynomial time and not exponentially.

But how wide are your lines – even mathematically? How sure are you they are single lines and not a small bundle of lines sharing a neighbourhood with infinitesimal spacing? And when does this vagueness start to matter? Doesn't it matter if your rigorous mathematical edifice must also fit a physical world were nonlinearity is in fact the generic condition?

So having signal processing to sharpen up your view of an uncertain world is great. Really useful. We can see why maths is "unreasonably effective" in that regard.

But that doesn't engage with the foundational issue of whether reality itself is vague or crisp at base. And hence what kind of ontology we are correct to import into our "picturing" of math's epistemology.

Your analogy betwen mathematics and theology is not apt.

One can disprove 'there exists an infinite set' by stating axioms that disprove 'there exists and infinite set'. The obvious choice for such an axiom is 'there does not exist an infinite set'.

Anyway, I never asked you to disprove anything at all. — TonesInDeepFreeze

I see your point. In reality there is one truth (e.g. God either exists or not). I have this romantic/naive notion that in Mathematics there is similarly one truth and one ultimate set of axioms that captures it. But as it is today that is not the case with mathematics. So we may both be right when I say infinite sets don't exist and you say that they do because maybe we're starting with different axioms. Point granted.

We don't intend or claim that a domain of discourse for set theory is a world such as a physical world of physical particles and physical objects. At the beginning of this discussion, if asked, I would concede that immediately. — TonesInDeepFreeze

In Set Theory we say 'There exists a set...'. What do we mean by this? I take it to mean that the set must literally exist somewhere and my understanding is that the mainstream view is that it exists in the Platonic Realm. This is the world I want to explore. While Hilbert's Hotel cannot exist in our world, it should have no problems existing in the Platonic Realm which is infinite. Same goes for Thompson's Lamp. And what I'm trying to get at is that in the Platonic Realm, where infinite operations are completed (i.e. each guest flicks the light switch) something breaks because Thomson's Lamp can't decide whether it should be on or off after all switches are flicked. Or in other words, if an inhabitant of the Platonic realm is computing a geometric series and Grandi's series simultaneously term by term, then they both must either sum to a number or not. We are not allowed to say that one does and the other does not. That would be a contradiction that would cause the Platonic Realm to explode. I want to argue that the Platonic Realm necessarily explodes leaving no place for infinite sets to exist.

You've described your notion of potential infinity a few times (in another thread especially). And I've replied about it each time. Now, you're coming back to restate it, but still not addressing the substance of my previous replies. As in another thread, this just brings us around full circle. — TonesInDeepFreeze

I apologize if I'm not addressing the substance of your previous replies. It's not intentional, I thought I was.

The thought experiment is suggestive of an analogy with set theory, but suggestiveness is not an argument about set theory itself. — TonesInDeepFreeze

As I mentioned in my other post, it's not just about analogies - it's about the Platonic Realm. It's the only place that can harbor infinite sets. It's also the only place that can harbor the thought experiments so they're inseparably linked.

why would we think they can be completed in reality?
— keystone

We don't! Set theory doesn't say there's a "completion in reality". Set theory doesn't have that vocabulary. — TonesInDeepFreeze

In the Platonic Realm, can infinite objects exist but never be constructed? My view of the Platonic Realm is that it's a world where infinite processes can be completed, where someone can add 1 + 0.5 + 0.25 + ... term by term and in a finite amount of time complete the calculation to yield exactly 2.

There's an Australian mathematician, Norman Wildberger, on YouTube who doesn't accept infinities.
Here's a link to one of his videos.
Difficulties with real numbers as infinite decimals
https://www.youtube.com/watch?v=tXhtYsljEvY
You might try contacting him. — Art48

Thanks. I have admired him for many years. I loved his videos on real numbers. I've reached out to him a couple of times but didn't get much of a response aside from a dismissive response when I posted a comment on one of his videos. I think he's a canary in the coalmine but his proposed solutions don't target what I believe is the heart of the issue - that points cannot be used to construct a continuum. He's also way too much of a ultra-finitist for me. I believe potential infinite is essential in mathematics.

How would you catalogue all continuous curves? That would be a starting "point". In order to have derivatives and integrals you would need some kind of function derivable from a catalogued example. Sorry, but the whole approach sounds absurd. — jgill

Can you explain what you mean by 'catalogue all continuous curves'?

I just now watched this video by Wildberger:

https://www.youtube.com/watch?v=U75S_ZvnWNk

In that video, he's an intellectually disorganized, sneaky, weaselly lying sophist and a fool. He's an insult to intelligence.

First, the title of the video is "Modern "Set Theory" - is it a religious belief system?" Yet he mentions nothing about that, let alone supporting it. Indeed, it is unsupportable.

Religions involve some combination of (1) a god, gods, deities, angels, demons and other non-physical beings - usually with personalities and utterances attributed to them, and acting causally, and usually miraculously, on the physical world, (2) explanations of the universe and creation, (3) human souls and accounts of what happens to those souls after death, (4) mythic narratives about human history, including religious founders and prophets, (4) divinely mandated moral codes, (5) prayers, chants and incantations that have power upon the physical world, (6) teleologies and (7) eschatologies. And I confined just to aspects of belief, since Wildberger didn't say that set theory is also a religious practice.

So, with that provocative title, coyly cast as an interrogative, what the sneaky sophist does is wet the water slide for thinking that set theory is religious while he does nothing toward engaging the very meme he's disseminating.

/

In the opening he says he's going to "dismantle" "what we currently are doing now".

Surprise: he doesn't.

He says he's going to "examine the details of what we're currently doing". But instead of "details", all we get in the video are atrocious oversimplifications, and not a single one of those articulates an actual problem with set theory.

Then he says the video presents a fair case that mathematics can have a solid foundation. But it turns out his "fair case" is just to wave his magic wand (actually he uses an old drum stick) and say that the natural numbers are the foundation for all the other branches of mathematics. But he says not a single word showing how that would be done except for a chart with 'the natural numbers' as the base of the pyramid of mathematical subjects. As if doing that carries even a mote of an argument. I'm going to make a chart with avocados at the base of a pyramid of the hierarchy of the U.S. federal government. And, presto, the avocado on my table is now the president, head of state and supreme commander of the U.S. armed forces.

Then he states his gravamen: "Set theory is a logically inadequate foundation for mathematics". But in the entire video he does not cite a single actual problem with the logic of set theory. His only criticism is that it's axiomatic. Yeah, set theory tells you what its axioms are - so you can take them or reject them - and then shows you exactly, step by algorithmically checkable step, the derivations of the theorems. So what would Wildberger's own foundation be if not to present axioms? He gives not a clue, except that pyramid chart.

* He is creative with that drum stick. He uses it not just as a pointer, but he holds it up to illustrate the real number line. And at other times he uses it like a magic he waves to declare that he's shown some supposed problem with set theory, analysis, and mathematics. Like Trump magically declassifying documents.

/

He says about the natural numbers, "as numbers become big, arithmetical problems arise". Okay, what is one them? He mentions no problems. This is from the guy who had a few minutes ago promised to get detailed in the video.

/

He says the problem with the continuum is how do we model what happens when we look closely at it. Well, that's merely a vague question. What do you mean by "model"? What do you mean by "happens"? What do you mean by "look closely at it"? He then asks, "What is the mathematical way of magniying and subdividing and looking more clearly and carefully at smaller and smaller subdivisions?" Um, how about doing that with proofs of theorems from axioms?

He says, "This is hugely problematic". Yes, Norman, your vagueness and lack of the details you promised is kind of a problem."

He mentions the irrationals, with an actual smirk on his face. Argument by facial expression. (The video is full of those kinds of smug faces, while his words are saying, at best, nothing really, if not, at worst, outright lying.) He says we are faced with how to set up an arithmetic for irrational numbers, and "This is hugely problematic. It's hugely problematic." (Like when Trump* says, "It's a terrible thing. A terrible thing", as we're supposed to be convinced by the mere fact that he says it, and twice.) Wildberger simply skips that mathematics does, from axioms, define the operations on the real numbers with proofs about them.

He says that when mathematicians did lay out foundations for the reals, "[This] is really where the difficulties lie. It's really where the difficluties lie." - Norman "Double Assertion" Wildberger.

* No political comparison is intended between Wildberger and Trump. Just that they're both crackpots and liars.

/

Here's the first lie:

He says that ZFC "was not really a framework based on precise defintions and clear theorems".

ZFC is nothing but utterly rigorous definitions and utterly rigorous proofs of theorems. One may reject the axioms, but it is beyond dispute that the defintions are rigorous and the proofs rigorously derived from the axioms.

So that's Wildberger lie #1.

He says the problem is that set theory is axiomatic. Of course it is. Because it's with the axiomatic method that one states rigorous definitions and show rigorous proofs.

Wildberger is not just lying; he's also showing himself to be completely confused.

Then, "It was as if we had abandonded the effort to try to set up this foundational issue very carefully and clearly."

ZFC was exactly an effort to more carefully use set theory than Cantor did. Even if one rejects ZFC, it is beyond dispute that ZFC is a full formalization where Cantor's work was not.

So that's Wildberger lie #2.

He says the set theorists said, "Let's just assume that it works".

The set theory mathematicians didn't just assume that set theory provides a foundation for analysis, they proved that it does, by constructing a complete ordered field and the foundational theorems about it, from which analysis can take over from that point.

So that's Wildberger lie #3. The set theory mathematicians didn't just assume that set theory provides a foundation for analysis, they proved that it does, by constructing a complete ordered field and the starting theorems about it, from which analysis can take over from that point.

Then, "That's really what happened. 'Let's assume that the kind of things we want to be true really are true'.

That's really what did not happen.

So, reiteration of Wildberger lie #3.

Then, "[They say] 'We'll dress this up as if it's an axiomatic framework'".

Argument by characterization rather than substance. They didn't "dress it up" as an axiomatic framework. It is an axiomatic framework.

Then, "The job of framing this all was outsourced to logicians, which is really philosophy, a branch of philosophy that overlaps with mathematics".

He's insinuating that the set theory was not really rigorous because it was done by philosophers not mathematicians.

(1) Actually they were mostly mathematicians. Many of them were not just set theorists but worked in other fields of mathematics too. Some of the greatest names in mathematics were key in the development of the logic, set theory and type theory: Hilbert, Godel, von Neumann, Tarski, Whitehead, et. al (2) Set theory and mathematical logic are mathematics. (3) Many were all three - mathematicians, logicians and philosophers. But mathematical logic is astoundingly rigorous; it's all about rigor

Then, "Its not too much of an exaggeration to say mathematicians outsourced the foundations of their subject to the philosophers".

First "the philosophers", as if there are two distinct sides - the mathematicians and the philosophers. Second, yes many of the logicians were both mathematicians and philosophers. But again, Zermelo, Fraenkel, Bernays, Godel, et. al, where mathematicians no matter what else they were or weren't.

So it's not too much of an exaggeration to say that Wildberger is lying on this point too, and to call it Wildberger lie #4.

Then, "There's kind of an agreement not to examine closely what this logical foundational of set theory is".

You have got to be kidding me! Set theory and its logic have been examined, discussed, critiqued, reworked and reworked until the cows home. In literally library stack upon library stack upon library stacks of books and journals, examining set theory from every angle it can be examined from. I guess Wildberger doesn't have access to even a single library, not even at the university where he teaches.

So that's Wildberger lie #5.

Then he wiggles his hand as he calls set theory, "crazy things".

Argument not just by hand waving, but hand wiggling too!

/

That's just some of it, in just one nineteen minute video.

This is Russell's argument. — apokrisis

You seem to be well read in math, philosophy, and science. Out of curiosity, what are you trained in?

But the reverse argument also applies. The representation can be sharper than what it represents. The right facial recognition algorithm could separate a dim CCTV image of Keystone in a hoody from all the other faces stored in a police data bank. Signal processing can extract structural information that stands behind any amount of confusing surface detail. — apokrisis

I disagree. While an algorithm can indeed tune an image to make it better suited for the intended audience (e.g. colorize by infrared for those who are unable to see in the infrared spectrum), it cannot add these colors if the infrared is not actually there in the first place. No computer processing can make the image higher resolution than reality. Or more mathematically, if I tell you that I'm thinking of a number that begins with 3.14159 there is no way that a computer can increase the precision of this number to tell me the next digit.

But how wide are your lines – even mathematically? — apokrisis

For a fully accurate depiction, the line width should be exactly 0. I can't accomplish that with a drawing. Perhaps I should have cut a paper instead, but that wouldn't work perfectly either as the paper would fold and bend and cast shadows and leave small gaps. Even better, I could have made a stereogram whereby each area pops out at a different height. But alas, I am lazy so you have to forgive that my lines have some width when in reality they should not.

How sure are you they are single lines and not a small bundle of lines sharing a neighbourhood with infinitesimal spacing? And when does this vagueness start to matter? — apokrisis

About as sure as I am that there is no teapot orbiting between the Earth and Mars revolving about the sun in an elliptical orbit. Give me a reason to believe otherwise. I don't believe in infinitesimals either so you'll need to also give me a reason to believe in them too. The figure (Apokrisis' face) is exact and so is the measurement (the photo of your face), it's just that they're not equivalent. Do you consider this mismatch to be vagueness?

Doesn't it matter if your rigorous mathematical edifice must also fit a physical world were nonlinearity is in fact the generic condition? — apokrisis

How does my view break down with nonlinearity?

But that doesn't engage with the foundational issue of whether reality itself is vague or crisp at base. And hence what kind of ontology we are correct to import into our "picturing" of math's epistemology. — apokrisis

Perhaps we need to figure out whether the objects of mathematics are vague or crisp at base. This might help us guide our investigations into reality. This is why I think it's so important to have a sound philosophy for mathematics.

But it turns out his "fair case" is just to wave his magic wand (actually he uses an old drum stick) and say that the natural numbers are the foundation for all the other branches of mathematics. But he says not a single word showing how that would be done except for a chart with 'the natural numbers' as the base of the pyramid of mathematical subjects. — TonesInDeepFreeze

I agree that this is the biggest weakness of his view. He complains about the foundations of math but his foundation is set of tick marks on a white board.

I'm answering very quickly, because I'm out of time now.

if we've only proved that the reals are an ordered field, then is it possible that we haven't proved that sqrt(2) is a number? — keystone

We proved they are a complete ordered field.

I answered that in exact detail in a previous post. Pretty much, you're asking for notes from the first week of Calculus 1.

When I think of A being equinumerous to B, I think that there exists a bijection AND no injection between A and B.
When I think of A being more numerous than B, I think that there exists an injection from B to A AND none from A to B.
When I think of A being less numerous than B, I think that there exists an injections from A to B AND none from B to A. — keystone

All bijections are injections. So you're confused to begin with.

What you mean is that you take equinumerous to mean: A and B are equinumerous iff there is bijection between A and B and no injection from one into a proper subset of the other.

Of course, in ZFC that does hold for finite sets but not for infinite sets.

B is more numerous than A iff there is an injection from A into B and no bijection between them. Same in set theory.

B is less numerous than A iff A is more numerous than B. Same thing with set theory.

You're wasting our time. We already know that in set theory, infinite sets differ in this salient way from finite sets. Galileo's paradox, Hilbert's hotel, Dedekind infinitude, and this latest point about equinumerosity are all variations on the same point: infinite sets map 1-1 with proper subsets of themselves. You don't need to keep giving examples. We already agree that infinite sets map 1-1 with proper subsets of themselves. For you, it's counterintuitive and you won't accept it. Fine. I have no motivation to convince you otherwise.

my intuition based on finite sets leads me to believe that infinite sets are all empty. — keystone

'Infinite sets are empty' is a contradiction. And set theory does not have that contradiction. So if it's your intuition, then set theory is not for you.

I'm not saying that set theory is wrong, I'm just proposing that set theory might not be about actually infinite sets, but instead the potentially infinite algorithms that describe the infinite sets. — keystone

I know. You've said that a dozen times.

Can you provide the simplest possible example in calculus where we need to assume that there are infinitely many points? — keystone

Day one of high school Algebra 1:

"Students, we start with the set of real numbers and the real number line."

And you don't have to tell me yet again your finitistic interpretation of that or how we can work with only a finite number of those points, etc. I know that's your view.

Can you explain what you mean by 'catalogue all continuous curves'? — keystone

When math starts with points that are then assembled into curves, there is a way of describing those points on the real line, identifying a point with .5 for example. If you are starting out with curves or geometric figures you need to be classify them, order them somehow, for you then wish to create points by intersections I suppose. You need rigorous definitions for curves, then an axiomatic structure. All of which seems far-fetched. But who knows?