My current understanding is that there exists indeed a detailed description of the infinite model(s) for real numbers but at this point I am unable to pierce through the dense vocabulary and concepts in order to develop a correct mental picture on the matter. — alcontali

I'm not sure exactly what you're looking for. To my knowledge, and I'm no specialist in these matters, the second-order theory of the real numbers is categorical, which means there is only one unique model up to isomorphism.

On the other hand set theorists do study alternate models of the reals that arise if you change the axioms of set theory. For example there's a famous example of Solovay in which, in the absence of the axiom of choice and the presence of an inaccessible cardinal, all sets of reals are Lebesgue measurable. This kind of thing may be of interest to you if you're curious about alternative models of the reals.

https://en.wikipedia.org/wiki/Solovay_model

The only reality that we can know is what we learn from experience. — aletheist

Didn't Plato point out that what we experience is but shadow on a cave? And that the true reality lies outside, unseen and unseeable by us?

But is what we're talking about simply the question of whether what we experience is the same as reality? Did Peirce argue that it is? But how can that be? Science is historically contingent; and the better equipment we have, the better experiments we can do. Our theories of the universe keep changing. The universe, presumably, stays the same.

The constructive reals aren't complete because there are too few of them, only countably many
— fishfry

Too few...or too many? The subset of computable total functions that correspond to the provably convergent Cauchy sequences form a countable and complete ordered field, that is a proper subset of the provably total functions. — sime

Too few, clearly. There are only countably many of them.

I do apprehend the point that the computable reals are computably uncountable, since there is no computable bijection between the computable reals and the natural numbers.

So what? The moment after Turing defined what it means to be computable, he showed that there are naturally-stated problems that are not computable. Point being that even computer scientists recognize the existence of noncomputable phenomena. See Chaitin's Omega, for example.

So yeah, there's no computable bijection. But there is a bijection, just as sure as there are only countably many Turing machines.

And no countable ordered field can be complete. It's a theorem.

rrational numbers are fascinating and there is a legendary story behind them. Back in time, over two thousand five hundred years ago, there was this incredibly brilliant mathematician named Pythagoras. — Michael Lee

Whatever you do, don't tell @Metaphysician Undercover. This information upsets him terribly.

We are conscious only of the present time, which is an instant, if there be any such thing as an instant. But in the present we are conscious of the flow of time. There is no flow in an instant. Hence, the present is not an instant.
— Peirce, c. 1895 — aletheist

Ok. But isn't he conflating human experience with reality? He's right that for humans, the present is an experience of flow. But we have no idea what the underlying reality is. Why should human experience be privileged above nature?

Just as it is strictly correct to say that nobody is ever in an exact Position (except instantaneously, and an Instant is a fiction, or ens rationis), but Positions are either vaguely described states of motion of small range, or else (what is the better view), are entia rationis (i.e. fictions recognized to be fictions, and thus no longer fictions) invented for the purposes of closer descriptions of states of motion; so likewise, — Peirce, 1906

Perfectly sensible. Our physics is an approximation or conceptual model to help us describe reality. It's not to be confused with reality. A point I've made many times.

Physical reality is a dynamical process of continuous motion, while psychical reality is an inferential process of continuous thought; more generally, continuous semeiosis. Positions and propositions are artificial creations for describing hypothetical instantaneous states of motion and thought/semeiosis, respectively. — aletheist

Yes of course. Perfectly well agreed. But if all Peirce is saying is that the map is not the territory, that our mathematical and conceptual models are useful fictions to help us manage or conceptualize reality. then of course he's right; but is that all there is? I thought this point was fairly well agreed, even in science. Einstein supersedes Newton supersedes Aristotle. You never get to the end of the process, you just get increasingly better models.

I had never read anything on model theory for real numbers. The materials I had run into were all about natural numbers. — alcontali

To add to the Wiki quote, something I mentioned earlier: The hyperreals are not Cauchy-complete. No non-Archimedean field can be. Which leads to one of my little hobby horses. The constructive reals aren't complete because there are too few of them, only countably many. The hyperreals aren't complete because there are too many of them, the reals plus an uncountably infinite cloud of infinitesimals about each real. The standard reals are the Goldilocks model of the reals. Not too small and not too big to be Cauchy-complete. They're just right. And are therefore to be taken as the morally correct model of the reals.

I'm saying a number "exists" only if it can be instantiated by something in reality. — Michael Lee

That seems unduly restrictive. By that criterion you would have rejected Riemann's non-Euclidean geometry in the 1840's because it was so obviously untrue about the world. Then when Einstein used Riemannian geometry to frame his general theory of relativity, you'd have had to change your mind. Except that by your logic, you'd have abandoned research into Riemann's work and Einstein would never have had the tool available.

Isn't it rather the job of math not to describe reality as we know it; but to provide concepts and tools that may be of use to future scientists? The existence of a mathematical object depends only on logical consistency and interestingness. Not on conformity to the limitations of contemporary knowledge of the world.

Suppose there is one and only one thing in the Universe and absolutely nothing else. Then the only number that exists is one. — Michael Lee

Wait, what? That's two things. The thing that exists and the number one. If a thing exists and there's no conscious entity around to comprehend it, there are no numbers. That would be my view. That numbers are an artifact of consciousness. There's a thing, but there's not the number one till someone experiences that thing; and moreover, evolves sufficient reason to count the thing. Counting's not an inherent part of the universe. It's something rational beings do. No experiencer, no numbers.

Calculus was just school exam material for me consisting of endless symbol manipulation. I didn't particularly "care" about it. It is not that I have read anything about calculus ever since. It doesn't appear in computer-science subjects either. So, what am I supposed to do with it? — alcontali

You invoked the extended real numbers and claimed it has something to do with L-S, which of course it does not. Unless I misunderstood your point.

Not sure about Aristotle, but Peirce indeed explicitly rejected the notion that continuous time is somehow composed of durationless instants. They are artificial creations of thought for marking and measuring time, just like discrete points on a line. — aletheist

Thanks. That would make sense. Physics would get more difficult I imagine.

Then, they say something very interesting but really complicated about the generalized continuum hypothesis in conjunction with real closed fields. I think that this is the kind of things that could shed light on the true nature of infinite cardinality in real-number theory. — alcontali

It means essentially that CH is equivalent to the fact that all models of the hyperreals are isomorphic. The idea is that the particular model of hyperreals you get depends on which nonprincipal ultrafilter you choose. If CH holds then all the models are isomorphic.

There's a Mathoverflow thread about this, let me see if I can find it. Ah here it is. Good luck reading. MO as you know is a site for professional mathematicians so the best one can hope for is to understand a few of the words on the page.

https://mathoverflow.net/questions/136720/why-does-ch-imply-that-there-is-a-unique-ultrapower-of-mathbbn

Also see:
https://mathoverflow.net/questions/88292/non-zfc-set-theory-and-nonuniqueness-of-the-hyperreals-problem-solved

I don't know the answers to all the good questions you raise, but I can't help thinking that you're overthinking things and letting yourself get confused by Lowenheim-Skolem.

I am confused myself over whether the completeness property is first or second order. I've seen explanations both ways. I believe it's second order. The hyperreals are a model of the first-order theory of the reals, but the hyperreals are not Cauchy-complete. That seems to imply that completeness must be second order.

In mathematics, a real closed field is a field F that has the same first-order properties as the field of real numbers. Some examples are the field of real numbers, the field of real algebraic numbers, and the field of hyperreal numbers. — Wikipedia on real closed fields which are a first-order theory

Makes perfect sense. The algebraic numbers are not Cauchy-complete but they are a real closed field, just as the real numbers (which are Cauchy-complete) are.

Still, the fact that an affinely extended real number system is possible, suggests that mathematical analysis may have exactly the same interpretation problem as Peano's arithmetic (PA), i.e. if one infinite cardinality satisfies the model, then all other upward infinite cardinalities also do. — alcontali

This really isn't true, since the standard reals with Cauchy-completeness are second order. They provably have cardinality $2^{\aleph_0}$. This is the part where you're confusing yourself.

Also the extended reals of analysis with $\pm \infty$ have nothing to do with any of this. The extra points don't participate in the field properties as I'm sure you know from calculus.

The Peirceans who are not on this forum, for starters; — aletheist

Makes sense. I've never met any besides here. Must not hang out among the right philosophers.

I can certainly see Peirce's objection that a true continuum could never be made up of individual points. Did Aristotle reject the notion of an instant of time? Or did Peirce? You could't accept instants if you reject points, I'd imagine.

That's the problem, I don't believe in the existence of any set. — Metaphysician Undercover

I've previously called your philosophy mathematical nihilism, and once again you confirm it. You start by saying you don't believe in the empty set; but it doesn't take long to get you to agree that you don't believe in the existence of any sets at all.

If you don't believe in sets, why go to the trouble of explaining why you don't believe in the empty set? I wonder if that shows that you haven't thought your idea through. Why bother to argue about the lack of elements, when you don't even believe in sets that are chock-full of elements?

That any set has real existence has not yet been demonstrated to me. And axioms which allow for the demonstrably contradictory "empty set" lead me away from believing that sets could be anything real. — Metaphysician Undercover

But this is a strawman argument, "... giving the impression of refuting an opponent's argument, while actually refuting an argument that was not presented by that opponent."

Nobody has claimed sets have "real" existence, whatever that is. Sets have mathematical existence, and that's the only claim I'm making.

I could easily take you down the rabbit hole of your own words. Is an electron "real?" How about a quark? How about a string? How about a loop? And for that matter, how about a brick? Are there bricks? When we closely examine a brick we see a chemical compound made of molecules, which are made of atoms, which contain protons, neutrons, and electrons, which themselves are nothing more than probability waves smeared across the universe.

Do you believe in the existence of bricks? Physics tells us that even bricks are nothing more than probability waves smeared across the universe. We see a brick in its location simply because that's the most likely location for it to be found. Once in a long while, a brick appears someplace else where it has a low probability of being found. I hope you know that this is standard doctrine of modern physics. Do you deny science along with math?

You are painting yourself into an ontological box. Not for the first time, I might add.

How is that relevant? As Mephist said, a set is identified by its elements. That's the reason why an empty set makes no sense. Clearly a closet is not identified by its elements.. — Metaphysician Undercover

Ok you're right. Closets and empty grocery bags aren't really on point, even though they can be helpful visualizations, such as a grocery bag containing an empty grocery bag to visualize $\{\emptyset \}$.

So how about an axiomatic approach? The axiom schema of specification says that if $P$ s a unary predicate, and $X$ is a set, then $\{x \in X : P(x) \}$ is a set.

Consider the unary predicate $x \neq x$. Let $X$ be any set whatsoever, say the natural numbers or the real numbers or whatever set you might happen to believe in. Then we can define

$\emptyset = \{x \in X : x \neq x \}$.

So if you believe in the existence of any set at all, and you accept the axiom schema of specification, then you must accept the mathematical existence of the empty set.

What say you?

That witless twit Prince Charles flew 16,000 miles in private jets in eleven days before posing for a photo with Greta. That's what turns people against the fake moral posturing of virtue-signaling environmentalism. I would never say fuck you to a 16 year old (unlike those lefties who said that and worse to the Covington kids); but to a vapid preening celebrity like Prince Charlie, I would.

https://www.dailymail.co.uk/news/article-7929735/Prince-Charles-flew-16-000-miles-just-11-days-proudly-posing-Greta-Thunberg-Davos.html

Do you see that this proposition denies the possibility of an empty set? The empty set has no identity as a set, and therefore cannot be a set. — Metaphysician Undercover

A closet is an enclosed space in which I hang my clothing.

One day I remove all the clothing from my closet.

Do I still have a closet?

Do I not in fact have a perfectly empty closet?

Yes, so be careful, any rashes, ringing in your ears, metallic taste in your mouth .....

or, Sensation of Blunt Trauma to Head. — Brett

I only get these feelings when @Mephist is trying to explain something to me.

When you and I agree on something, that's really something to be afraid of; better move the hands on the doomsday clock. But I think the appearance of agreement is based in different principles, so there's really nothing to worry about. — Metaphysician Undercover

LOL. @Mephist was making the point that one can do "set theory without elements" as in Lawvere's elementary theory of the category of sets, which unfortunately doesn't have a Wiki page. But basicaly you can do most of set theory in a purely categorical way. As I understand it you get a slightly weaker version of set theory than the standard theory.

some argue that the real numbers are not truly continuous, — aletheist

Who argues that, exactly, besides the Peirceans on this forum? I've actually never run across this point of view except for here. And how does that square with the intuitionist continuum, which has even fewer points than the standard reals? They can't all be right.

Actually, this vector field is a good example of a dependently-typed function. The domain of the function is the surface of the sphere, but what is it's codomain? For each point of the sphere the codomain is a different vector space. But all these vector spaces are identical, except for the fact that they are associated to a different base point. This in type theory is called a parametric type: a type that depends on a parameter in an "uniform" way. And the value of the function is the vector representing wind's direction and velocity, that of course vary with the point on the sphere. — Mephist

Ah. Thank you. That was very interesting and helpful.

Hair Loss — Brett

So THAT's how it happened.

if we have an infinite number of sets, and a set that includes all of them and itself, — Gregory

If you have a set that includes itself as an element, you're no longer in the realm of standard set theory, in which self-membership is forbidden by the axiom of regularity.

https://en.wikipedia.org/wiki/Axiom_of_regularity


There are in fact sets that contain themselves, but not in standard set theory.

https://en.wikipedia.org/wiki/Non-well-founded_set_theory

In my opinion, the misleading part of that example is that the tangent planes seem to have some points in common, since they are immersed in an ambient 3-dimensional space. That's not true! The tangent vector spaces are completely separated from each-other (no points in common). — Mephist

Oh I see. Good point. Funny but it never occurred to me to be confused by that. The tangent planes are conceptual thingies attached to each point but they don't "intersect in 3-space" at all. The technical condition is that the total space is the disjoint union of the fibers. I suppose I like this example because it's nice and concrete. For example a vector field is a choice of a single vector from each fiber. So if we have a vector at each point of a sphere that gives the wind direction and velocity at that point, that's a section of a fiber bundle. In set-theoretic terms a section is a right inverse of the projection map. That's how I think about all this.

In set theory class many moons ago I proved that "every surjection has a right inverse" is equivalent to the axiom of choice. That makes sense because it says we can always make a simultaneous choice of a tangent vector from each tangent plane. When I found out that a section is what differential geometers call a right inverse, I was enlightened.

Yes, of course it is! — Mephist

That's the intuition I'm working with at the moment, special case that it may be.

Unfortunately, it's not downloadable for free — Mephist

Yes I just checked that out. I'll keep searching around for an insight.

The usual intuition is more like an "airbrush" ( https://en.wikipedia.org/wiki/Fiber_bundle ). The fibers are seen as stick wires coming out from a common surface; they are separated from each other. — Mephist

I stand by my remark. The tangent bundle of a sphere is most definitely a fiber bundle.

https://en.wikipedia.org/wiki/Tangent_bundle

Can you give me the link you want me to look at? There's been so much back and forth and so many links.

Good! thanks for wasting my time you giant Philip K. — Bartricks

Have you seen any of the movies Minority Report, Blade Runner, Total Recall, The Adjustment Bureau, or A Scanner Darkly?

Well, OK, never mind. However, the book that I gave you the link is very clear and contains proofs and exact definitions. Surely that's easier to understand than my explanations... — Mephist

Ok I will have a look. Many links have been posted recently. Can you repost the one you want me to look at please?

But I especially chose a finite set to make it crystal clear — Mephist

Not clear to me. I literally and honestly did not understand what you said in this post. Perhaps it's a lost cause.

A fiber bundle is like the collection of tangent planes to a sphere. Somehow, one can replace the tangent planes with logical structures of some sort, and the points of the sphere with .... something, and intuitionist logic drops out. Perhaps it's not explainable in elementary terms. But I couldn't relate what you wrote with any attempt to clarify this point.

Taking your .10101010..., how long is it? How many zeros and ones? As many as there are counting numbers? Or more? ℵo or ℵ1? — tim wood

There's one decimal place for each natural number. A decimal expression .abcdef... means a/10 + b/100 + c/1000 + ... There's one place for each negative power of 10.

I'm thinking the number of digits must be countable. And I'm thinking my listing, then, being ordered, is also countable. It's all countable. But clearly that's not correct. — tim wood

Your list is countable. You've listed all the FINITE bitstrings. Where is .10101010101010... on your list? It's not there.

How? I'm not embarrassed. He should be, with a name like that.

Anyway, how about actually addressing the OP rather than telling me about dead science fiction authors with silly names — Bartricks

Asshole. LOL. No more from me on this.

No, I am unsure who he was — Bartricks

Hence embarrassing yourself. Use the Google, Luke. Philip K. Dick was a prolific writer of science fiction. He's greatly revered.

fishfry I don't think we're living in the past. I thought you were saying that it was his opinion that we were. If he agrees with me, then his surname is unjust. — Bartricks

In some of his later writings he expressed that idea. I can't imagine slurring the guy for his name. Do you even know who he is? I'm going to let this go. Sorry I mentioned it.

↪fishfry
Philip K. Dick was of that opinion.
— fishfry

Then he deserved his surname. — Bartricks

Uh ... he's agreeing with you. I'm not sure I follow your point, and I disagree strongly with your apparent criticism of the man.

If that's true, then doesn't that mean we are subject to a systematic illusion of the present? — Bartricks

Philip K. Dick was of that opinion.

— tim wood

Let's start a list of them all.
.1
.01
.11
.001
.011
.101
.111
.0001
.0011
.0101
.0111
.1001
.1011
.1101
...
You get the idea.

This list will eventually take in all the numerals of denumerable length. — tim wood

No, your idea only lists all the bitstrings of FINITE length, of which there are only countably many. For example 1010101010101010... never appears on your list.

There are no points in a truly continuous line, period. — aletheist

That's a Peircean view and not a standard mathematical view; and I think it's important to make that distinction when explaining things. The standard mathematical view is that "the continuum," "the real line," and "the set of real numbers" are synonymous. Philosophical considerations do not alter the conventional mathematical meanings.

consider sets to be more "fundamental" than their elements. — Mephist

I'm afraid I share @Metaphysician Undercover's misgivings about this remark. I understand the categorical viewpoint of sets, but I would not characterize that viewpoint via this particular way of phrasing it.

I see that there is a misunderstanding between us on what it means "a logic has a model". — Mephist

I found a paper that indicated the the fibers are "L-structures." Not too sure what those are, or what the base set is. I'm not sure I entirely believe it's a discrete topological space. I'm thinking you've probably explained this point to me several times over but I still don't get it. My apologies for giving you a hard time out of frustration at my inability to understand how fiber bundles can be used to model logical structures.

From page 4 of the the text referenced:
1) "Since the cardinality of the set R of reals is the same as that of the powerset P(N) of the set of natural numbers."

Please help me out? — tim wood

The powerset of a set is the set of all subsets of the set. So for example $\mathcal P(\{1,2,3\}) = \{\emptyset , \{1\}, \{2\}, \{3\}, \{1,2 \},\{1,3\}, \{2,3\},\{1,2,3\}\}$.

It's easy to show that $| \mathbb R | = | \mathcal P(\mathbb N)|$. For any subset of $\mathbb N$, create a bitstring that has 1 in the n-th position if n is in the subset, 0 otherwise. Put a binary point in front of the string and you have the binary expression of a real number in the unit interval, and vice versa.

For example the real number whose binary representation is .10101010101... corresponds to the set {1, 3, 5, 7, ...}. So we have a bijection between the real numbers (in the unit interval) and the subsets of the natural numbers. (For convenience I'm numbering positions to the right of the binary point starting at 1, and excluding 0 from the natural numbers).

[We can ignore dual representations (.5 = .4999...) because there are only countably many of those and countable sets don't make any difference to the cardinality of an infinite set].

2) Is there an error in thinking of a representation of a powerset as all the permutations of the elements of the original set? — tim wood

That doesn't work because for example the set of permutations of {1.2.3} is 123, 132, 321, 312, 213, and 231. It's not the same thing.

3) if 1 and 2 are correct (and if 2 is correct, then I'm thinking 1 obviously follows), then the question of the cardinality of the continuum, c, becomes the question of the existence of point on the line to which no real number can be applied - for some reason: is this a correct way to think of it? — tim wood

No not really. We just proved above that the cardinality of the reals is $2^{\mathbb N}$, the set of functions from $\mathbb N$ to the set {0,1} (which we can think of as the set of bitstrings). Now if the transfinite cardinals are $\aleph_0, \aleph_1, \aleph_2, \dots$, the question is which Aleph is $2^{\mathbb N}$? The claim that it's $\aleph_1$ is the Continuum hypothesis. For all we know it's some other Aleph, perhaps a very large one. The answer is independent of the usual axioms of set theory.

4) But if 3, and there is no such point on the line, then (it appears to me) that c = P(N). — tim wood

I'm not sure I follow your idea of a particular point on the line.

5) And it cannot be that simple. which implies there are points on the line that cannot be numbered. — tim wood

I don't follow your idea but that's not what CH is. CH is just the question of which Aleph is the cardinality of the reals.

6) By "number on the line," I am assuming that to each point on the line is assignable some unique number representable as, say, some numeral in binary form, all of which points/binary numerals represented in the set of permutations of all the zeros and ones. — tim wood

The points on the real line are just the real numbers and vice versa.

Is 5 the true statement, that there are points on the line to which no real number can be applied? — tim wood

I'm afraid I don't follow this idea. The "real line" is just a synonym for the set of real numbers.

I apologize for interrupting a productive flow of thought. But I was curious what you guys were talking about. Seems pretty esoteric. — jgill

I can summarize. Short answer is that these days you can do logic via category theory; and when you do that, you get intuitionist logic (denial of the law of the excluded middle (LEM) and all that) in a natural way.

This relates to topology via the idea of fiber bundles from differential geometry if you know what those are. If not just ask). The examples presented so far aren't clear to me and I haven't worked through @fdrake's promising-looking examples. @Mephist may or may not have presented coherent examples but there's a gap between his expositions and my understanding that only gets worse over time. The fault may all be mine.

There's a very nice illustration of how this works if you consider any topological space and restrict your attention to the open sets. The "complement" of an open set in a topology, if you only care about open sets, is the interior of the complement of the set. (In other words in general the complement is not an open set, but if we only look at the open sets, it makes sense to define funny complements this way).

So it is in general NOT true that the complement of the (true) complement of an open set is going to give you back the original open set. This corresponds to a failure of the law of the excluded middle.

For a very nice overview of the history and meaning of all this I recommend the prologue of Mac Lane's Sheaves in Geometry and Logic. One need not understand the details to get the big picture from this very clearly written book.

Intuitionism was developed in the 1930's but didn't get any mindshare in mainstream math. Now with the advent of computers (where the complement of a noncomputable set of natural numbers may also be noncomputable), denial of LEM is back in fashion, especially in computer science and categorical logic. I call this Brouwer's revenge. Brouwer invented intuitionism in the old days but it didn't catch on. His form of intuitionism had a touch of mysticism to it, but the modern versions are mathematically solid. Fifty years from now (or sooner) they'll be teaching this to undergrads and set theory will be a relic of the past like Euclidean geometry. Set theory of course won't become wrong, just out of fashion.

Another thread of development is that mathematicians want to use computers to check their proofs for accuracy. It turns out that intuitionist type theory (which I know nothing about) is the key. In homotopy type theory one uses the idea of homotopy from topology (continuously deforming one path into another) to do intuitionist logic in such a way that you can build working computerized proof assistance for professional mathematicians. Also see intuitionistic type theory.

These are the broad outlines I've picked up, but I haven't spent much if any time on the details. One name you'll hear a lot is Vladimir Voevodsky, a Fields medal winning mathematician who became frustrated at longstanding errors in published proofs and devoted himself to the project of computerized proof assistants. He died tragically young just recently, in 2017.

Voevodsky's contribution is the Univalent foundations of mathematics. The idea here is that mathematicians routinely conflate equality and equivalence whenever it's convenient. For example there's only one cyclic group of order four, even though there are lots of isomorphic copies of it that are not equal as sets. For example the integers mod 4 and the powers of the imaginary unit $i$ are the "same" group.

I should mention that this kind of thing bothers @Metaphysician Undercover greatly, and he's right that mathematical equality can sometimes be stretched past what he would consider true equality. The answer to this is that if it quacks like a duck it's a duck, and if it's isomorphic to the cyclic group of order four, it doesn't matter what representation we choose. They're all the "same" in the appropriate technical sense.

In category theory this conflating of equality and equivalence becomes semi-formalized via universal properties. Any two objects that satisfy the same universal property are isomorphic and are regarded as the same thing.

Univalent foundations takes this one step farther by formalizing an axiom that says that equivalent things are equal. My high-level understanding is that the univalence axiom makes mathematically precise the informal practice mathematicians have been accustomed to for decades.

https://en.wikipedia.org/wiki/Univalent_foundations

All of what I've written is of course hopelessly vague and should not be relied on as gospel. But I've hit most of the buzzwords and major concepts in their broad outlines.

ps -- The immediate subject of the thread recently is to see how we can view intuitionist logic as an example of a fiber bundle. Maybe that was the short answer to the question.