How To Build Infinite Numbers


I had missed it, as mentioned in the above tweet. The link is to a page on the Form And Formalism blog, reprinting a translation of one of Georg Cantor’s papers in which he founded the modern understanding of sets, of infinite sets, and of infinitely large numbers. Although it gets into pretty heady topics, it doesn’t actually require a mathematical background, at least as I look at it; it just requires a willingness to follow long chains of reasoning, which I admit is much harder than algebra.

Cantor — whom I’d talked a bit about in a recent Reading The Comics post — was deeply concerned and intrigued by infinity. His paper enters into that curious space where mathematics, philosophy, and even theology blend together, since it’s difficult to talk about the infinite without people thinking of God. I admit the philosophical side of the discussion is difficult for me to follow, and the theological side harder yet, but a philosopher or theologian would probably have symmetric complaints.

The translation is provided as scans of a typewritten document, so you can see what it was like trying to include mathematical symbols in non-typeset text in the days before LaTeX (which is great at it, but requires annoying amounts of setup) or HTML (which is mediocre at it, but requires less setup) or Word (I don’t use Word) were available. Somehow, folks managed to live through times like that, but it wasn’t pretty.

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Author: Joseph Nebus

I was born 198 years to the day after Johnny Appleseed. The differences between us do not end there.

8 thoughts on “How To Build Infinite Numbers”

        1. Ah, I see. I wouldn’t have thought to connect the topics quite that way, although it’s possible I’m just thinking too heavily of how it happened to be done the semesters I took linear algebra, which were pretty heavily biased towards the sorts of matrix and vector space stuff that would be helpful in physics. Maybe I failed to read the chapters the professor chose to skip.

          (I didn’t have much choice: I lost my textbook after the first exam and couldn’t buy or borrow a second copy. Luckily homeworks were assigned by actually writing out the problems, rather than just ‘Chapter 2.3 3-9 odds, 12, 14’, so I could keep up, but it was tougher than it needed to be. I’m not positive the professor wasn’t kind to me with my final grade, or whether having to pay extremely close attention to definitions and proofs in class was better for me than trusting I could check the details in the textbook later on.)

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          1. Yes, the lecture was mainly matrices, vector spaces, and tensors. The set of sets and Cantor’s diagonal argument etc. were mentioned in one of the first chapters if I recall correctly. Russell’s proof (or some version of it) required mapping elements of a set onto their power set (or something ;-)) so this was introduced right after surjective and injective linear maps.

            I am too lazy now, but I could check – I still do have this textbook, but it is literally falling apart!

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          2. … and as you speak about Real Analysis this is maybe the time to ask a perfectly stupid question, but blame it on differences in our educational systems and my ignorance thereof: When I was a student of physics, I had two math classes in the first year, Linear Algebra and Real Analysis (two semesters each, first and second of my “undergrad” studies, though we had no bachelor degrees back then, only masters – this was just the first year of five).
            So I always thought “Calculus” = “Real Analysis”. But it isn’t, right?
            I know this may sound dumb but I have tacitly made this assumption so often, so I admit my blunder publicly now :-)

            Real analysis was mainly theorems and proofs, “building math from scratch”, series and functions, their properties – continuous, differentiable etc.
            Is “Calculus” more about learning rules how to integrate and differentiate, but without all those detailed proofs? I started thinking about it when I read a book by a science writer (an English major) who tought herself calculus later. It seems it had not been mandatory in her high school. Then I’d understand why colleges would have to teach calculus to make sure everybody has the same background. I can remember I had a few colleagues who came from a high school not at all specialized in science. We have e.g. something like “business highschools”, with accounting classes and the like…. but those students were unlikely to pick a science degree at the university so perhaps nobody cared that they had a really hard time with that rigorous, proof-based math right from day 1.

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            1. I had to think about this one a bit, but I believe there is a subtle difference between Calculus and Real Analysis. Real Analysis is the study of real-valued functions — how to define them, how to use them, how to manipulate them. But the most interesting stuff to do with real-valued functions that you can teach with the sorts of proofs that new students can follow or reconstruct are generally the things that we get in intro calculus: finding maximums and minimums, finding derivatives, integrating, that sort of thing. So Real Analysis tends to look like “Intro Calculus, only this time you have to do the proofs”.

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