We can formalize the process of filling in the holes with various technical constructions of the reals. There are several, the two best known being Dedekind cuts and Cauchy sequences. The details aren't of interest. The point is that it can be done within set theory and it allows us to found calculus in a logically rigorous way, something that escaped Newton and Leibniz. We can also axiomatically define the reals as "the unique Cauchy-complete totally ordered infinite field." When you unpack the technical terms, you end up with an axiomatic system that's satisfied within set theory by the Dedekind cuts or Cauchy sequences. It's all very neat. One need not believe in it or care. It must be frustrating to you to both not believe in it, yet care so much! — fishfry
TD;LR: we should teach ZF in high school and then add C later for pure maths students. — boethius
as small as you want ... but not infinitesimal — boethius
As for the intricacies of the real number system, I wonder — jgill
Second, if your complaint is with pedagogy it's not about math. — fishfry
It would be fun to teach ZF to SOME high school students, the especially mathematically talented ones. The mainstream, no. I wonder what you are talking about here. Again, the axiom of choice is not needed to defined or construct the reals. — fishfry
It's worth noting that the pedagogy retraces the history. — fishfry
Dating from 1687, the publication of Newton's Principia, to the 1880's, after Cantor's set theory and the 19th century work of Cauchy and Weirstrass and the other great pioneers of real analysis; it took two centuries for the smartest people in the world to finally come up with the logically rigorous concept of the limit. For the first time we could write down some axioms and definitions and have a perfectly valid logical theory of calculus. — fishfry
Logical properties
The method of constructing infinitesimals of the kind used in nonstandard analysis depends on the model and which collection of axioms are used. We consider here systems where infinitesimals can be shown to exist.
In 1936 Maltsev proved the compactness theorem. This theorem is fundamental for the existence of infinitesimals as it proves that it is possible to formalise them [...]
There are in fact many ways to construct such a one-dimensional linearly ordered set of numbers, but fundamentally, there are two different approaches:
1) Extend the number system so that it contains more numbers than the real numbers.
2) Extend the axioms (or extend the language) so that the distinction between the infinitesimals and non-infinitesimals can be made in the real numbers themselves.
[...]
In 1977 Edward Nelson provided an answer following the second approach. The extended axioms are IST, which stands either for Internal set theory or for the initials of the three extra axioms: Idealization, Standardization, Transfer. In this system we consider that the language is extended in such a way that we can express facts about infinitesimals. The real numbers are either standard or nonstandard. An infinitesimal is a nonstandard real number that is less, in absolute value, than any positive standard real number.
Calculus textbooks based on infinitesimals include the classic Calculus Made Easy by Silvanus P. Thompson (bearing the motto "What one fool can do another can"[12]) [...]
Another elementary calculus text that uses the theory of infinitesimals as developed by Robinson is Infinitesimal Calculus by Henle and Kleinberg, originally published in 1979.[16] The authors introduce the language of first order logic, and demonstrate the construction of a first order model of the hyperreal numbers. The text provides an introduction to the basics of integral and differential calculus in one dimension, including sequences and series of functions. In an Appendix, they also treat the extension of their model to the hyperhyperreals, and demonstrate some applications for the extended model.
The problem is that no math course has enough time to really take the time. Usually it's just "here's the proof, there, I showed it to you, now use this algorithm".My beef is that the real numbers are introduced too early in education. Infinite processes and essentially 100% of numbers being infinitely complex are, though perhaps can be dealt with abstract rules, too difficult to conceptually grasp for most high school students. — boethius
With this clear distinction the complications of maths fade away. — A Seagull
my feeling is that you can't really do pure maths without set theory, — boethius
I presume that you are referring to the idea that set theory provides the 'foundation' to mathematics. — A Seagull
But pure mathematics is abstract and doesn't need any foundations apart from its axioms which introduce the symbols and define the rules. (And admittedly these axioms are more implicit than explicit). — A Seagull
And as for the real numbers, they become necessary when one looks to divide (for example) 10 by 4. (10/4). although the task is in the domain of integers the answer is outside. — A Seagull
So if you want to get back on track, answer my questions concerning the real numbers i — boethius
Perhaps you could state them succinctly. — fishfry
You have an ax to grind and I've only succeeded in upsetting you. — fishfry
The whole point of my post is that high school students would have no way of stating their questions succinctly as you demand, but they are in my view meaningful questions to ask. — boethius
Instead of accepting the conclusion that root 2 is irrational (not a rational number) — boethius
Instead of accepting the conclusion that root 2 is irrational — boethius
Please demonstrate how this infinite numerator and denominator either does not get counted in Cantor's diagonal proof, does not represent an irrational value, or there is something preventing finding and placing all the digits of suitable real numbers into a numerator and denominator to solve for root 2. — boethius
What axioms does a high school student possess to avoid the above issue of concluding root 2 is rational? — boethius
Followup question (as I believe this is what interests you to demonstrate) what axioms does a university student possess to avoid the above issue and how? — boethius
But this isn't what the words "constructing a number" suggest to me. Any light for the darkness, here? — tim wood
Very, and very clear. Especially this.Was this helpful? — fishfry
Because this is the answer to a question I'll ask pro forma below.Construct in this context means build a thingie within set theory that behaves exactly like we want our thingie to do. — fishfry
Then for the application to the 'real world' (applied maths) one takes a particular part of mathematics and applies a mapping between the abstract symbols and concepts that apply to the 'real world'. — A Seagull
Why set theory? Set theory is pretty uninteresting really, apart from Venn diagrams which are fun and useful. — A Seagull
The main application of relational algebra is providing a theoretical foundation for relational databases, particularly query languages for such databases, chief among which is SQL. The relational algebra uses set union, set difference, and Cartesian product from set theory, but adds additional constraints to these operators. — Wikipedia on relational algebra
I'm familiar with the axioms (I'm a child of new math, if you know what that 1960s fad was), — tim wood
but the lub - well, that's not so clear. Maybe because it's too obvious - that happens.
From online
"Let S be a non-empty set of real numbers.
1) A real number x is called an upper bound for S if x ≥ s for all s ∈ S.
2) A real number x is the least upper bound (or supremum) for S if x is an upper bound for S and x ≤ y for every upper bound y of S."
1) is pretty clear. With respect to integers only, given the set (1,2,3) 3 is an upper bound. Now here's maybe the question the answer to which will help me out. 3 is an upper bound, but is it correct to say that all x, x>3 are also upper bounds, and that 3 is the least upper bound? — tim wood
And the idea that the rationals do not provide a lub for the square root of two simply means that although there is no end of upper bounds, for any upper bound that seems a candidate for the lub, a better candidate can always be found, in the rationals. If this is it, then I understand the lub. — tim wood
Here the pro forma question, though it's evolved since the first paragraph. And even this you've already answered. It seems to me that to question the existence of a measure, or distance, or number corresponding to the square root of two is the greatest nonsense, because it is so easily modeled, and a fortiori any other irrational. Almost as easily is π modeled, so with transcendentals. — tim wood
Nor is a numeral for any of these lacking, if by "numeral" is meant a name. Of course an exact decimal representation is a problem, But then so is my idea of the perfect woman (pace wife). But it appears that the proof of these things ignores the obvious: irrational numbers are easily proved to exist. For me, I guess, the question is, what is (was) the problem? What the need for the thingie? (If for some arcane application, that's enough of an answer: likely I could not follow anything more than that.) And ty, btw. — tim wood
Some years ago the New Math was in vogue. As a mere instructor at the time I was given a text on College Algebra having a lengthy first chapter devoted to an axiomatic approach to the subject. It was not a good experience for instructor or student. — jgill
We're in deep and complete agreement on this. The mathematical definition of the real numbers is far beyond high school students; in analogy with the difficulties Newton and Leibniz had, which needed to wait 200 years for resolution. — fishfry
ps -- Note well The irrationality of the square root of 2 does NOT introduce infinity into mathematics. All the irrationals familiar to us are computable, and have finite representations. The noncomputable reals do introduce infinity into math; but plenty of people who don't believe in noncomputable reals nevertheless DO believe in the square root of 2. Namely, the constructive mathematicians. — fishfry
Euclid's proof of the irrationality of 2‾√2 has nothing at all to do with Cantor's discovery of the uncountability of the reals. The rest of this paragraph, I confess, is not intelligible to me. — fishfry
None whatever. In high school we mumble something about "infinite decimals" while frantically waving our hands; and the brighter students manage not to be permanently scarred for life. — fishfry
The teaching of mathematics in the US public schools is execrable. How many times do I have to agree with you about this? — fishfry
A university student in anything other than math: None. — fishfry
A well-schooled undergrad math major? Someone who took courses in real and complex analysis, number theory, abstract algebra, set theory, and topology? They could construct the real numbers starting from the axioms of ZF. They could then define continuity and limits and I could rigorously found calculus. It's not taught in any one course, it's just something you pick up after awhile. The axiom of infinity gives you the natural numbers as a model of the Peano axioms. From those you can build up the integers; then the rationals; and then the reals. Every math major sees this process once in their life but not twice. Nobody actually uses the formal definitions. It's just good to know that we could write them down if we had to. — fishfry
Lol. Well, they took it back so I guess that the cadre was very small. And as Fishfry commented earlier, this experiment wasn't just limited to my country (Finland), but the US too. I'd suspect that we copied the 'new trends' during those progressive times from the US. From the viewpoint of teaching small children math, starting with counting sheep is the way to go. It is the natural way, I'd argue.Did this set theory experiment simply not work at all, or did it produce some small cadre of math geniuses? — boethius
I think this is a bigger philosophical problem for mathematics. Basically mathematics has evolved from the necessity of counting, calculating and computation. Hence 'applied math' came first. Abstract mathematical thought has emerged only later. This makes us start mathematics from the natural numbers.In teaching maths, I think it is important to make a clear distinction between pure (abstract) maths and applied (to the real world) maths. It is the conflation of the two that causes problems.
Of course children first learn maths with the conflated maths; counting sheep etc. But perhaps around the time they enter secondary school ( around age 13) the distinction needs to be emphasised. — A Seagull
However, it's also necessary to deal with questions like "why can't we just have rational numbers with infinite numerator and denominator; seems unfair to allow infinite decimal expansion but not infinite numerators?". Of course, we can have rational numbers with infinite numerator and denominator but it's then needed to explain how these aren't the "real" rational numbers we're talking about when we say the square root of two is irrational. — boethius
Starting mathematics from the natural numbers is pretty natural. If you begin with nothing but the empty set and the sole sufficient operator of joint denial, the simplest new operator you can build is disjunction, and the simplest thing you can disjoin with the empty set is the set containing itself, and hey look that’s the first iteration of the successor function and if you keep doing that you end up with the natural numbers. — Pfhorrest
I think the main thing to understand here is that decimal numbers with infinite decimals can be considered as an extension of "regular" decimal numbers (finite list of digits), but infinite natural numbers (infinite list of digits) cannot be considered as an extension of "regular" natural numbers, since you cannot define on them arithmetic operations compatible with the ones defined on the "regular" natural numbers. Then, you can't build fractions with infinite integers because you cannot build infinite integers in the first place. In my opinion this is quite easy to understand. Did I miss something? — Mephist
I think the infinities and infinitesimals of mathematics are the things that make it become more "magic" and interesting. The problem with teaching in my opinion is more related to the fact that the "magic" of the fact that infinities and infinitesimals really work is not explained, or worse, explained by pretending to have a simple logical explanation that, however, is not part of the school program. — Mephist
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