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To identify the real numbers with the computable numbers would then be a contradiction. And in fact, Cantor's diagonal argument is constructive, in the sense that given a bijection between the real numbers and natural numbers, one constructs a real number that doesn't fit, and thereby proves a contradiction.

We can indeed enumerate algorithms to construct a function T, about which we initially assume that it is a function from the natural numbers onto the reals.

But, to each algorithm, there may or may not correspond a real number, as the algorithm may fail to satisfy the constraints, or even be non-terminating T is a partial function , so this fails to produce the required bijection. In short, one who takes the view that real numbers are individually effectively computable interprets Cantor's result as showing that the real numbers collectively are not recursively enumerable.

Still, one might expect that since T is a partial function from the natural numbers onto the real numbers, that therefore the real numbers are no more than countable. And, since every natural number can be trivially represented as a real number, therefore the real numbers are no less than countable.

They are, therefore exactly countable.

However this reasoning is not constructive, as it still does not construct the required bijection. The classical theorem proving the existence of a bijection in such circumstances, namely the Cantor—Bernstein—Schroeder theorem , is non-constructive and no constructive proof of it is known.

Axiom of choice[ edit ] The status of the axiom of choice in constructive mathematics is complicated by the different approaches of different constructivist programs.

One trivial meaning of "constructive", used informally by mathematicians, is "provable in ZF set theory without the axiom of choice. In intuitionistic theories of type theory especially higher-type arithmetic , many forms of the axiom of choice are permitted. For example, the axiom AC11 can be paraphrased to say that for any relation R on the set of real numbers, if you have proved that for each real number x there is a real number y such that R x,y holds, then there is actually a function F such that R x,F x holds for all real numbers.

Similar choice principles are accepted for all finite types. The motivation for accepting these seemingly nonconstructive principles is the intuitionistic understanding of the proof that "for each real number x there is a real number y such that R x,y holds".

According to the BHK interpretation , this proof itself is essentially the function F that is desired. Of course these are based on the course material that was available when I attended, so these statements do a reflect the course at the time I took it and b are my subjective opinions. It covers user stories, test driven development, agile techniques, working with legacy code, and many more things that are often left out in academical education but are so crucial to being a good programmer.

In fact I would say that this was the course that transformed me from a shitty hacker into a proud and proper software engineer. Links to course offering of part 1 and part 2 ; links to certificates that I completed part 1 and part 2. After this course you will have a good background knowledge to create data models and setup databases on your own. Link to course offering ; link to my Statement of Accomplishment.

CSx: Quantum Mechanics and Quantum Computation Taught by: Umesh Vazirani UC Berkeley Brief description: the course gives a good background about the mathematics behind quantum mechanics and the concepts needed to understand qubits and quantum algorithms.

It is very challenging as the subject is not an easy one, but given the complexity of the topic I think the course does a very good job in giving you all the necessary means to understand the material. I definitely gained a lot of insight into this amazing world that somehow governs how reality works but is still so far from our everyday experience. I would love to see quantum computing turning into real life applications within my lifetime.

It gives a good overview of these topics and encourages to think about these fundamental concepts. I very much liked the course. It consists of several formal problems and paradoxes like the Zeno Paradox, the Monty Hall problem and the theory of voting around Arrows Theorem.

As it does not require a mathematical background it is well suited for beginners but as the covered topics are often quite capturing it is still interesting for someone with a math major.

Technology Entrepreneurship Taught by: Chuck Eesley Stanford University Brief description: the course gives a basic introduction into business model creation, evaluation of market potential and entrepreneurial planning in general. It very much follows the lean start up mentality and focuses very much on team assignments and communication within the course community.

Most material did not go much in depth but the course is a proper overview for people new in the entrepreneurship world. An Introduction to Operations Management Taught by: Christian Terwiesch University of Pennsylvania Brief description: this course gives an introduction into basic topics of operations management like queuing theory and calculation of batch sizes. With a background in mathematics most of these materials are very trivial calculations but it is interesting to see some of the applications and to get some understand of how and when certain methods can be applied in practice.

Computational Investing, Part I Taugh by: Tucker Balch Georgia Institute of Technology Brief description: the course explains some methods for evaluating portfolios and for creating models for financial markets.

It did not go much in depth and the covered material was not very advanced. Furthermore the whole course was more of a work in progress and did not just lack some polishing.