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mathwithbaddrawings.com | ||
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thatsmaths.com
The numbers are usually studied in layers of increasing subtlety and intricacy. We start with the natural, or counting, numbers $latex {\mathbb{N} = \{ 1, 2, 3, \dots \}}&fg=000000$. Then come the whole numbers or integers, $latex {\mathbb{Z} = \{ \dots, -2, -1, 0, 1, 2, \dots \}}&fg=000000$. All the ratios of these (avoiding division... |
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jdh.hamkins.org
I have been reading Alan Turing's paper, On computable numbers, with an application to the entsheidungsproblem, an amazing classic, written by Turing while he was a student in Cambridge. This... |
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www.jeremykun.com
It's often that a student's first exposure to rigorous mathematics is through set theory, as originally studied by Georg Cantor. This means we will not treat set theory axiomatically (as in ZF set theory), but rather we will take the definition of a set for granted, and allow any operation to be performed on a set. This will be clear when we present examples, and it will be clear why this is a bad idea when we present paradoxes. |
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dominiczypen.wordpress.com
Let $latex \omega$ denote the first infinite cardinal - that is, the set of non-negative integers. Let $latex p_0 = 2$ be the smallest prime number, and let $latex (p_n)_{n\in\omega}$ enumerate all prime numbers in ascending order. Let $latex \mathcal{U}$ be a free ultrafilter on $latex \omega$. We consider the field $latex F = \big(\prod_{n\in\omega}\mathbb{Z}/p_n\mathbb{Z}\big)/{\mathcal... |
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