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xorshammer.com | ||
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unstableontology.com
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| | | | | (note: one may find the embedded LaTeX more readable on LessWrong) The Löwenheim-Skolem theorem implies, among other things, that any first-order theory whose symbols are countable, and which has an infinite model, has a countably infinite model. This means that, in attempting to refer to uncountably infinite structures (such as in set theory), one "may... | |
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billwadge.com
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| | | | | The famous mathematician Kurt Gödel proved two "incompleteness" theorems. This is their story. By the 1930s logicians, especially Tarski, had figured out the semantics of predicate logic. Tarski described what exactly was an 'interpretation' and what it meant for a formula to be true in an interpretation. Briefly, an interpretation is a nonempty set (the... | |
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jdh.hamkins.org
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| | | | | I'd like to share a simple proof I've discovered recently of a surprising fact: there is a universal algorithm, capable of computing any given function! Wait, what? What on earth do I ... | |
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peterbloem.nl
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| | | [AI summary] The text provides an in-depth explanation of the Fundamental Theorem of Algebra, which states that every non-constant polynomial of degree $ n $ has exactly $ n $ roots in the complex number system, counting multiplicities. It walks through the proof by first establishing that every polynomial has at least one complex root (using the properties of continuous functions and the complex plane), then using polynomial division to factor the polynomial into linear factors, and finally addressing the nature of roots (real vs. complex) and their multiplicities. The text also touches on the conjugate root theorem, which explains why complex roots of polynomials with real coefficients come in conjugate pairs. | ||
