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blog.sigfpe.com | ||
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xorshammer.com
Let $latex \mathrm{PA}$ be Peano Arithmetic. Gödel's Second Incompleteness Theorem says that no consistent theory $latex T$ extending $latex \mathrm{PA}$ can prove its own consistency. (I'll write $latex \mathrm{Con}(T)$ for the statement asserting $latex T$'s consistency; more on this later.) In particular, $latex \mathrm{PA} + \mathrm{Con}(\mathrm{PA})$ is stronger than $latex \mathrm{PA}$. But certainly, given that... |
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unstableontology.com
(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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www.lesswrong.com
The Löwenheim-Skolem theorem implies, among other things, that any first-order theory whose symbols are countable, and which has an infinite model, h... |
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thatsmaths.com
The Riemann Hypothesis Perhaps the greatest unsolved problem in mathematics is to explain the distribution of the prime numbers. The overall ``thinning out'' of the primes less than some number $latex {N}&fg=000000$, as $latex {N}&fg=000000$ increases, is well understood, and is demonstrated by the Prime Number Theorem (PNT). In its simplest form, PNT states that... |
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