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Robert'); DROP TABLE *; wrote:Are there some theorems that cannot be written down
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Robert'); DROP TABLE *; wrote:I know that there are an uncountably infinite amount of irrational numbers, and so an uncountably infinite number of theorems along the lines of proving every irrational number to be irrational. However, there are only a countably infinite number of strings of potentially infinite length. Are there some theorems that cannot be written down, or have I missed something?
In math, you have to specify how you are writing things down - what language. If the language you are using is consistent, then there are, by necessity, true things which cannot be written in that language. For every two consistent languages, there is another language that can say by itself everything that either of the two says on their own. No matter how sophisticated you make your system, there will always be things you cannot say, or else you will be struck by inconsistencies."Suppose the set T of number which cannot be written down is nonempty. Choose some t in T. As t is a written form an element of T, it has been written down, and so cannot be in T. Contradiction."
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Big freaky cereal boxes of death.
Yakk wrote:Multiple choice quote goes here!
Yakk wrote:Multiple choice:
1) There are only a countable number of numbers that can be picked out and distinguished from other numbers.
2) There are only a countable number of numbers that can be distinctly described.
3) Not all sets you can describe in English exist.
4) The Axiom of Choice is not true.
5) Zorn's Lemma is true.
6) S(5)
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