Simple paradox

  • #1
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It's simple for you mathematicians, but I'm a physician, I don't know much about set theory or logic and such, so it's difficult for me.

Let M be the set of all integers that can be described in English in, say, ten lines of text. For example, "fourteen" or "seventy minus eight" or "832832541872 to the power of 784315" are all numbers belonging to M. Let k be the largest number in this set. Since in ten lines of text you have a large, but finite, combination of characters, and since not all combinations are meaningful in English, and certainly not all combinations describe a number, then k exists and it's finite.

Let m be k plus one.

I have described, in less than ten lines of text, a number that it's larger than the largest number that can be described in ten lines of (bad, I'm sorry) English text.

Explanations?
 
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Answers and Replies

  • #2
Hurkyl
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This is one of the standard paradoxes of self-reference. A more famous one is "this statement is false".

The standard approach to mathematical logic doesn't permit self-reference. Roughly speaking, logic that talks about objects of interest is "first-order logic". Logic that talks about first-order logic is called "second-order logic", and so forth.

So, "one plus the largest number that can be described in ten lines of first-order logic" is a number defined by second-order logic, and there is no contradiction!
 
  • #3
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What about this: answer "true" or "false" to this statement:

"Your answer will be "false""

or to this other statement:

"Your answer will be false"

is it again a matter of self-reference?
 
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