Claim: Every natural number can be expressed using only the words: 'the', 'least', 'natural', 'number', 'which', 'cannot', 'be', 'expressed', 'using', 'only', 'words'.
Proof: Assume (to reach a contradiction) that this is not the case. Then there is a least natural number which cannot be expressed thusly. Consider the expression "The least natural number which cannot be expressed using only the words: 'the', 'least', 'natural', 'number', 'which', 'cannot', 'be', 'expressed', 'using', 'only', 'words'." We have expressed said number thusly. Contradiction.
Claim: Every natural number can be expressed using only precisely two copies of each word amongst: 'the', 'least', 'natural', 'number', 'which', 'cannot', 'be', 'expressed', 'using', 'only', 'precisely', 'two', 'copies', 'of, 'each', 'word', 'amongst'.
Proof: Assume (to reach a contradiction) that this is not the case. Then there is a least natural number which cannot be expressed thusly. Consider the expression "The least natural number which cannot be expressed using only precisely two copies of each word amongst: 'the', 'least', 'natural', 'number', 'which', 'cannot', 'be', 'expressed', 'using', 'only', 'precisely', 'two', 'copies', 'of, 'each', 'word', 'amongst'." We have expressed said number thusly. Contradiction.
In particular, there are at most 34! natural numbers.
2 comments:
The claim that "'the', 'least', 'natural', 'number', 'which', 'cannot', 'be', 'expressed', 'using', 'only', 'words'" is describing a number is paradoxical. Since this number cannot be expressed using only words, it is impossible for you to have described it using only words. By definition, there is no way you could have expressed the number.
I feel this logic is analogous to me saying that everything that exists can be sensed with human senses and then saying the thing that exists but I couldn't sense it actually was sensed because I sensed that I couldn't sense it.
It is intended to be paradoxical. This was just a modification of "Berry's Paradox".
If it were the case that,
"The least natural number which cannot be expressed using only the words: 'the', 'least', 'natural', 'number', 'which', 'cannot', 'be', 'expressed', 'using', 'only', 'words'."
didn't designate any natural
number, then it would *by definition* designate zero. Contradiction.
So you aren't going to get out of it that way. It's not enough to say "oh, it's broken when you do that, don't do that". Why is it broken?
I don't understand the sentence "Since this number cannot be expressed using only words..."
Every number can be expressed using only words,
991 - "nine hundred ninety one"
42 - "forty two"
In the second paragraph you have merely conflated two distinct notions of the word "sense".
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