Here are 10 numbered statements. How many of them are true?

1. Exactly one of these statements is false.
2. Exactly two of these statements are false.
3. Exactly three of these statements are false.
4. Exactly four of these statements are false.
5. Exactly five of these statements are false.
6. Exactly six of these statements are false.
7. Exactly seven of these statements are false.
8. Exactly eight of these statements are false.
9. Exactly nine of these statements are false.
10. Exactly ten of these statements are false.

Originally posted by Pianoman1 Here are 10 numbered statements. How many of them are true?

1. Exactly one of these statements is false.
2. Exactly two of these statements are false.
3. Exactly three of these statements are false.
4. Exactly four of these statements are false.
5. Exactly five of these statements are false.
6. Exactly six of these statements are false.
7. Exactl ...[text shortened]...
9. Exactly nine of these statements are false.
10. Exactly ten of these statements are false.

Of necessity (they contradict one another) at most one. And therefore, Reveal Hidden Content

exactly one, since if it were zero, #10 would contradict itself. Hence number 9 is the only true ststement.

Originally posted by Shallow Blue Of necessity (they contradict one another) at most one. And therefore, [hidden]exactly one, since if it were zero, #10 would contradict itself. Hence number 9 is the only true ststement.[/hidden]

Originally posted by Pianoman1 Here are 10 numbered statements. How many of them are true?

1. Exactly one of these statements is false.
2. Exactly two of these statements are false.
3. Exactly three of these statements are false.
4. Exactly four of these statements are false.
5. Exactly five of these statements are false.
6. Exactly six of these statements are false.
7. Exactl ...[text shortened]...
9. Exactly nine of these statements are false.
10. Exactly ten of these statements are false.

Originally posted by Shallow Blue That doesn't work. If none of them are true, all of them are false, which makes #10 true, which contradicts your assumption.

Richard

A sentence is truth-apt if there is some context in which it could be uttered (with its present meaning) and express a true or false proposition. Sentences that are not apt for truth include questions and commands, and, more controversially, paradoxical sentences of the form of the Liar (‘this sentence is false&rsquoðŸ˜‰; or sentences (‘you will not smoke&rsquoðŸ˜‰ whose apparent function is to make an assertion, but which may instead be regarded as expressing prescriptions or attitudes, rather than being in the business of aiming at truth or falsehood. See expressivism, prescriptivism.

Reminds me of the old dilemma of having a piece of paper with the text "the statement on the other side of this paper is false" printed on both sides. =)

Originally posted by Pianoman1 Here are 10 numbered statements. How many of them are true?

1. Exactly one of these statements is false.
2. Exactly two of these statements are false.
3. Exactly three of these statements are false.
4. Exactly four of these statements are false.
5. Exactly five of these statements are false.
6. Exactly six of these statements are false.
7. Exactl ...[text shortened]...
9. Exactly nine of these statements are false.
10. Exactly ten of these statements are false.

What would be the case as i -> infinity, that is, as the number of statements approaches infinity, each statement saying "Exactly i of these statements is/are true"?

n statements, n-1 'th is true, where n = 2, 3, 4, ... so we'd get

lim n-1 as n-> oo as the ordinal of the statement that is true, and the ordinal approaches infinity as the number of statements increases without limit?

Originally posted by Shallow Blue That doesn't work. If none of them are true, all of them are false, which makes #10 true, which contradicts your assumption.

Richard

It works fine. If they are not truth-apt to begin with, then they are all neither true nor false.

At any rate, the OP presumes that such statements are truth-apt (not explicitly, but in the spirit of the exercise as it was given). I rather doubt that. (But this is debatable.)

If I am wrong and indeed such statements are truth-apt, then I would agree with the solution you gave.

Originally posted by LemonJello It works fine. If they are not truth-apt to begin with, then they are all neither true nor false.

That is only true (and the term "truth-apt" only meaningful) in one specific branch of logic. Real life is not that branch, and neither are logical puzzles unless explicitly stated.

Originally posted by SwissGambit A sentence is truth-apt if there is some context in which it could be uttered (with its present meaning) and express a true or false proposition. Sentences that are not apt for truth include questions and commands, and, more controversially, paradoxical sentences of the form of the Liar (‘this sentence is false&rsquoðŸ˜‰; or sentences (‘you will not smoke&rsquoðŸ˜‰ whos ...[text shortened]... expressivism, prescriptivism.

Originally posted by Shallow Blue That is only true (and the term "truth-apt" only meaningful) in one specific branch of logic. Real life is not that branch, and neither are logical puzzles unless explicitly stated.