# Infinite choice question

The assertions in the following infinite list are all the same, true or false.
$~\vdots$
T$~~$F $~~$ I have a black Model T.
T$~~$F $~~$ I have a black Model T.
T$~~$F $~~$ I have a black Model T.
$~\vdots$

The object is to concoct an infinite list of assertions where: $\strut$
•   The assertions are identical. $\strut$
In every possible consistent scenario where each assertion is either true or false: $\strut$
•   At least one assertion must be true. $\strut$
•   At least one assertion must be false. $\strut$
And at least one consistent scenario is possible.

In this open-ended challenge infinitely many answers are possible.
Try to find one (or more) in any (or none) of these categories. $\strut$
a.  Exactly one assertion must be true.
b.  Exactly one assertion must be false.
c.  Infinitely many assertions must be true
while infinitely many must be false.
d.  Some other quirky condition.
e.  Some other quirky condition.
f.   Some other quirky condition.
$~\vdots$

Feeling competitive?$~$ Try for the fewest words.

Here's an answer that is incorrect because the truth values could be all true:
$~\vdots$
T$~~$F $~~$ This assertion is true.
T$~~$F $~~$ This assertion is true.
$~\vdots$

Here's an answer that is incorrect because the assertions can be neither true nor false:
$~\vdots$
T$~~$F $~~$ This assertion is false.
T$~~$F $~~$ This assertion is false.
$~\vdots$

• What is the exact objective? You could clarify the Q a bit. – ghosts_in_the_code Jan 4 '16 at 6:47

Answers without a ** next to them work only if the list is countably infinite and ordered (although an ordering could be enforced in the assertion at the cost of brevity).

a**:

Other assertions are False

If this is true for any assertion, all other assertions are false. If this is false for any assertion, a true assertion must exist. The combination of these two facts guarantees exactly one true assertion.

b**:

Another assertion is False

If this is false for any assertion, then all other assertions are true. If this is true for any assertion, a false assertion must exist. The combination of these two facts guarantees exactly one false assertion.

c:

Next assertion is False

If the first assertion is true, the next is false, which makes the next true... etc. The inverse is also possible, but either way, infinite assertions are both true and false.

d**:

Other [property] assertions are False

This allows you to make any single assertion true. For example, if you want only assertion 18 to be true, pick the property [18th]. This is vacuously true for the 18th assertion, since no other 18th assertions exist. Since the 18th assertion is true, every other assertion is false.

Another [property] assertion is False

This allows you to make any single assertion false. For example, if you want only assertion 18 to be false, pick the property [18th]. This is false for the 18th assertion, since no other 18th assertions exist to have truth values. Since the 18th assertion is false, every other assertion is true.

[X] other [property] assertions are True

Through similar logic, this allows you to make any finite number of specific assertions true, with the rest false.

[X] other [property] assertion are False

Through similar logic, this allows you to make any finite number of specific assertions false, with the rest true.

And finally, the universal (if boring) e**:

[Property] assertion

This lets you cause any describable set of assertions to be true, with the rest false. It also implies some cheesy ways of doing a, b, and c.

a:

First assertion

Obviously, true only of the first assertion.

b:

Subsequent assertion

True of all but the first assertion.

c:

Prime assertion

True and false for infinitely many assertions.

• Right on for a, b and c, for which you didn't even exploit numbering. And thank you for d, d, d and d! Gotta give full credit for your efficiency with words, too, though you might want to begin your answer for $~$a$~$ with "All..." so that it couldn't be interpreted as "Some ... ." Your first d answer, though similar, seems to completely avoid this ambiguity. – humn Jan 4 '16 at 7:54
• I went for word efficiency over clarity, and I couldn't think of a better 4-word way to phrase that logical statement. The universal is at least a reasonable interpretation, though unfortunately not the only one. – Zerris Jan 4 '16 at 7:57
• Nice way to get c with numbers, @Zerris. I take it your use of the word prime is numerical, not some sense of foremost as in "prime directive." – humn Jan 5 '16 at 3:54
• Correct. You could also select "even", "cubic", "Fibonacci", or any other infinite sequence. – Zerris Jan 5 '16 at 3:57
• To be even cheesier, you could select a subset of the sequences on OEIS. – Jack Tiger Lam Feb 6 '16 at 7:36

I propose:

.
.
.
The next assertion is false
The next assertion is false
The next assertion is false
The next assertion is false
.
.
.

The assertions alternately take the truth values true, false, true, false, .... This is consistent with what they are asserting, since the ones that are true say that the next one is false, which is indeed the case, and the ones that are false say that the next one is false, which is not the case, so those assertions are indeed false.

• Yes! That's the shortest I've found for category c. – humn Jan 4 '16 at 6:00
• Actually, I have a problem assigning truth values to these assertions. There is, in fact, no way to verify the truth value of any of these assertions without externally establishing the truth value of one of them. Upon doing so, yes, the whole thing collapses into alternating T/F values, but before doing so, these assertions lack a truth value. Unless I'm misunderstanding something in either the problem or the answer...? – João Mendes Jan 4 '16 at 13:20
• @JoãoMendes well, the OP didn't say precisely what the rules of the game were, so I assumed what he/she meant was that there needs to be at least one self-consistent assignment of truth values to each of the assertions. But you are correct that if the requirement were the stronger one that there has to be precisely one such assignment, then my solution would be wrong. – user17947 Jan 4 '16 at 15:21
• Well, the question says specifically "Each assertion is either true or false." Your assertions, as they stand, do not have a truth value. – João Mendes Jan 4 '16 at 15:55
• True. $~$ Now the statement includes "... in every possible consistent scenario ..." – humn Jan 4 '16 at 19:11

In the spirit of Zerris's other-logic above:

Category f. $~$ Exactly $n$ assertions must be true, $n \ge 1$ .

$~\vdots$
Fewer than $n$ other assertions are true.
Fewer than $n$ other assertions are true.
$~\vdots$

trues $\le n~~~~$ or else any supposedly-true assertion would be self-contradictory.
trues $\ge n~~~~$ because infinitely many false assertions remain and claim that $~$ trues $< n~$.

Category g. $~$ Exactly $n$ assertions must be false, $n \ge 1$ .

$~\vdots$
At least $n$ other assertions are false.
At least $n$ other assertions are false.
$~\vdots$

falses $\le n~~~$ or else the claim of each supposedly-false assertion would be satisfied.
falses $\ge n~~~$ because infinitely many true assertions remain to say so.

Category h. $~$ An odd number of assertions must be true.

$~\vdots$
An even number of other assertions are true.
$~\vdots$

trues $\ne$ 0 $~~~~$ or else the claims of the supposedly-false assertions would be satisfied.
trues is odd $~$ because at least one true assertion implies this.
(Let's agree that 0 is an even number and infinity is not.)

In a slightly different spirit, with a messier solution (for now?), for an ordered bi-infinite list:

Category i. $~$ Infinitely many consecutive assertions must be true while infinitely many consecutive assertions must be false.

$~\vdots$
Infinitely many consecutive assertions are false while this assertion and the next have the same truth value.
$~\vdots$

The reasoning is similar to those for the preceding lists, with the first step being to establish that the assertions cannot be all false.