Set theory (true or false)?
My answer is false but I'm not sure. But my reasoning appears to not make sense: Although it cannot be counted in a trillion years, there may be one point where you can count them.
But I'm assuming in this case which might be true or false, right? So, I need an expert's opinion about this. Thanks in advance!
- PuzzlingLv 71 month agoFavorite Answer
Take a smaller time frame.
"If the elements in a set cannot be counted in 1 minute, the set is infinite"
That's clearly FALSE.
I could make it 1 hour, 1 day, 1 year, 1 million years, 1 billion years, 1 trillion years, etc.
Just because a set has an extremely large number of elements and it is more than I could possibly ever count, it may still be finite.
For example, the number of grains of sand on a beach is more than I could ever count, but it is still finite.
So I agree with you, that statement is FALSE.
- MorningfoxLv 71 month ago
If the elements could be counted in a trillion + 1 years, then the set is a finite set.
Assuming that the counting rate is finite. If you can count an infinite number of elements in a finite time, then all bets are off.
- az_lenderLv 71 month ago
Your instinct is correct. The proposition is false. "Large" and "infinite" are not the same thing. Maybe it takes a quadrillion years to count up a particular (finite!) set. One cannot judge that a set is infinite just because it could take a long time to count the elements.
It's easier to prove that there are infinitely many points on a one-inch line segment. Let's say you designate these points by some decimal fraction of an inch. Like, start with 0.253 inches and 0.907 inches. Now pick some point in between. (ANY point in between.) And write down its decimal representation. Now pick a point that's in between the last two points you wrote down.
Now repeat that last instruction forever. It is obvious that you cannot "run out" of points.