In Search of Useless Truth

I heard a new phrase in class yesterday. The situation reminded me of when I first heard the phrase "vacuously true." It was just casually dropped, only briefly defined before moving on.

"Infinitely often true." Roll the phrase around in your brain for a sec. It's nice, no? It was used in contrast to when something is always true.

A good example would be even numbers. It would not be correct to say that it is always true that numbers are even....because you can probably think of an odd number. However it is correct to say that it is infinitely often true that numbers are even.

The test for truth is simple, if as you head towards infinity there will always be another true instance, then it is infinitely often true. In the case of even numbers, since there will always be another even number it's infinitely often true that numbers are even. The same can be said of prime numbers....they just get very sparse as you head towards infinity.

I think I really enjoy these concepts because they're mostly useless. Harmless if you will. It's not an especially powerful concept. It's not paradigm shifting. It's just a splitting the hairs on a concept that people probably don't care about to start with. In short, you can find useless trivia no matter where you look.

I'm still missing software on my new machine. There is a cool symbol for infinitely often true, but you'll have to wait for it. Yeah, yeah, I know. You're crushed. In the mean time I included a picture of a bird with a penchant for infinity.