On: A Bad Trip to Infinity

[bryanrasmussen]

let's consider two potentially infinite sentences:

Buffalo buffalo buffalo buffalo -> infinite

and

Police Police Police Police -> infinite

certainly Buffalo and Police sentences can both be infinite but the Buffalo sentence is obviously larger than the Police sentence because Buffalo has 7 letters and Police only 6.

https://en.wikipedia.org/wiki/Buffalo_buffalo_Buffalo_buffal...

[CrazyStat → bryanrasmussen]

This is a bad example. In fact those sentences have the same cardinality, much like the set of even natural numbers has the same cardinality as the set of all natural numbers even though only half the natural numbers are even.

[bryanrasmussen → CrazyStat]

they may have the same cardinality, but one infinite sentence is obviously longer than the other infinite sentence.

on edit: noting depends on if we consider as elements the words, or the letters.

[srean → bryanrasmussen]

Longer/larger according to what ?

If its according to cardinality then no, they have the same cardinality, same number of elements, presumably letters. If its according to a measure, then sure, one can be longer than the other -- the subset of [0,1] of the real line has the same cardinality, same number of 'points' as [0, 100] but the latter is longer when you compare according to the Lebesgue measure, i.e. 'length'.

[CrazyStat → bryanrasmussen]

Things that are "obviously" true are often false.

The only way to compare the lengths of the two infinite sentences is by comparing their cardinalities. There is no other meaningful definition of sentence length for an infinite sentence. They have the same cardinality, so they're the same "length."

[tromp → bryanrasmussen]

> Buffalo sentence is obviously larger than the Police sentence

It obviously isn't, since the infinite repetition of "Police " is

THE SAME as the infinite repetition of the longer "Police Police ".

[bryanrasmussen → bryanrasmussen]

there are some amusing other parts to this of course.

Since Buffalo -> is infinite and Police -> is infinite, but Buffalo is larger than Police how much larger is it?

It is obviously infinitely larger.

[jdkee → bryanrasmussen]

Those two infinite statements have the same cardinality.

[bryanrasmussen → jdkee]

they may have the same cardinality, but one infinite sentence is obviously longer than the other infinite sentence.

on edit: noting depends on if we consider as elements the words, or the letters.

[srean → bryanrasmussen]

If one of them, presumably the buffalo sentence, indeed has more letters, then could you give us a invertible scheme that maps to each letter of the police sentence, a letter in the buffalo sentence, that covers all the letters in the police sentence, but has letters in the buffalo sentence still left over.