It gets even worse than that if you want to keep your intuitions (which are actually partially formalized as the concept natural density). Imagine that T is the set of all Unicode text strings. Most of these strings, like โ๐พโจ๊ โงฬโฉถ๐โ, are gibberish, while some are valid sentences in various languages (such as โThe five boxing wizards jump quickly.โ, โprint โHello, world!โโ, โแผฯฯฮฑฯฮฟฯ แผฯฮธฯแฝธฯ ฮบฮฑฯฮฑฯฮณฮตแฟฯฮฑฮน แฝ ฮธแฝฑฮฝฮฑฯฮฟฯฮโ, or โืืงืจืืชื ืืฉื ืืืืืื ืืื ื ืืงืจื ืืฉื ืืืื ืืืื ืืืืืื ืืฉืจ ืืขื ื ืืืฉ ืืื ืืืืืื ืืืขื ืื ืืขื ืืืืืจื ืืื ืืืืจโ). The interesting strings for this problem are things like โ42โ, โ22/โ7โ, โeโ, โ10โโ(10โโ10)โ, or even โThe square root of 17โ. These are the strings that unambiguously describe some number (under certain conventions). As we havenโt put a length limit on the elements of T, we can easily show that every natural number, every rational number, and an infinite number of irrational numbers are each described by elements of T. As some elements of T donโt unambiguously describe some number, our intuitions tell us that there are more text files than there are rational numbers.
However, a computer (with arbitrarily high disk space) would represent these strings encoded as sequences of bytes. If we use a BOM in our encoding, or if we use the Modified UTF-8 used in Javaโs DataInput interface, then every sequence of bytes encoding a string in T corresponds to a different natural number. However, given any common encoding, not every byte sequence corresponds to a string, and therefore not every natural number corresponds to a string. As encoding strings like this is the most natural way to map strings to natural numbers, there must intuitively be more natural numbers than strings.
We have thus shown that there are more strings than rational numbers, and more natural numbers than strings. Thus, any consistent definition of โbiggerโ that works like this canโt be transitive, which would rule out many potential applications of such a concept.
EDIT: Fixed an error arising from my original thoughts differing from the way I wanted to explain them
It gets even worse than that if you want to keep your intuitions (which are actually partially formalized as the concept natural density). Imagine that T is the set of all Unicode text strings. Most of these strings, like โ๐พโจ๊ โงฬโฉถ๐โ, are gibberish, while some are valid sentences in various languages (such as โThe five boxing wizards jump quickly.โ, โprint โHello, world!โโ, โแผฯฯฮฑฯฮฟฯ แผฯฮธฯแฝธฯ ฮบฮฑฯฮฑฯฮณฮตแฟฯฮฑฮน แฝ ฮธแฝฑฮฝฮฑฯฮฟฯฮโ, or โืืงืจืืชื ืืฉื ืืืืืื ืืื ื ืืงืจื ืืฉื ืืืื ืืืื ืืืืืื ืืฉืจ ืืขื ื ืืืฉ ืืื ืืืืืื ืืืขื ืื ืืขื ืืืืืจื ืืื ืืืืจโ). The interesting strings for this problem are things like โ42โ, โ22/โ7โ, โeโ, โ10โโ(10โโ10)โ, or even โThe square root of 17โ. These are the strings that unambiguously describe some number (under certain conventions). As we havenโt put a length limit on the elements of T, we can easily show that every natural number, every rational number, and an infinite number of irrational numbers are each described by elements of T. As some elements of T donโt unambiguously describe some number, our intuitions tell us that there are more text files than there are rational numbers.
However, a computer (with arbitrarily high disk space) would represent these strings encoded as sequences of bytes. If we use a BOM in our encoding, or if we use the Modified UTF-8 used in Javaโs DataInput interface, then every sequence of bytes encoding a string in T corresponds to a different natural number. However, given any common encoding, not every byte sequence corresponds to a string, and therefore not every natural number corresponds to a string. As encoding strings like this is the most natural way to map strings to natural numbers, there must intuitively be more natural numbers than strings.
We have thus shown that there are more strings than rational numbers, and more natural numbers than strings. Thus, any consistent definition of โbiggerโ that works like this canโt be transitive, which would rule out many potential applications of such a concept.
EDIT: Fixed an error arising from my original thoughts differing from the way I wanted to explain them