Anyway, Set Theorists determine that the size of the set of irrational numbers is great than the size of the rational numbers through a reverse reasoning. They start with the assumption that the Real numbers are at a higher level of infinity than the rationals. The term irrational number refers to a real number that is not rational. As the reals are the union of rational and irrational numbers, set theorists conclude that the irrational numbers must be part of this higher level of existence shared with the real numbers. In fact, the theory is essentially saying that the irrational numbers are so dense that they give the real line dimension.
I need to repeat that. The rational numbers do not have any dimension. As you know, a point has no dimension. If you took all of the rational numbers together, you would not have any dimension. The supposition is that the irrational numbers are so dense. The irrational numbers are at a higher level of existence. There are so many of them that they actually create dimension. If you draw a line from point A to Point B, you do not see any rational numbers. If you look at the line going from point A to B below. You don't see any rational numbers...you only see irrational numbers:
A _________ B
Let's jump out of dialectical metaphysics, and get back on track.
Premise A: I understand that set theorist hold to the belief that all irrational numbers can be expressed with a unique infinite string of digits. (NOTE, this is a supposition that may or may not be true).
Anyway. If you hold to premise A, then you come across a very interesting point. There is a rational number between any two irrational numbers. This falls instantly from premise A. If two irrational numbers have a unique digital representation, then there must be some digit in their representation that they differ. You can create a rational number simply by finding this point where the two numbers differ and truncate the larger of the two digits to that precision. Voila. You have a rational number that is between your two irrational numbers.
This is easier to discuss in binary than in decimal mode. Binary numbers are expressed as strings of 0s and 1s. You can use binary digits to express numbers less than 1. For example 0.1 is 1/2, 0.01 is 1/4. The nth digit in the string is 1/2^n. The number .1011 is 1/2 + 1/8 + 1/16. This happens to equal 11/16.
The rules for binary digits follows are the same as the rules for decimal numbers. Rational numbers will
either end in an infinite string of zeros, or in a repeating pattern. .01010101... Where 01 repeats forever is equal to 1/3.
Okay, so we have two irrational numbers a and b where a < b. By definition neither a nor b end in an infinite string of zeros. If I find the first digit where a and b differ and simply truncate b to that digit, then I have a rational number that is between a and b. This is a demo:
Note, the number ...11101101010101011 is between a and b. Not only that. I can append an infinite combinitions of repeating strings (rational numbers) to the string and show that there is an infinite number of rational numbers between each irrational number. (BTW, it is possible to show that there is an infinite number of irrational numbers between each pair of rationals using the same technique.)
Back to Premise A.
If I hold to the position that irrationals can be represented by a unique infinite string--the keyword being unique. Then I must conclude that for irrational number a, there is some digit at which a differs from all other irrational numbers. Truncating at that digit, I get a rational numbers. With a little bit of finagling, I can create a theoretic 1-1 mapping between rationals and irrationals. It is a theoretic mapping because we are dealing with strings so large that they are transcomputational, but I know of no limits that says I can't imagine a set of irrational numbers larger than the number of quantum states on the planet earth.
As dialecticians are granted the divine right to change definitions at a whim. The route around this dilemma is probably just to say that two numbers might have the exact same digital representation, but still be unique.
Lesser folk, like myself, simply stare and wonder we've pushed this convoluted dialectical system into the foundations of modern mathematics.
The integers, rational numbers and real numbers are all different types of numbers. This transfinite theory that tries to claim that the rational numbers are just a different type of integer is really one of the most bogus theories ever to be forced on man.