I had four examples of two principles. Please clarify?
Aranfan just disproved all four of your examples, so it's kind of a moot point.
In any case, you have
asserted that the set of all positive even integers is exactly half as large as the set of all positive integers. But as Aranfan pointed out, for every positive integer, there is a corresponding positive even integer. Namely, two times the original integer.
You cannot point to a single integer and say "this is an integer that does not come with a corresponding even integer." Because
every integer can be doubled to create a corresponding even integer.
Therefore, the set of positive even integers must be as large as the set of all positive integers. If it weren't, it would be impossible to assign one positive even integer to every positive integer.
If we can hand every child in a group a lollipop, and no child is left without a lollipop, logically there must be as many lollipops as children. We can hand every positive integer a positive even integer, and there is no positive integer that doesn't get its own positive even integer.
Q. E. D.
A similar argument applies to your other examples under (1).
The case of "positive integers divisible by 3" is trivially obvious and I will not go into detail.
And just as every integer can be assigned a corresponding even integer by doubling it, every half-integer (numbers of the form N+0.5 where N is an integer) can be assigned a corresponding odd integer by doubling it, and so the set of all integers plus all half-integers cannot be larger than the set of all integers.
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(Infinity plus one) times 26 is equal to (infinity plus 26), even though (infinity times 26) is just (infinity). This is how you can have infinite sets that contain each other, but are still greater together than they are apart - you simply count the markers you used to distinguish them.
Notably, your first sentence concedes that your claim that infinities can be compared by ratios is erroneous: you say "even though infinity*26 is just infinity."
But aside from that, there is a separate error on your part in this new claim; namely that you are trying to apply arithmetic and the distributive property to infinite quantities when, as noted, that doesn't work.
In high school math, 26*(X+1) = 26X + 26. But that is based on the assumption that X is a
finite number on which the arithmetic operations "add, subtract, multiply, divide," and so on can be performed.
If you add one object to an infinite set, you have an infinite set. If you then multiply the infinite set 26-fold you have an infinite set, regardless of whether you previously added one object to it or not. There is no "infinity plus one" size, and no "twenty-six times infinity" size. Infinity is not a number. it is a concept.
What you're doing is like arguing that 4*[(2/0)+1] equals four, or for that matter [(8/0)+4]. It's simply not the case, because these are not quantities that are capable of being calculated by conventional arithmetic.
But if you follow that axiom when comparing infinities, you wind up discarding the ability to compare them proportionally. Why would you do that?
Because the belief that you can compare infinite sets proportionally is a delusion created by treating infinite sets as if they were finite sets. It is
factually incorrect, and I can prove it, and have already done so, as has
@Aranfan .
Therefore, whether it would be useful or not to be able to compare infinite sets proportionally, one cannot do so.
No. Demanding an authoritative source is the evil twin of arguing from authority.
The problem is, Horatio, your argument fails on its own merits.
If you backed up your argument with an authoritative source (e.g. someone with a degree in higher math), it would provide us with reason to think that someone who does in fact understand set theory and higher mathematics can help compose the argument, which would give us less reason to dismiss this out of hand, the way we would casually dismiss someone who maintains that rockets fly by pushing against the air and can't possibly work in space.
