In fact, this problem has been deeply troubling us:
1. Are “infinite sets” in present set theory “actual infinite sets” or “potential infinite sets”?
2. Are infinite elements in infinite sets “actual infinite many” or “potential infinite many”? If they are “actual infinite many”, how can we conduct the quantitative cognitions to them; and if they are “potential infinite many”, how can we conduct the quantitative cognitions to them? If Set A has more eliminates than Set B, can we say “Set A is more infinite than Set B”? can we really have “infinite”, “more infinite”, “less infinite”, “more more infinite”, “less less infinite”, “more more more infinite” “less less less infinite”, “more more more more infinite”, “less less less less infinite”, “more more more more more infinite”, “less less less less less infinite”, “more more more more more more more infinite”, “less less less less less less infinite” and “more more more more more more more more infinite”…?
3. What kind of mathematical tool of “one-to-one correspondence” is? When we conduct the quantitative cognitions to different infinite sets with “one-to-one correspondence” tool, is it “one element corresponding to one element” or “many elements corresponding to one element” or “many elements corresponding to many elements”, is it “potential infinite many elements corresponding to potential infinite many elements” or “actual infinite many elements corresponding to actual infinite many elements” or “potential infinite many elements corresponding to actual infinite many elements”?
4, can we have many different bijection proofs with different one-to-one coresponding results between two infinite sets? If we can, what conclusion should people choose in front of two opposite results, why?”
I don't quite understand the problem you are trying to address. However, in set theory a bijection (one to one correspondence) is a map f:X->Y. f is a bijection is if it is an injection, i.e. if f(x1)=f(x2) then x1=x2 and a surjection that is for any y in Y there is an x in X such that f(x)=y. So that eliminates all but one of the cases in you item 3.
In dealing with infinite sets one is quickly drawn into the properties of the cardinal numbers. The smallest class of infinite sets is the natural numbers, N. Any set X that where there exist a bijection between N and X has the same carnality (same number of elements). Such a set is called countable. It can easily be shown that there is a bijection of N into the rational numbers, that is the rational numbers are countable by using Cantor's dragonalization argument.
The same argument shows that the algebraic numbers of the rationals is countable (an algebraic number) is a root of a polynomial with rational coefficients.
http://aircoversolutions.com
Cantor established that the real numbers are uncountable.
https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument
So while the rational numbers are countable, the reals are larger which means there is no bijection between the rationals and reals. This gives rise to the cardinal numbers which are the cardinality (number of elements) of sets.
So why the rational numbers are dense in the reals ( for every real number we can find a rational that is arability close to it) the reals are a much larger set. Sets that can be a one to one correspondence with the reals are called uncountable.
https://en.wikipedia.org/wiki/Cardinal_number
Hopefully this helps.
Thank you for your idea dear Mr. Truman Prevatt，
If Set A has more eliminates than Set B, can we say “Set A is more infinite than Set B”? Can we really have “infinite”, “more infinite”, “less infinite”, “more more infinite”, “less less infinite”, “more more more infinite” “less less less infinite”, “more more more more infinite”, “less less less less infinite”, “more more more more more infinite”, “less less less less less infinite”, “more more more more more more more infinite”, “less less less less less less infinite” and “more more more more more more more more infinite”…?
Best regards,
Geng
Check the references on cardinal numbers in my original. Cardinal numbers are the way we measure the size of sets - and yes one infinite set can be larger than another. The reals are "more infinite" than the natural numbers or rational numbers. If X is a set the carnality of the power set of X which is the set of all subsets of X is larger cardinality that X. The cardinal number alpha-null is the carnality of the natural numbers, rationals, algebraic numbers over the rationals, etc. The cardinality of the continuum is the cardinality of the real numbers, normally represented by c.
Cantor address all the issues you discussed in what is known as Cantor's theorem.
https://en.wikipedia.org/wiki/Cantor%27s_theorem
In other words there is an infinite number of infinities and they are expressed by the cardinal numbers.
Thank you for your idea dear Mr. Truman Prevatt.
I agree with you that wikipedia has been always telling us much information from books.
So far as I know, no body in the world (including Cantor and you) has ever answered above 4 questions under this topic and a new question was produced from your above answer which can never be checked from wikipedia:
What is the definitions for “infinite”, “more infinite”, “less infinite”, “more more infinite”, “less less infinite”, “more more more infinite” “less less less infinite”, “more more more more infinite”, “less less less less infinite”, “more more more more more infinite”, “less less less less less infinite”, “more more more more more more more infinite”, “less less less less less less infinite” and “more more more more more more more more infinite”…?
Best regards,
Geng
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