@Karthikdeva @freemo @Surasanji

I just finished reading (and then re-reading, up and down) an excellent article.

https://cosmosmagazine.com/mathematics/some-infinities-are-bigger-than-others

Until about halfway into the article I felt my previous belief on sets being not comparable, not larger or smaller if both are infinite.

"It is tempting to say there are twice as many natural numbers as even numbers. But that’s wrong.

When we say two sets of objects are equal, we put them into correspondence on a one-by-one basis. For example, if I claim I have the same number of fingers as toes, I mean that for every one finger there corresponds one toe, with no toes left over and no fingers left unmatched at the finish.

Now do the same for natural numbers and even numbers: pair 1 with 2, 2 with 4, 3 with 6, and so on. There will be exactly one even number for every natural number. The fact that each series forms an infinite set means the sets of numbers are the same size, even though one set is contained within the other! "

Aha, I thought!! 😃

But... Read on...

"This is best understood by envisaging a continuous line, labelled by equally spaced natural numbers: 1, 2, 3 and so on. There will be an infinite number of points between 1 and 2, for example, with each point corresponding to a decimal number. No matter how small an interval on that line and how much you magnify it, there will still be an infinite number of points corresponding to an infinite number of decimals.

It turns out that the set of all points on a continuous line is a bigger infinity than the natural numbers; mathematicians say there is an uncountably infinite number of points on the line (and in three-dimensional space). You simply can’t match up each point on the line with the natural numbers in a one-to-one correspondence. "

So, the Learn New Thing of the Day is that there ARE sizes of infinities. Owa. 😲

By all means read the article --- it's NOT hard to follow, really well explained and authored by a Uni professor.