Kinda left of center observation. — Yohan
We have two rows of identical bananas and each row stretches out to actual infinity. The two rows are lined up so that there is a one-to-one correspondence between them. According to Cantor, the two sets of bananas are therefore identical sets.
We add one banana at the start of the second set and then shift all the bananas right one place so they are lined up again - they are in one-to-one correspondence again - so Cantor would claim they are identical sets.
We then remove every second banana from the second set and then shift all the bananas in the second set to the left so they are lined up with the first set again. Again they are in a one-to-one correspondence - so Cantor would claim they are still identical sets. — Devans99
How does one add a banana at the beginning of a row of infinite bananas? There is no beginning, and therefore no second banana, in a infinite row of bananas. There is no beginning or end with infinity. You're simply misusing terms. — Harry Hindu
We have two rows of identical bananas and each row stretches out to actual infinity. The two rows are lined up so that there is a one-to-one correspondence between them. According to Cantor, the two sets of bananas are therefore identical sets. — Devans99
3. So all bananas are in one-to-one correspondence
4. So the sets are identical — Devans99
Help me out here: where did any sane person aver that there were any actual infinities of anything? There are certainly ideas of them, very many of them, in all kinds of sizes. And even potential infinities, but actual? (There are those who might say that a line segment is made up of an actual infinity of points. To my way of thinking the operative phrase is "made up of." The line segment itself is just a line.)I still maintain however that my argument highlights the absurdity of actual infinity. — Devans99
Firstly, if all the elements of the sets are identical, then they just have one element. Sets are defined by what distinct elements belong to them; a set is a collection of distinct objects — fdrake
Two sets being in a one to one correspondence says nothing about whether they are identical sets. The odds are in a one to one correspondence with the evens, but even numbers are necessarily not odd. — fdrake
Help me out here: where did any sane person aver that there were any actual infinities of anything? — tim wood
For example, some folks believe that past time is actually infinite; implying an actually infinite collection of moments in the past.
Or some hold that space is continuous, implying an actually infinite collection of distinct spacial positions in a unit of space. — Devans99
Help me out here: where did any sane person aver that there were any actual infinities of anything? — tim wood
A little more rigour, please. First, what "some folks believe" is no standard for anything (than perhaps that some folks may believe anything). — tim wood
Back to the line segment. It's just a line. Is there "an actual" infinity of points? Depends on your purposes and definitions - but then you're beyond what it is. — tim wood
Help me out here: where did any sane person aver that there were any actual infinities of anything? — tim wood
I called the groups of bananas 'collections' rather than 'sets' to get around the issue of sets having to be composed of distinct objects. — Devans99
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