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# Can one infinite set of things be bigger than another infinite set of things?

Wed 28 Jan, 2015 02:36 pm
I contend this is possible.

Moreover, just between 0 and 1 on the number line there are MORE points than there are counting numbers (counting numbers -- that is, numbers like 1,2,3,4,etc.).

Anyone disagree?
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fresco

2
Wed 28 Jan, 2015 02:44 pm
@Kolyo,
Of course there is agreement ...provided that the "things" are types of number and that "infinity" is defined with respect to a particular "counting operation".
There is no paradox because "infinity" is not "a number" in the traditional sense.
Google "Cantor" for definitive discussion of this.
Kolyo

1
Wed 28 Jan, 2015 03:05 pm
@Kolyo,
There is agreement among mathematicians, at least those of us who make set theory the foundation of mathematics.

I'm unfamiliar with category theorists' definition of cardinality, but they probably agree as well.

I was mainly interested in people's intution. My very gifted student challenged me on this, and said there was only one infinity. I tried to show him that he was wrong by precisely defining what it meant for one set to be "bigger" than another without going into what ordinals and cardinals were.

However, perhaps I was wrong to just shoot him down, because he was right, in a way, according his own concept of infinity -- probably based on distance traveled along a number line. Perhaps I had as much to learn from his view as he had to learn from me.
Kolyo

1
Wed 28 Jan, 2015 03:07 pm
@fresco,
fresco wrote:

There is no paradox because "infinity" is not "a number" in the traditional sense.

Then again his hang-up may have derived from as simple a misunderstanding as that.
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contrex

1
Wed 28 Jan, 2015 03:14 pm
@Kolyo,
Kolyo wrote:
I was mainly interested in people's intution. My very gifted student challenged me on this, and said there was only one infinity. I tried to show him that he was wrong by precisely defining what it meant for one set to be "bigger" than another without going into what ordinals and cardinals were.

Did you succeed in convincing your student? Although as a young man I had never been exposed to the work of Cantor, it made perfect sense to me the first time I read about his diagonal argument.
fresco

1
Wed 28 Jan, 2015 03:19 pm
@Kolyo,

On the subject of "right" and "wrong" perhaps you should heed Paul Cohen's proof that there both is and is not an "infinity" of magnitude between two of Cantor's infinities ! (For that he received the Fields Medal)
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Kolyo

1
Wed 28 Jan, 2015 04:06 pm
@contrex,
contrex wrote:

Did you succeed in convincing your student? Although as a young man I had never been exposed to the work of Cantor, it made perfect sense to me the first time I read about his diagonal argument.

Oh, once I started talking about "infinities" as the sizes of infinite sets he changed his mind immediately and agreed with me.

Now the problem is convincing him that sets which look like they might be different sizes (such as the set of integers, the set of even numbers, and the set of all "fractions") are in fact the same "size" (in terms of cardinality).

(His argument for why I was *right*, half a second after I told him there were multiple infinities, was that the size of {...,-1, 0, 1,2,3,...} couldn't be as large as sets got because you could always find new numbers between of your set's numbers, making the set steadily larger. But that kind of thinking would him lead to believe there were more rationals than integers, and I didn't want that.)

Pondering it aloud and reading the comments, I think the best approach might be to stick with the difference between countable and uncountable infinities. Then I can use the diagonal argument to show him the number of points in [0,1] is uncountable. It didn't make sense to me the first time I saw it, but it does now, and he'll probably understand it immediately.

I would have had a lesson plan ready if I'd known we would be talking about this at all, but this was all an aside to a discussion about probability theory. I had to explain what a countable union of events was. (If people are wondering, this is "enrichment tutoring" not "remedial tutoring".)
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tsarstepan

2
Fri 30 Jan, 2015 01:39 pm
@Kolyo,
According to Hazel Grace Lancaster from The Fault in Our Stars (a famous YA novel by John Green and Hollywood movie from 2014)? Yes.

Quote:
Can one infinite set of things be bigger than another infinite set of things?

The infinite set of numbers between 1 to 100000000000000000000000000000000000000000000000000000(etc...) >
The infinite set of numbers between 1.000000000000000000000000000000000000000000000000000(etc...) to 2.00 (etc...)

Watched The Fault in Our Stars last night. Shailene Woodley was truly snubbed from her deserving lead actress nomination.
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