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navetzz t1_j28qm4u wrote

No! There is infinitely less of them. (Also, some people in the comments are mixing numbers and integers)

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farineziq t1_j292xsd wrote

Op is right. Cardinality of the Continuum: "Between any two real numbers a < b, no matter how close they are to each other, there are always infinitely many other real numbers, and Cantor showed that they are as many as those contained in the whole set of real numbers. In other words, the open interval (a,b) is equinumerous with {R} ."

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Angry_Guppy t1_j29aopr wrote

How can that be true? For each real number between 0 and 1, there is a real number that is that value +1, +2, +3, etc. all the way up to infinity?

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hacksoncode t1_j29byu4 wrote

Ultimately it's a semantics question about how you measure the "number" of things in an uncountably infinite set.

Consider "every real number" x in (1, +∞). There is a corresponding real number in (0,1) which is 1/x. So the sets are exactly the same size.

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JoePoe247 t1_j29qncu wrote

How is that true though? For every number between 1 and 3, there is only one correlating number between 0 and 1 (1/x). But for every number between 0 and 1, there are two correlating numbers (x+1 and x+2)

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M8dude t1_j29t6r5 wrote

you have to use a different 'correspondence' function for (0,1) and (1,3), which in this case is 2*(x+1/2).

notice how __every__ number in one set has to be matched with __exactly__ one of the other set.

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JoePoe247 t1_j2ayejy wrote

Ok thanks, didnt realize that with the different corresponding function. My logic would be that y=2*(x+1/2) can be used to define every number in both sets (where y is 1-3 and x is 0-1). But there are numbers outside of the set 1-3, say 15. So if I made a different set, inclusive of 1-3 and 15, then there are more numbers in this new set.

I guess I understand that I'm wrong since mathematicians smarter than me have come up with theories/proofs to what you're saying, but I think there's logic in my argument.

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M8dude t1_j2bv9x8 wrote

yeah, there's plenty logic in your argument, there are many different so-called 'measures' to quantify sets of numbers, example the 'distance measure' (i think) of an interval [a, b] is just denoted by b - a.

This is makes the sets have different measure (and more useful ones than just "infinite"), even though they have the same number of numbers.

the measure we would have used before is called the 'counting measure', telling us we'd have to count to infinity for both sets, but that doesn't mean they have the same number of elements (see cantor's second diagonal argument, or yours with the 15), so it has to be shown using a so-called 'bijective function' (our correspondence function), which thank god is pretty easy to construct for any two intervals.

But anyway, good thinking and yep you are right about the example set including [1, 3] and 15, for the counting measure.

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hacksoncode t1_j2adzng wrote

> But for every number between 0 and 1, there are two correlating numbers (x+1 and x+2)

There are an uncountably infinite number of ways to correlate numbers in any open set of real numbers... they all have the same "count" in that they can all be correlated to all of the other reals with unlimited simple relationships.

Example: 1/2x is also in (0,1) just like x+2 and x+3 exist outside of that range. So is 1/(pi*x)...

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moomerator t1_j29chbu wrote

So as somebody relatively well versed in math but definitely not full blown mathematician (engineer with a math minor) - I accept that this is a conclusion that a lot of people smarter than myself have agreed on but it still always bothered me.

I’m not disagreeing that it’s how the math falls out but I feel like there’s something fundamentally wrong with it. It’s like (and more or less related to) our understanding of quantum mechanics.. it feels like our current understandings are a case of the kid who got the right answer despite solving it incorrectly and like there’s a significant error/piece missing.

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farineziq t1_j29u4qj wrote

I think finding a one to one relation between elements of one set to elements of another set is a convincing way to know if they have the same amount of elements without counting them. Like if you have two heaps of rocks, you could make two lines of rocks where rocks are side by side and if both lines end together, both sets have the same amount of rocks.

Doing the same with infinite sets is useful because we can't count the elements, but might be able to find a function that associates each element of one set to one and only one element of the other. (Regarding the rocks example, every possible way to put them side by side is such a function.)

For example, there are as many positive integers as positive and negative integers because f(x) = (x % 2) * (-x / 2 - 1 / 2) + (1 - x % 2) * (x / 2) from 0 to ∞ gives 0, -1, 1, -2, 2, -3, 3, ...

Now regarding uncountable infinite sets like the real numbers, someone else in the comment showed that ]0, 1] has the same amount of elements as [1, ∞ [ using f(x) = 1 / x. And regarding what we're actually looking for, let's say [0, 1] has the same amount of elements as ℝ, I didn't really pay attention but this seems convincing.

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Ok_Substance_1560 t1_j28x5jd wrote

Yes. It would be infinitely less. There are different measures of infinity too!

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