superbyrd22000 t1_iuacu6h wrote on October 29, 2022 at 8:13 PM

So there are infinite many infinity but let's talk about the two most common one, countable and continuous.

Countable is any thing one can count, think of the number 0,1,2,3,4...78810836689017,.... This will go on forever thus infinite, but in a infinite amount of time one could count all of the numbers (this is not possible for human because we have finite time).

Continuous think of decimal pick two decimal call the larger one B and the sampler one A, then pick a decimal C; where C is in-between A and B, then repeat (C will now be B and one will pick another C) This will go one forever and you can always find another decimals that we didn't account for. One can't "count" all of the decimals because you can always pick another decimals between A and B.

Follow up point, the reason the first infinite is countable is because you can create a function to get to all of the number by adding one (in the example above), but you can make a function to get all of the decimals.

(Not quite el5 because of the math I know, but it's the best I know how to do)

Chromotron t1_iuap2ai wrote on October 29, 2022 at 9:41 PM

> One can't "count" all of the decimals because you can always pick another decimals between A and B.

This is unrelated to "counting". The rationals satisfy the very same property, yet can be counted. But all decimals, i.e., real numbers, cannot be counted, they are "uncountable".

Conversely, there are uncountable "discrete" ordered sets where nothing is between a number and its two neighbours. Hence the property you speak of and being (un)countable are independent, neither implies the other.

breckenridgeback t1_iuaziy4 wrote on October 29, 2022 at 11:02 PM

> Conversely, there are uncountable "discrete" ordered sets where nothing is between a number and its two neighbours.

That's not true, and it's (relatively) easy to prove.

Consider a set S that is a subset of the reals, with the property that for each s in S there exist two numbers u and l (for "upper" and "lower") such that u < s < l and there are no numbers x for which u < x < s or s < x < l. In other words, u is the "next biggest" number and l is the "next smallest" (this formalizes the idea you've stated informally).

For each s in S, consider the radius R = min(d(s,u), d(s, l)) (of course, R, u, and l all depend on s, but reddit markdown means I'm gonna skip the subscripts). This radius is basically just the "minimum spacing" around s. Such an R exists for each s, and is strictly positive. Since R is strictly positive, so is R/2. And since the rationals are dense in the reals, we can find two rational numbers a and b (again, also dependent on s) such that s - R/2 < a < s < b < s + R/2. In other words, we can find an interval of rational numbers (a,b) that does not overlap the corresponding interval for any other s in S.

Now, consider the function f: S -> (Q x Q) that takes each element s in S and maps it onto the ordered pair of the interval generated by the process in the previous paragraph. This function is clearly injective, since none of the intervals (a,b) overlap (so they certainly cannot be the same), but the set (Q x Q) is a Cartesian product of countable sets and therefore countable. Since we have an injection from S to a countable set, S is itself (at most) countable.

Chromotron t1_iue4c7b wrote on October 30, 2022 at 5:16 PM

> Consider a set S that is a subset of the reals,

That's where you went wrong: the set is simply not required to be contained in the reals. You can give any set X a total ordering such that the induced topology is discrete; similarly, we can give X a metric that makes it discrete, the discrete metric.

breckenridgeback t1_iuenf2y wrote on October 30, 2022 at 7:21 PM

Oh, I thought you were claiming an uncountable discrete subset of the reals (since your comparison was the rationals). Yes, obviously you can give any set the discrete topology (although the discussion of "between" suggests something more like an order metric?)

snoias t1_iuav9h0 wrote on October 29, 2022 at 10:28 PM

> So there are infinite many infinity but let's talk about the two most common one, countable and continuous.

I think you mean "uncountable", not "continuous". Or possibly "the continuum", which is sometimes used as a fancy word for the real numbers.

> Countable is any thing one can count, think of the number 0,1,2,3,4...78810836689017,.... This will go on forever thus infinite, but in a infinite amount of time one could count all of the numbers (this is not possible for human because we have finite time).

I don't think it's helpful to talk about what you could do in an "infinite amount of time", because like you said that's not possible, and it's not really obvious what we might be able to do given infinite time.

A more concrete way to talk about this is to say that, if we have a countable set of objects, we can come up with a way of listing them that will eventually reach any given object. For example, if we list the positive integers like 1,2,3,4,..., then you can pick any positive integer you like and it will show up in our list eventually.

> Continuous think of decimal pick two decimal call the larger one B and the sampler one A, then pick a decimal C; where C is in-between A and B, then repeat (C will now be B and one will pick another C) This will go one forever and you can always find another decimals that we didn't account for. One can't "count" all of the decimals because you can always pick another decimals between A and B.

But this just shows that your particular approach to counting them didn't work. Maybe there is a way of counting them that doesn't keep zeroing in on a smaller and smaller interval. In fact, for the set of all numbers with finite decimal expansions, there is a way.

superbyrd22000 t1_iubgkx1 wrote on October 30, 2022 at 1:19 AM

No I meant continuous with this definition "A continuous data set is a quantitative data set representing a scale of measurement that can consist of numbers other than whole numbers, like decimals and fractions. Continuous data sets would consist of values like height, weight, length, temperature, and other measurements like that. They're things that can be measured in fractions and decimals. Usually a tool, like a ruler, measuring tape, scale, or thermometer, is required to produce the values in a continuous data set." But yes the superset is an uncountable which would be the more encompassing definition but more difficult to understand.

As far as a method it's a el5, thus a master level proof would not be appropriate. Regardless the method I explained is the baseline for proving that a infinite is not countable via the proof of contradiction.