cokernel

(Edited to try to answer the objection more clearly.)

The intervals overlap heavily. Each of the intervals R_i contains infinitely many rational numbers, and all but finitely many of those will have corresponding intervals which are completely contained within the interval R_i. As a result, the estimate obtained by the construction described is always strictly smaller than the chosen e.

Slogan: "rational numbers exist but do not take up space."

This is in a measure-theoretic sense, so literally 100% (not just 99.5%). On the number line, the measure of an interval is the (absolute) difference between its endpoints (the "length" of the interval). For any measurable set, you can approximate its size from above by covering its points with intervals and shrinking them as much as possible.

But because there are only countably many rational numbers, you can cover all of the rationals with a set of intervals with a finite total sum of lengths.

To be a bit more specific, if your rationals are r_1, r_2, r_3, ..., then for any small number e > 0, you can form the set of intervals

(r_1 - e/2^2, r_1 + e/2^2), (r_2 - e/2^3, r_2 + e/2^3), (r_3 - e/2^4, r_3 + e/2^4), ...

which have a total sum of lengths e/2 + e/4 + e/8 + ... = e and contain all of the rational numbers.

So for any positive number e > 0, the measure of the rational numbers is less than e, meaning that the measure of the rational numbers is 0.

To turn this into actual probabilities requires a little bit more work, since the measure of the number line, and hence that of the irrational numbers, is infinite, not "100%". But you could look at the probability of selecting a rational number in the interval [0, 1] and use the same reasoning as above to get a probability of 100% for an irrational number and 0% for a rational number.

The other three items listed for Socrates are:

* Knowledge of the universe does not affect our behavior.

* We should pity the perpetrator of injustice, not the victim of injustice.

* No one really knowingly does wrong.

Many libraries will purge checkout records once a patron returns books, but the increasing reliance on centralized cloud-based ILS vendors is making this harder to do reliably.

I'd appreciate it if you could name the online system your county uses. (Or if you can provide a link.)

Yup, you never get to \omega that way. Cardinality is a coarser notion than "ordinality". So your example shows that Card(\omega * \omega) = Card(\omega) = \aleph_0, even though \omega * \omega and \omega have different order types.

EDIT: Just wanted to add that an order isomorphism has two requirements:

(1) it needs to be a bijection (so order-isomorphic objects have the same cardinality); and

(2) it needs to preserve all inequalities (so a strict inequality among items in one object turns into a strict inequality in the same direction among the corresponding items in the other object).

The article suggests using nonviolent communication to clarify which sort of support is needed:

> Beyond that, ask for what you want, and check with your partner about what they want. If you want your partner’s support about something, let them know whether you’re looking for understanding or advice. Don’t make them guess. Don’t wait to see what they do. Tell them what you want.

I can see that the "Default to emotional support" can seem to suggest that you must provide emotional support whenever asked, but if you can't provide emotional support (whether from low EQ or from being drained), you can't. The suggested strategy of making the request for support unambiguous allows a response along the lines of (but probably not worded this way) "I'd like to provide support, but I can't."

The distinction between "stop giving X" and "stop asking for X" is important — in the latter case, someone is asking for what they need, whether or not the need can be fulfilled, while in the former case, someone is providing X, regardless of whether it is desired.