I quiz ChatGPT about math

>Not every set has a measure. In order for a set to have a measure, it must satisfy certain properties, such as being countably infinite or having a well-defined notion of “size” or “length.” For example, the set of real numbers has a measure (namely, its Lebesgue measure), but the set of all subsets of the real numbers does not have a measure. In general, only certain types of sets can be assigned a measure in a meaningful way.

This response is just nonsense. Very interestingly the question is phrased using colloquial terminology and ChatGPT answers in a way where it talks about the colloquial phrase as if it were real mathematical terminology, in turn getting extremely confused and making stuff up.

(The reason the response is wrong is because the central point is lost. Being measurable is a property of the measure. ChatGPT claims some set isn't measurable while not naming under which measure, which is just nonsense. In fact the real numbers might "have no measure" e.g. under the empty measure and the set of all subsets of the real numbers might "have a measure" e.g. under the counting measure.)

Gotta hand it to ChatGPT. It's not wrong.

As it gets smarter and actually gets better reasoning, and capability- the way to keep AI sane and kind, will be to try to teach it like a person, with it putting importance on lessons from designated teachers even more, than it's weight in casual conversation , generally.

I think i forsee teachers being hired to teach AI, not just humans. Though how much AI will benefit from it will depend on their level But we'll have to get to that point first, as i read attempts to teach ChatGPT having odd results , which of course reveal what it can't do-yet.

At some point, we're going to have to intentionally educate AI- i feel like giving them knowledge in a similar manner is not talked about enough, even if some feel teaching AI like we do people isn't necessary due to AI's other abilities

I don't think i've seen others here consider the idea of seeing the o

JFZ: how do we know there are infinitely many primes

ChatGPT: One way to show that there are infinitely many prime numbers is to use the fact that the set of prime numbers is infinite to show that there must be an infinite number of composite numbers as well. Composite numbers are positive integers that have at least one positive integer divisor other than 1 and itself. For example, 4 is a composite number because it can be divided by 2 (and is not prime), and 6 is a composite number because it can be divided by 2 and 3 (and is not prime).

To show that there are infinitely many composite numbers, consider the sequence of numbers 2, 4, 6, 8, 10, 12, 14, 16, and so on. Each of these numbers is composite because it can be divided by 2. Now consider the sequence of numbers 3, 6, 9, 12, 15, 18, 21, 24, and so on. Each of these numbers is composite because it can be divided by 3. And so on. For every prime number, there is an infinite sequence of composite numbers that can be generated by multiplying that prime number by other positive integers.

Since there are infinite sequences of composite numbers, and each composite number is the product of at least two prime numbers (for example, 12 is the product of 2 and 6, which are both prime numbers), there must be an infinite number of prime numbers as well. This is one way to show that there are infinitely many prime numbers, though there are other ways to prove it as well.

(seems hand wavey, and probably wrong though I can't figure out where the flaw is)

JFZ: is A5 a solvable group

Yes, the group A5 (also known as the alternating group on 5 elements) is a solvable group. A group is solvable if it can be constructed from simpler groups using only group extensions and group quotients, and if it has a composition series in which each factor is a simple group. (wrong)

The group A5 is the group of even permutations of five objects. It has 60 elements, and it is isomorphic to the group of rotations of a regular dodecahedron. A5 is a simple group, which means that it has no proper normal subgroups (subgroups that are normal in the group but not equal to the group itself). This means that A5 cannot be constructed from simpler groups using group extensions or group quotients. However, it can be shown that A5 has a composition series in which each factor is a simple group, so it is solvable.

In general, many groups that arise in mathematics and physics are solvable, and the study of solvable groups is an important area of research in group theory.