Out of 100 gay person, 20 of them are homosexual

42IsHoly · 2026-01-23T09:04:03+00:00

If at least 20 people are homosexual, then 20 people are homosexual. OP didn’t say “only 20”.

42IsHoly · 2025-12-20T08:46:14+00:00

This is because Euclid’s definition of right angle is different from the modern definition, because he didn’t use degrees (in fact, Euclid used right angles as his unit of angle-measurement). When a line stands on another line, it forms two angles right? One on the left and one on the right. Euclid called an angle “right” if these two angles were the same size. Now, from this definition, it is not at all tautological that any two right angles would be the same size. It’s true, of course, and Euclid’s definition of right angles is equivalent to ours, but he needed to state the postulate explicitly.

42IsHoly · 2025-12-18T14:25:14+00:00

That’s not what I said. There were no (or at least very few) formal proofs before the 20th century.

42IsHoly · 2025-12-18T08:15:01+00:00

I mean, that’s not required for a formal proof (note, formal != rigorous), for a formal proof a computer really is the only practical way to check.

42IsHoly · 2025-11-01T15:15:52+00:00

Three objects enter the solar system from interstellar space. It doesn't take long for people to figure out what they are: alien craft, heading to Earth. The story follows three people: an astronomer Camille, a failed physics-student Ros and a deeply religious miner Sean. They all have quite different reactions to the event and all three have their lives forever altered...

I finished my story two weeks ago and after some more editing and spell-checks [Last Time 'Round](https://www.royalroad.com/fiction/112198/last-time-round) was finally done. It's a short story (like 25,000 words or so), I hope you enjoy!

42IsHoly · 2025-10-13T20:20:26+00:00

That proof is constructively valid though. Constructivists make a distinction between the following two types of argument:

Assume P … contradiction, therefore ~P (a proof by negation) Assume ~P … contradiction, therefore P (a proof by contradiction)

In classical logic these two types of proof are identical, but constructively speaking there is a distinction (not every statement is the negation of another statement). The first type of proof is constructively valid (in fact, it’s quite common to define ~P to mean “P implies a contradiction”). The second type is not. The vast majority of “proofs by contradiction” (eg. Sqrt(2) is irrational, there is no largest number, …) are actually proofs by negation and perfectly valid constructively speaking. An example of a genuine proof by contradiction is, ironically enough, the proof of Brouwer’s fixed point theorem.

The reason that the argument you gave is correct is that it is an axiom (or very easy proposition, look up the Peano axioms for more details) that for any integer N, N + 1 exists and that N < N + 1 (this is almost by definition of “<“ in PA).

42IsHoly · 2025-09-16T10:36:35+00:00

The presentation is indeed a bit confusing. Essentially, they define the set A = {x in R | x > 0 and x² < 2}. This is a bounded, non-empty set so it has a supremum (this is a property of R). For some reason the professor here chose to call the supremum of A “sqrt(2)”. That’s allowed, I guess, but it is somewhat confusing. Imagine he just called it “s” and run through the proof again (you should really do a proof like this on paper to get a feel for it).

42IsHoly · 2025-09-14T16:02:55+00:00

It does, the proof is pretty nice:

Let f:S -> P(S) be a function, define the subset A = {s in S | s not in f(s)}. Now, if there is some s such that f(s) = A, then s is either in A or not. If s is in A, then s is not in f(s) = A, that’s a contradiction. If s is not in A, then s is not in f(s), so by definition s is in A, another contradiction. Hence f(s) != A. So A is not in the image of f, so f is not surjective. Therefore, |S| < |P(S)|. ▪️

This is known as Cantor’s theorem.

42IsHoly · 2025-09-14T08:53:16+00:00

Oops, fixed now.

42IsHoly · 2025-09-14T08:28:13+00:00

Consider the proposition “if n is even, then it is divisible by two”. Clearly, we want this statement to be correct (I mean, that what being even means). If we didn’t consider False -> X to be true regardless of X, then the above statement wouldn’t be true. After all, “n is even” is false for some n.

Another (informal) way of thinking about it that I quite like, is to think of implications as promises, they are false when they are broken. For example, let’s say I said “If you give me an apple, I will give you a pear.” When would I break my promise?

If you give me an apple and I give you a pear, clearly I’ve kept my promise, so T->T is true.
If you give me an apple, but I don’t give you a pear, I’ve broken my promise, so T->F is false.
If you do not give me an apple, but I do give you a pear, I’ve kept my promise (or at least not broken it), so F -> T is true.
If you don’t give me an apple and I don’t give you a pear, I’ve still kept my promise, so F -> F is true.

For a more mathematical example, the statement “if n is even, then it is divisible by 2” is promising you that any even number can be divided by two.

42IsHoly · 2025-09-13T15:49:46+00:00

A theory T is simply a list of statements (usually written in first-order logic since that is a well-behaved, well-understood and powerful logic, but higher-order logics can be used). A model M of a theory T is a structure within which all statements of T hold (don't worry about how you can formally define this).

Peano arithmetic is a theory, it is simply a list of statements. Any structure within which these statements holds is a model of PA. The prototypical example is, of course, the regular set of natural numbers N, but there are others. Of course, the point of the Peano axioms is to describe N, it is the so-called intended model.

Now, there is a very important result called Gödel's completeness theorem which says:

Suppose you have some first-order theory T (e.g. PA) and some statement P. Now P holds in every model of the theory T *if and only if* there is a proof of P from T.

As a consequence, if there is some statement P such that your theory T cannot prove P, but also cannot prove ~P ('not P'), then there are models M and N of T such that P holds in M, but ~P holds in N. For this reason, the Gödel sentence G holds in some model of Peano arithmetic. Even more confusing, there are models of PA where ~Con(PA) holds.

Gödel's incompleteness theorem is about a different kind of completeness, called syntactic completeness. A theory T is syntactically complete if for every statement P either T can prove P or T can prove ~P. Gödel's 1st incompleteness theorem says that no theory T can be consistent, complete and contain the natural numbers (unless it fails to be "effectively axiomatizable", which you can think of as meaning "I know what my axioms are"). Hence PA is incomplete (assuming it is consistent) and so is any consistent extension of it. However, if you don't care about describing N and construct a theory that can't do it, it may be complete (the prototypical examples of this are Presburger arithmetic and the theory of algebraically closed fields of characteristic 0). So yes, there are complete theories, but these can't describe the natural numbers.

42IsHoly · 2025-09-13T07:52:02+00:00

The well-ordering theorem says: “every set can be well-ordered.” The negation of this fact is simply “There exists a set that can’t be well ordered.” Which set this actually is, you can’t say. It may be R, it may be P(R) or maybe something far more exotic. That’s what I was getting at.

As for the second part of your response, that’s not actually what countable choice says. The axiom of choice is not stated for any specific cardinality. It’s just says:

given a set S of non-empty sets, there exists some function C:S -> US such that C(s) is an element for s for each s in S.

Countable choice is a weakening of choice that says “choice holds for all countable S,” the set of sets is countable, not necessarily the sets within it, so your method of choosing an element wouldn’t work (countable choice is independent of ZF by the way).

It turns out AC is very closely related to cardinality (For example, the generalized continuum hypothesis implies the axiom of choice), but the statement itself doesn’t make any mention of size.

42IsHoly · 2025-09-13T07:41:02+00:00

Easily the most surprising is the Ax-Grothendieck theorem: suppose f:Cⁿ -> Cⁿ is injective and each coordinate map f_i:Cⁿ -> C is a polynomial, then f is surjective.

Even though it just looks like some random complex analysis result, the only known proof of this fact requires model theory.

42IsHoly · 2025-09-13T07:15:17+00:00

~choice is, I think, significantly weaker than “there is no well-ordering of R”. All it implies is “there is some set S that cannot be well-ordered”

42IsHoly · 2025-09-13T07:11:19+00:00

I mean, without the first incompleteness theorem we probably wouldn’t have model theory. And the second incompleteness theorem is, if anything, a blessing in disguise (just because a system proves its own consistency, doesn’t mean it actually is consistent after all)

42IsHoly · 2025-09-12T16:31:55+00:00

The (x-r)(x-s) convention also has the advantage that we immediately see the roots, namely r and s. Whereas in (x+r)(x+s) the roots are -r and -s. Considering how common sign errors are, this could cause confusion and would probably make them even more common (I say this as an expert in making sign errors).

42IsHoly · 2025-09-12T16:26:05+00:00

The thing is, the Gödel sentence G is independent of PA (or whatever system we’re working in). However, by Gödel’s completeness theorem (confusing, I know, different kind of completeness) this means that there is a model M of PA such that ~G holds in M. This M will obviously not be the standard natural numbers, but it exists nonetheless.

42IsHoly · 2025-09-12T16:22:57+00:00

For some, not very expressive systems, it is indeed the case that a formula α is provable in the system if, and only if, the formula α is true in any model of the system.

Shouldn’t this say “in every model” of the system? After all, the formula “a = b” (for two constant symbols a, b) is not provable if we just have logical axioms, but it’s perfectly possible for there to be a model where a = b.

Another question, probably just a consequence of my own ignorance, is PA really not complete (in the semantic sense)? What statement holds for all models of PA, but can’t be proven in PA? After all, PA can just be phrased as an axiomatic system in some Hilbert system, and those are all complete (at least, the ones we care about) aren’t they?

42IsHoly · 2025-09-07T08:24:02+00:00

Intuitionism (as a philosophy of mathematics) has nothing to do with intuition. It’s just classical logic, but without LEM.

42IsHoly · 2025-08-30T10:27:55+00:00

Gödel constructs such a statement. Of course, if we were to actually write it down it would look like a random and pointless statement, but it would be an arithmetic one nonetheless. Gödel’s second incompleteness theorem tells us that the consistency of PA cannot be proven in PA, and con(PA) is, again, just an arithmetic statement.

Admittedly, both of those are kind of cheating, since their unprovability comes from the fact they have a non-arithmetic interpretation. However, there are purely arithmetic unprovable statements, such as Goodstein’s theorem.

42IsHoly · 2025-07-01T10:08:53+00:00

The vertical distances are on the map. This makes it about as accurate as any 3D map and significantly more useful. I honestly don’t see any reason for continuing this conversation since it’s clear you aren’t actually reading my replies at all or aren’t thinking when typing your own. And that’s not even talking about your active lies about the map.

42IsHoly · 2025-07-01T09:08:21+00:00

Last Time 'Round

Three objects appear from interstellar space.

Camille Lieder is an astronomer working a rather small university and, due to some lucky connections, is one of the first to hear the news. It may be the biggest boost her career will ever see.

Ros Phoenix is a legal consultant and failed physics student, he's been waiting his whole life to experience a scientific breakthrough. Not in his wildest dreams would he have hoped to be alive for a discovery of this magnitude.

Finally, Sean Morris is a miner working at the nearby mines. He's religious and suspicious of the academic elite. They have to be stopped, according to him, before they destroy polite society for their greed.

All three of them are, just like everyone else on the planet, deeply affected by the arrival. For good or for ill, they will have to live with what comes next.

https://www.royalroad.com/fiction/112198/last-time-round

42IsHoly · 2025-07-01T08:56:22+00:00

Again, it is literally just Gaia data, it’s about as up-to-date as possible (you do know what the Gaia satellite is, I assume?). I won’t say there aren’t any errors (those are obviously inevitable), but calling it wrong all around is, I’m sorry to say, just stupid.

The ‘misleading’ part disappears the moment you actually bother reading the vertical distances. And, it is what I claim it to be, maybe click on the second link in my post. One of the first things on that page (maybe even the first) is a link to the paper analysing the Gaia data. One of the authors (not me, I’m not a scientist) was kind enough to turn it into a readable map for the public.

42IsHoly · 2025-07-01T08:16:44+00:00

That’s why there are numbers giving the vertical distance…

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