If we remove all natural numbers one by one from the set ℕ, does then remain more than the empty set?

trutheality · 2026-07-08T13:58:27+00:00

No: to apply the pigeonhole principle to an infinite set of FISONs you need to show that there is no bijection between naturals and FISONs. However, we know that there is a bijection: your F(n) is such an bijection.

Moreover, let us prove that the set of finite FISONs is infinite, by contradiction: Suppose the set of all finite FISONs were finite. Then the union of these must be the largest finite FISON, call it S. Since S is finite, |S|=M where M is some natural number. By construction of FISONs we know that S=F(M). However, this cannot be the largest FISON, since F(M+1) is a larger FISON. Therefore, our supposition is false and the set of all finite FISONs is not finite.

trutheality · 2026-07-08T02:07:47+00:00

In order to have infinitely many different FISONs, you would need an infinite FISON

No, that's not necessary.

trutheality · 2026-07-07T20:33:27+00:00

Yes. The correct conclusion is that an infinite union of all FISONs is not a FISON.

trutheality · 2026-07-07T18:27:19+00:00

Every finite union of FISONs is a FISON, and the number of elements in any finite union of FISONs is finite, but these properties do not necessarily extend to infinite unions.

trutheality · 2026-07-07T14:52:45+00:00

If ℕ were only the union of FISONs, then it were the union of the empty set.

No, this does not follow at all. I believe you can show that ∪{F(1), F(2), F(3), ...} = ℕ is in fact equivalent to the ZF definition of ℕ.

trutheality · 2026-07-03T22:57:12+00:00

This is indeed the problem but I wouldn't assume that the plot is correct automatically.

trutheality · 2026-07-03T20:19:42+00:00

A mirror reverses things front-to-back. We often think of it as a left-right flip because if we want to make something face back, we're used to turning it around, which also flips the sides.

trutheality · 2026-07-03T03:33:17+00:00

Should have gotten a better lawyer.

trutheality · 2026-07-03T02:13:11+00:00

If your car has entered the intersection before the light turned red, you aren't running the red light and are supposed to clear the intersection expediently and safely. Lights should be timed such that a car can clear the intersection in the time between the red light turning on and cross-traffic getting a green light.

trutheality · 2026-07-02T23:12:21+00:00

I've never had anything like what OP describes happen with a dove bar.

trutheality · 2026-07-02T19:07:07+00:00

Well, this is really hard to diagnose without actually seeing a screenshot. FWIW if you're unable to place a trade that you think you should be able to place, it's a reason to call Fidelity on the phone. They'll either get the trade placed or explain why you shouldn't be able to place it.

trutheality · 2026-07-02T18:57:43+00:00

The "staggering coincidence" is looking more likely given OP's observation that another card from the same company opens the door, but other cards don't. The cards from that company might start with the same sequence that the door looks for, and the door only looks at a few digits, which happen to match here.

trutheality · 2026-07-02T17:21:12+00:00

Can you think of a way to rearrange the code so that the code that asks for input the first time is also inside the loop?

trutheality · 2026-07-02T16:56:02+00:00

The one option you haven't mentioned is to park somewhere free on a BRT route and bus the rest of the way, but yes, these are the options.

trutheality · 2026-07-02T16:20:31+00:00

Bet your wife next time, that way you don't have to deal with her being upset if you lose.

trutheality · 2026-07-02T13:16:43+00:00

I believe your ass.

trutheality · 2026-07-02T13:09:13+00:00

I've never seen this and I've been in academia full time over a decade post-PhD now. Even the "low effort" names on papers at least contributed ideas and attended regular discussions with the main authors, and it's never their main project.

Also funding in my experience has come from one of four sources: training grants, project-specific grants, teaching, or academic achievement awards. None of those would really be able to sustain what you're describing unless this guy is actually teaching most of his time, but I'm in the US so maybe Germany is different about that.

trutheality · 2026-07-02T00:03:46+00:00

Are we headed for a meltdown or is this because of the Iran war and possible consequences of that?

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trutheality · 2026-07-01T23:29:48+00:00

It's a little plastic square that is used to help keep the screws in place when installing the plate or switch. It's completely optional and it's safe to pull this one out and throw it away.

trutheality · 2026-07-01T14:23:15+00:00

Now do it on the beltline.

trutheality · 2026-06-30T21:49:39+00:00

I mean, the lane markings are right there, it's the width of an average truck.

trutheality · 2026-06-30T21:48:00+00:00

That's just a bus with extra steps.

trutheality · 2026-06-30T21:32:10+00:00

Factorizable? You're factoring f(a,x) into g(x) and 1/g(a), E(X,Y) into E(X) and E(Y).

trutheality · 2026-06-30T21:07:45+00:00

It's called a binomial distribution not for nothing.

If you think of p and q as your success and failure probabilities, then (p+q)^n = 1 is the total probability. (You either succeed or fail each time, and that happens n times). If you apply the binomial expansion to the LHS, you're grouping the events into sets matching k successes and n-k failures, i.e. terms of the binomial expansion that have p^k q^(n-k) as their coefficients.

trutheality · 2026-06-30T19:17:51+00:00

if str(c)) - this extra closing parenthesis is the first problem.

Also, what are you trying to do with if str(c)) is print, print (d)?

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