Search found 602 matches

by Cauchy
Wed Feb 03, 2016 10:43 pm UTC
Forum: Mathematics
Topic: Well-ordered Subset of Reals
Replies: 8
Views: 3211

Re: Well-ordered Subset of Reals

All countable ordinals are isomorphic to some subset of the rationals with the standard ordering. (So yeah, ω 1 is the upper bound.) I've convinced myself that this it true. My argument uses transfinite induction, leveraging that there's an order-preserving "shrinking" map that takes the ...
by Cauchy
Sun Jan 10, 2016 12:22 pm UTC
Forum: Logic Puzzles
Topic: Blue Eyes with Superrationality
Replies: 48
Views: 11046

Re: Blue Eyes with Superrationality

The superrational part is that the islanders know everyone else must arrive at the same conclusions with the same given information. Even the original solution required superrationality as the islanders moved en masse and even waited together based on the same piece of information. But the islander...
by Cauchy
Tue Dec 15, 2015 4:53 pm UTC
Forum: Logic Puzzles
Topic: Computing a function without knowing the arguments
Replies: 6
Views: 3267

Re: Computing a function without knowing the arguments

It sounds like you're stretching the definition of "communication between" very far; I would argue it's past the breaking point. I assume the answer you're looking for is something along the lines of "you create a magic function that takes the 2n inputs, and then reports back to each ...
by Cauchy
Tue Dec 15, 2015 7:02 am UTC
Forum: Logic Puzzles
Topic: Computing a function without knowing the arguments
Replies: 6
Views: 3267

Re: Computing a function without knowing the arguments

This is not possible for n >= 2. For n = 2, consider the function that looks at the two pairs of boolean values and treats them as integers between 0 and 3, inclusive; the function returns 0 when the sum of the two integers is congruent to 0 or 1 mod 4, and 1 when their sum is 2 or 3 mod 4. Both fri...
by Cauchy
Fri Dec 11, 2015 10:41 pm UTC
Forum: Mathematics
Topic: Describe an Inconsistent System.
Replies: 44
Views: 8318

Re: Describe an Inconsistent System.

1) If you were treating a gerund like a present participle when the lesson was about not confusing different similar-looking parts of speech, I'd say you were having a negative impact on everyone else in the room and should be talked to. 2) I'm talking between classes, while the student is away. I'm...
by Cauchy
Fri Dec 11, 2015 5:30 pm UTC
Forum: Mathematics
Topic: Describe an Inconsistent System.
Replies: 44
Views: 8318

Re: Describe an Inconsistent System.

If I have a student who doesn't understand the difference between a noun and a verb, I don't spend hundreds of words (most of which the student also doesn't understand, most likely) detailing the distinction between gerunds and present participles. My words were for Xanthir, not Treatid, so I don't...
by Cauchy
Fri Dec 11, 2015 4:18 am UTC
Forum: Mathematics
Topic: Describe an Inconsistent System.
Replies: 44
Views: 8318

Re: Describe an Inconsistent System.

I don't see too much difference between saying an inconsistent Peano arithmetic proves Santa Claus is real, and Treatid's talking about axiomatic mathematics as containing contradictions. If we're going to give him trouble for conflating two levels, then we should certainly hold ourselves to a highe...
by Cauchy
Fri Dec 11, 2015 3:42 am UTC
Forum: Mathematics
Topic: Describe an Inconsistent System.
Replies: 44
Views: 8318

Re: Describe an Inconsistent System.

I'm not sure that playing fast and loose with definitions is the best course of action right now, given that looseness of definitions is basically the entire problem. So I'm very reticent to conflate first-order logic with one of its models.
by Cauchy
Thu Dec 10, 2015 9:47 pm UTC
Forum: Mathematics
Topic: Describe an Inconsistent System.
Replies: 44
Views: 8318

Re: Describe an Inconsistent System.

Xanthir wrote:You're allowed to say that "A" means "Santa is real". Standard FOPL stuff, no predicates or anything.


This is certainly a model of first-order logic, which we're trying to divorce from the formal system itself in this discussion.
by Cauchy
Thu Dec 10, 2015 8:58 pm UTC
Forum: Mathematics
Topic: Describe an Inconsistent System.
Replies: 44
Views: 8318

Re: Describe an Inconsistent System.

Logical systems are more self-contained than you're making them out to be, Treatid. A system comes equipped with: 1) Its alphabet, that is, the symbols that can be used to make its well-formed formulas (the logical sentences that could or could not be provable) 2) Its grammar, which is the collecti...
by Cauchy
Thu Dec 10, 2015 7:10 pm UTC
Forum: Mathematics
Topic: Describe an Inconsistent System.
Replies: 44
Views: 8318

Re: Describe an Inconsistent System.

Logical systems are more self-contained than you're making them out to be, Treatid. A system comes equipped with: 1) Its alphabet, that is, the symbols that can be used to make its well-formed formulas (the logical sentences that could or could not be provable) 2) Its grammar, which is the collectio...
by Cauchy
Tue Dec 01, 2015 3:41 pm UTC
Forum: Mathematics
Topic: Describe an Inconsistent System.
Replies: 44
Views: 8318

Re: Describe an Inconsistent System.

Consider the system with one axiom, A (with A not equal to ~A), and no rules of inference. This system is consistent, because the only statement that can be proved is A. It's an axiom, so it's proved by default, and since there are no rules of inference, nothing else can be proved, and hence ~A cann...
by Cauchy
Tue Dec 01, 2015 3:28 pm UTC
Forum: Mathematics
Topic: hotel infinity and countably infinite
Replies: 31
Views: 5943

Re: hotel infinity and countably infinite

I think the thing that's leading you astray, phillip, is that the diagonalization arguments all end with "Here's a single real number you missed.". While that's true, it might feel unsatisfying, like "Oh, I only missed one? I'll just add it on. What then?". Actually, your propose...
by Cauchy
Mon Nov 30, 2015 1:19 am UTC
Forum: Mathematics
Topic: hotel infinity and countably infinite
Replies: 31
Views: 5943

Re: hotel infinity and countably infinite

What's wrong with the standard proof? Take a bus of real numbers (that is, a list of real numbers that has a real number for each positive integer). We'll look at the binary expansions of the real numbers on this bus, and we'll list the terminating rational numbers twice, once under each expansion. ...
by Cauchy
Sat Nov 21, 2015 6:08 pm UTC
Forum: Mathematics
Topic: Reverse carry-less binary multiplication problem
Replies: 5
Views: 3863

Re: Reverse carry-less binary multiplication problem

If we treat the binary string a_na_{n-1}...a_0 as the polynomial a_nx^n + a_{n-1}x^{n-1} + ... + a_0x^0, then your multiplication is multiplication of these polynomials in the polynomial ring over Z/2Z, that is, the integers modulo 2. Since Z/2Z is a field, Z/2Z[x] is a unique factorization domain, ...
by Cauchy
Fri Nov 20, 2015 2:27 am UTC
Forum: Logic Puzzles
Topic: number pyramid
Replies: 1
Views: 1658

Re: number pyramid

Can you explain what a "number pyramid" is? The only puzzle type I could think of is one where you have a triangle of numbers such that each number above the bottom layer is the difference of the two numbers directly below it. But this doesn't have the right number of numbers on the middle...
by Cauchy
Mon Nov 16, 2015 3:27 am UTC
Forum: Mathematics
Topic: Least numbers discarded when dealing cards
Replies: 30
Views: 6766

Re: Least numbers discarded when dealing cards

There was a similar problem in (I think it was) the Logic Puzzles subforum about minimizing the expected number of rolls of an n-sided die to emulate a k-sided die. I recall that a good strategy (though maybe not the best strategy?) was to interpret the numbers as digits in the base-n expansion of a...
by Cauchy
Sun Nov 15, 2015 1:20 pm UTC
Forum: Mathematics
Topic: Combinatorics
Replies: 4
Views: 1622

Re: Combinatorics

I've never seen this before in my life, so I don't know where you'd find it in the literature. Just write the proof, it's not that bad. If you rewrite it in the graph theory way, then you can pass off a lot of stuff to graph theory. Specifically, the "no loops implies a vertex of degree one&quo...
by Cauchy
Sat Nov 14, 2015 9:13 pm UTC
Forum: Mathematics
Topic: Combinatorics
Replies: 4
Views: 1622

Re: Combinatorics

We'll consider the 2n sets as n pairs, each A_i paired with its A_i union {x_i}. Start with the set A_1, and denote it B_0. If no other set is A_1, we're done! Otherwise, A_1 appears again in one of the pairs, either as some other A_i or as some A_i union {x_i}, in each case with i != 1. In either c...
by Cauchy
Fri Oct 30, 2015 8:32 am UTC
Forum: Logic Puzzles
Topic: Room count of the cross-wired tower.
Replies: 2
Views: 1980

Re: Room count of the cross-wired tower.

Pick a starting room, and choose one direction as "forward". We say that two rooms are linked if their light switches toggle the other room's state. For each positive integer N, in sequence: Note the state of the starting room. Walk forward N rooms, flip the light switch, and walk bac...
by Cauchy
Wed Oct 14, 2015 3:11 am UTC
Forum: Mathematics
Topic: Prob/Expected Value Problem
Replies: 4
Views: 2023

Re: Prob/Expected Value Problem

When you say "1", I assume you're talking about the random variable "the number of times Event B occurs before Outcome 1 happens", and similarly for "2", "3", "4", and "5". In that case, the total number of times that Event B occurs is not ...
by Cauchy
Sat Oct 10, 2015 8:17 am UTC
Forum: Mathematics
Topic: Prob/Expected Value Problem
Replies: 4
Views: 2023

Re: Prob/Expected Value Problem

The second one is correct. In the first, what are the five random variables you're applying linearity of expectation to? And why is their sum the number of times Event B occurs? For the second, the five variables are: X_1 = the number of occurrences of Event B until the first of the five outcomes fo...
by Cauchy
Mon Oct 05, 2015 8:44 am UTC
Forum: Logic Puzzles
Topic: How many rooms are there in the tower?
Replies: 26
Views: 5968

Re: How many rooms are there in the tower?

I find all the weird case analysis to speed up the algorithm rather inelegant. My preferred solution is cleanly stated, even if it's slow. For each integer N in sequence, starting from 1: Note the current state of the room you're in. Walk in one direction N rooms, toggle the switch there, and th...
by Cauchy
Fri Oct 02, 2015 2:36 am UTC
Forum: Mathematics
Topic: Composite numbers and polynomials
Replies: 9
Views: 2802

Re: Composite numbers and polynomials

Doesn't P(n) = n cover all the composite numbers?
by Cauchy
Wed Sep 30, 2015 6:17 am UTC
Forum: Mathematics
Topic: Factoring numbers of the form p^x+q^y
Replies: 3
Views: 2082

Re: Factoring numbers of the form p^x+q^y

Using modular arithmetic will help greatly here. If we're testing whether a prime r divides the sum, then Fermat's little theorem tells us that we can replace x and y with their remainders mod r-1 for the test, which will make the calculations much more tractable. Separately, if x and y share an odd...
by Cauchy
Fri Sep 25, 2015 7:24 am UTC
Forum: Logic Puzzles
Topic: Anti-Gambler's Fallacy
Replies: 162
Views: 27952

Re: Anti-Gambler's Fallacy

The warden never speaks. You know the method the warden used to give out the hat colors. See? In the original puzzle, everyone is told that they know the warden used a method that made getting a red or a blue hat equally likely, and nobody doubts it. The strategy that people came up with works in a...
by Cauchy
Fri Sep 25, 2015 1:16 am UTC
Forum: Logic Puzzles
Topic: Anti-Gambler's Fallacy
Replies: 162
Views: 27952

Re: Anti-Gambler's Fallacy

You're still fundamentally missing the problem. If I see those 8000 red hats, why should I keep believing that the hats were picked with a distribution of 70-30? If anything, your variation has the problem even more to the forefront: why should I take this warden's word as the absolute truth of the ...
by Cauchy
Thu Sep 24, 2015 6:26 pm UTC
Forum: Mathematics
Topic: Convergence of sets in a geometrical sense
Replies: 27
Views: 5137

Re: Convergence of sets in a geometrical sense

Your condition 3 is weird, which makes sense since it ends up being irrelevant. For any sequence with a limit and any point in one of the sets, we can swap that individual point into the sequence without changing the limit, so trivially every point is part of some sequence with a limit. Your definit...
by Cauchy
Wed Sep 16, 2015 1:12 am UTC
Forum: Mathematics
Topic: I don't seem to understand my own brain teaser, please help!
Replies: 22
Views: 5177

Re: I don't seem to understand my own brain teaser, please h

I don't know it seams simple to me. Your expected life after X trials is 10*(2^(0.55*x))*(0.3^(0.45*x)). It is easy to show that as X->infinity expectancy->0. This is not your expected life. This is the exponential of the expected log of your life, which is not the same thing. Alternately, this is ...
by Cauchy
Mon Sep 14, 2015 7:19 pm UTC
Forum: Logic Puzzles
Topic: 5-uples around a table (hard puzzle)
Replies: 12
Views: 4319

Re: 5-uples around a table (hard puzzle)

So, just to be absolutely clear: the goal is to maximize the shortest distance between any two brothers (and then the target value of d is one less than this shortest distance)?
by Cauchy
Mon Sep 14, 2015 12:43 am UTC
Forum: Logic Puzzles
Topic: 5-uples around a table (hard puzzle)
Replies: 12
Views: 4319

Re: 5-uples around a table (hard puzzle)

So you mean a distance between two seats? For example, the 5-tuples within distance 5 of a given 5-tuple are the ones that are up to five seats in either direction? That's what I thought you had meant, but then: In that case, what do you mean by the maximal distance? If distance isn't a derived quan...
by Cauchy
Sun Sep 13, 2015 11:35 pm UTC
Forum: Logic Puzzles
Topic: 5-uples around a table (hard puzzle)
Replies: 12
Views: 4319

Re: 5-uples around a table (hard puzzle)

Goahead52 wrote:Definition 2 :

We call distance d the number of seats at left and at right of some 5-uple sitting around the table.


I do not understand you definition of distance here. Doesn't a given position have just one seat to its left and to its right?
by Cauchy
Sun Sep 13, 2015 11:29 pm UTC
Forum: Logic Puzzles
Topic: The castle
Replies: 9
Views: 3375

Re: The castle

I have the solution for all rectangular castles. I prove that the mathematician's location can be identified after the number of moves specified, and that it cannot be identified after fewer moves. The answer is that 25 moves (so that they've visited 26 rooms) is the least number at whic...
by Cauchy
Sat Sep 12, 2015 11:38 am UTC
Forum: Logic Puzzles
Topic: Anti-Gambler's Fallacy
Replies: 162
Views: 27952

Re: Anti-Gambler's Fallacy

I feel like you're looking at a different problem from the rest of us, Vytron. While I'm solving the problem as written, you've made a mental substitution, switching out "You know for certain that the coin is biased in your favor: 70 to 30." for "the coin, when flipped, will always co...
by Cauchy
Sat Sep 12, 2015 6:38 am UTC
Forum: Logic Puzzles
Topic: Anti-Gambler's Fallacy
Replies: 162
Views: 27952

Re: Anti-Gambler's Fallacy

I do have another question though. Why is the normal distribution defined over the entire domain rather than just within 6 standard deviations? Anything falling outside that range indicates an error in how the expected value was determined. Why not simplify things and say the data range can be no b...
by Cauchy
Fri Sep 11, 2015 8:13 pm UTC
Forum: Logic Puzzles
Topic: Anti-Gambler's Fallacy
Replies: 162
Views: 27952

Re: Anti-Gambler's Fallacy

I guess the thread can be summarized as follows: Most people, after witnessing very unlikely events happening by chance, assume there's some outside force intervening and making those events happen. I'll now say that our reality is as inconsistent, at least according to the Many-Worlds interpretati...
by Cauchy
Fri Sep 11, 2015 8:38 am UTC
Forum: Logic Puzzles
Topic: Anti-Gambler's Fallacy
Replies: 162
Views: 27952

Re: Anti-Gambler's Fallacy

Cradarc wrote:My bad Cauchy. Forgot about that post.
Do you consider yourself a frequentist?


No, I don't think I'd say I'm a frequentist. I think I fall pretty close to the Bayesian camp, that outside of the quantum level, a probability just represents my ignorance of what's going on.
by Cauchy
Fri Sep 11, 2015 8:26 am UTC
Forum: Mathematics
Topic: I don't seem to understand my own brain teaser, please help!
Replies: 22
Views: 5177

Re: I don't seem to understand my own brain teaser, please h

Starting with 33 years: You flip the coin: You have a 55% chance that you get heads, in which case your remaining life becomes 66 years; or 45% chance of tails, in which case your remaining life become 9.9 years. Then you flip again. If your first flip is heads and your second flip is heads, your l...
by Cauchy
Fri Sep 11, 2015 5:37 am UTC
Forum: Logic Puzzles
Topic: Anti-Gambler's Fallacy
Replies: 162
Views: 27952

Re: Anti-Gambler's Fallacy

I'd like to point out that I presented a bound lower than 8000 in the very first reply in this thread . There is no earthly way that you could convince me to keep playing after 8000 consecutive losses, because nothing could grant me a prior certainty strong enough to fight that data Strange. Let's f...
by Cauchy
Fri Sep 11, 2015 4:36 am UTC
Forum: Mathematics
Topic: I don't seem to understand my own brain teaser, please help!
Replies: 22
Views: 5177

Re: I don't seem to understand my own brain teaser, please h

Coyne, I think you're misunderstanding expected value. Consider the game where you roll a six-sided die. On a 1-5, you lose $1. On a 6, you gain $6. The expected value for this game is $1/6, even though you're more likely to lose than to win. The small chance of a large payoff skews the expected val...

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