Math / Logic Discussion

[Strikingwolf]

But the conclusion of your argument is that one element in any set that satisfies your (rather nebulous) requirement must be false. If you use that number (which is irrational) to fill in the elements of a multiset as described in the first line of the post you quoted, you'll find that all the elements are true. There can be no false elements because there are no odd digits in the number.

How, precisely, is that not a counterexample? (if it's not, tell me why- I'm trying to figure this out)

That is a counterexample, I literally said it was

My argumentation however is correct, and thus it is an inherent logical contradiction, those are bad anyway, but y'know.

Also, if I were to make my self-referential definition more precise in order to avoid contradiction I could say "A set's elements are not self-referential if their true-false values are not generated in the same way (from the same source or otherwise) or depend on each other"

 

[BrickVoid]

That's a contradiction to the argument, it does not show the proof/argument is false, those are separate things

AKA, counterexamples/contradictions mean nothing if the deduction points to a conclusion

Your "proof", such that you claim it to be, doesn't actually test IF a variable is true or false, it simply assumes either "A or (not A)" is present, without actually going through even one iteration of actually checking whether said variable being tested is actually true or false. It is also a deduction that points to a conclusion, the statement is self evident in what you wrote, as your disjunction is what actually points to your conclusion:

Consider an infinite set of true/false statements {A, B, C, D...}. To prove that at least one of them is false we must construct the following disjunction `(not A) or (not B) or (not C) or (not D)`. We can assume that given any X, X or (not X) is true. Therefore, given A we can assume A or (not A) is true. Then to prove our conclusion we must consider both scenarios, A or (not A). For the second scenario, we already have proven our conclusion, one of the values is false. For the first scenario, we invoke B or (not B), and then repeat the process infinitely. We then never run out of things that would cause the conclusion to be true, and thus it must be true as we cannot prove that given that A is true that B is true also and so on for all the set as they are not self-referential

Therefore what you wrote originally adds up to nothing, and doesn't prove anything! It's an interesting theory, though, try running it past an algebra professor and see how many flaws are sent back to you.

There is another thing your proof doesn't take into account: Most people don't have the time to go checking an infinite set of true/false statements, most would rather you brought them a filtered set of finite true/false statements to evaluate. I am, after all, someone with only a finite amount of time, how do you reasonably expect me to consider an infinite set of true/false statements? What if the very first true/false statement actually IS false, and not true, as you claim it to be? What if an infinite set of true/false statements are all false, even though infinite? What you wrote fails to take into account what someone might expect to find, or even actually find, it just ties one algebraic expression up neatly if one rather convenient variable is actually what you claim it to be.

Cheers ...

BrickVoid

 

[Strikingwolf]

Your "proof", such that you claim it to be, doesn't actually test IF a variable is true or false, it simply assumes either "A or (not A)" is present, without actually going through even one iteration of actually checking whether said variable being tested is actually true or false.

You don't need to manually check if the variable is true or false. A or (not A) is always true as if A is true, then the first branch of the or is satisfied, if it is not then the second branch is. This only assumes A is a true/false statement or value, which is given in the information

It is also a deduction that points to a conclusion, the statement is self evident in what you wrote, as your disjunction is what actually points to your conclusion:

A proof is deduction from given statements pointing to a conclusion, and the disjunction is not used in the proof, as the disjunction is the conclusion

Therefore what you wrote originally adds up to nothing, and doesn't prove anything!

Not true as pointed out earlier

It's an interesting theory, though, try running it past an algebra professor and see how many flaws are sent back to you.

Ran it past a geometry teacher (someone skilled in proofs by necessity, especially given his teaching style), while we didn't go over it in detail he agrees that it is an interesting concept, and that my proof is the basic idea of what a proof for this would be, although it would need slight modification.

There is another thing your proof doesn't take into account: Most people don't have the time to go checking an infinite set of true/false statements, most would rather you brought them a filtered set of finite true/false statements to evaluate. I am, after all, someone with only a finite amount of time, how do you reasonably expect me to consider an infinite set of true/false statements?

This is a logic problem, not a practicality problem

What if the very first true/false statement actually IS false, and not true, as you claim it to be? What if an infinite set of true/false statements are all false, even though infinite?

I never claimed that the statement was true, I said that A or (not A) was true, which would follow from it being false

What you wrote fails to take into account what someone might expect to find, or even actually find, it just ties one algebraic expression up neatly if one rather convenient variable is actually what you claim it to be.

I use no variables besides the true/false statements and the set themselves, and I don't initialize the former or the latter, although I make stipulations on the latter in order to make the proof work, but that's equivalent to any given information in proof.

 

[Rubyheart]

How's about you mathlethes put your skills to the test and find out the probably of a vanilla world generating without any lapis (which is apparently more rare than diamond), which is apparently possible, but infinitesimally rare.

 

[BrickVoid]

How's about you mathlethes put your skills to the test and find out the probably of a vanilla world generating without any lapis (which is apparently more rare than diamond), which is apparently possible, but infinitesimally rare.

Have you checked to see if the config for the installation in question actually specifies a default value for lapis generation? Have you checked on the Mojang Atlassian bug reporting site for the version of vanilla Minecraft in question, to check whether there is actually a problem with lapis not being generated properly, in that version of Minecraft?

I currently play modded Minecraft, and I do indeed find lapis ore pockets occasionally. I also know that it's probably modified in the ore generation configs to generate at certain desired amounts for certain mods that actually use lapis in their recipes.

Where vanilla Minecraft is concerned, you need to check if you're mining between the correct y-level upper and lower bounds for where vanilla Minecraft generates lapis ore. Only if you are actually digging at the correct level for that ore will you run into any.

Lapis was also recently added to the trading list for Minecraft Villagers, because of the need for it in enchanting. It was, however, added after they added in requiring lapis for enchanting. Are you on the correct snapshot, if the lapis traidng is only currently present in a snapshot, and not in a released update of Minecraft?

There are lots of variables that control lapis generation, and while not infinite they are many. You need to check them all and make sure you're using the correct version of Minecraft for your needs.

Cheers ...

BrickVoid

 

[Rubyheart]

I don't have a lapis shortage, I was giving the math geeks a way to talk about math and still have it relate to minecraft.

 

[Strikingwolf]

How's about you mathlethes put your skills to the test and find out the probably of a vanilla world generating without any lapis (which is apparently more rare than diamond), which is apparently possible, but infinitesimally rare.

This isn't math it's logic

 

[RavynousHunter]

Set theory could be considered part of math, I suppose. I'm more of a hard numbers guy, though. Don't much care for proofs.

 

[BrickVoid]

This isn't math it's logic

Although logic can seem like it's not math, the two share a deeply intertwined relationship. A much better understanding of one comes from an understanding of the other and people who know both equally well benefit more than those who only know one or the other or who have an unequal grasp of either.

Here's a question: Which concept do you think came first? Learning that one plus one equals two or learning that a stick with a piece of flint stuck in the end of it represents a food gathering tool?

Logic cannot by itself throw rocks at something, that requires math in order to determine the size of rock one can throw without killing oneself! ;-)

Cheers ...

BrickVoid

 

[Strikingwolf]

Although logic can seem like it's not math, the two share a deeply intertwined relationship. A much better understanding of one comes from an understanding of the other and people who know both equally well benefit more than those who only know one or the other or who have an unequal grasp of either.

Logic is a superset of math, math is just logic with some axioms

 

[Strikingwolf]

How's about you mathlethes put your skills to the test and find out the probably of a vanilla world generating without any lapis (which is apparently more rare than diamond), which is apparently possible, but infinitesimally rare.

impossible b/c infinite world

 

[Rubyheart]

Near infinite, and certainly larger than Earth, but not infinite. ...unless that was changed in an update at some point.

 

[Someone Else 37]

Near infinite, and certainly larger than Earth, but not infinite. ...unless that was changed in an update at some point.

Nope, it's still technically finite. The x and z coordinates limited by Integer.MAX_VALUE = 2^31 - 1 ~= 500 million in each direction, if I remember correctly, for a total surface area of (2^32)^2 = 2^64 square meters, discounting hills, caves, etc.

This includes the quite large area once occupied by the Far Lands where physics starts to break down, but excludes the area beyond that where the chunks just stop existing.

I'm not sure what the chances of a lapis vein spawning in any random chunk are, but the probability of getting none of them anywhere is within a few orders of magnitude of one in 10^18.

 

[Heliomance]

That is a counterexample, I literally said it was

My argumentation however is correct, and thus it is an inherent logical contradiction, those are bad anyway, but y'know.

Also, if I were to make my self-referential definition more precise in order to avoid contradiction I could say "A set's elements are not self-referential if their true-false values are not generated in the same way (from the same source or otherwise) or depend on each other"

Failing to prove something false does not prove it true, and a single counter example is in fact enough to disprove a theorem. It is entirely possible to have an infinite, non self referential set where every element is true, and several examples have been provided. "Every element must be either true or false; if an element is true, check the next element which will either be true or false; given infinite elements you must encounter one which is false" is not a proof. The last step does not follow from anything. You have not offered any proof that eventually you must find an element for which (not p) is true.

EDIT: On a proper computer now, not my phone, so I'll break down your proof:

Consider an infinite set of true/false statements {A, B, C, D...}. To prove that at least one of them is false we must construct the following disjunction `(not A) or (not B) or (not C) or (not D)`. {Not actually true. To prove that at least one of them is false, you need to... demonstrate that at least one of them is false.}

We can assume that given any X, X or (not X) is true. {Valid, but tautological. It's not an assumption though, it's true by axiom.}

Therefore, given A we can assume A or (not A) is true. {Valid.}

Then to prove our conclusion we must consider both scenarios, A or (not A). For the second scenario, we already have proven our conclusion, one of the values is false. {Valid.}

For the first scenario, we invoke B or (not B), and then repeat the process infinitely. {Valid.}

We then never run out of things that would cause the conclusion to be true, and thus it must be true as we cannot prove that given that A is true that B is true also and so on for all the set as they are not self-referential {Not valid. While it is true that this process fails to prove that every result is true, it also fails to prove that any result is false. Simply because you have an infinite quantity of elements which may theoretically take one of two values, it does not follow that there must be an element that takes a given value.}

Consider an infinite set that consists of randomly selected integers. Each element of that set may be either imaginary (we'll call that P) or real (not P). The integers are randomly selected, so they're not self referential. Does it follow that at least one of those integers must be imaginary?

Alternatively, let's take that same set again. Each element can be said to be either a number (not P), or a giraffe (P). I don't think you're going to find any giraffes.

 

[RavynousHunter]

Nope, it's still technically finite. The x and z coordinates limited by Integer.MAX_VALUE = 2^31 - 1 ~= 500 million in each direction, if I remember correctly, for a total surface area of (2^32)^2 = 2^64 square meters, discounting hills, caves, etc.

This includes the quite large area once occupied by the Far Lands where physics starts to break down, but excludes the area beyond that where the chunks just stop existing.

I'm not sure what the chances of a lapis vein spawning in any random chunk are, but the probability of getting none of them anywhere is within a few orders of magnitude of one in 10^18.

Oddly, that's a bit of a pet peeve of mine: people calling finite things infinite, or "endless," or some other synonym. A multiverse is infinite, the amount of detail you can put into a digital tree is not. Besides, its not like English is wanting for words to describe finite, but arbitrarily large things: gargantuan, colossal, vast, immense, enormous...

 

[sgbros1]

Oh my god...

I was not kidding about this turning into a math thread.

Please, a moderator (like @Padfoote) do something...

Not only is all this unrelated, I'm supposed to be on school vacation...

Jesus guys have some respect.

 

[Lethosos]

Shhh! We don't get a civilised argument like this every day. I'm enjoying the heck out of this right now.

Sent from my Puzzle Box of Yogg-Saron using Tapatalk 2

 

[erindalc]

Although I'm not particularly well versed in what they're talking about, I'm laughing and enjoying this as well.

Sent from my XT1028 using Tapatalk

 

[Padfoote]

Oh my god...

I was not kidding about this turning into a math thread.

Please, a moderator (like @Padfoote) do something...

Not only is all this unrelated, I'm supposed to be on school vacation...

Jesus guys have some respect.

Due to this:

Shhh! We don't get a civilised argument like this every day. I'm enjoying the heck out of this right now.

Sent from my Puzzle Box of Yogg-Saron using Tapatalk 2

I'm not going to say knock it off. So long as you all don't actively try to prevent people from being on topic, it's fine. Although, three pages of it starts to raise the question of if it's time for a new thread. I may split it off into a new thread, but I'm currently too busy to do that right now.

 

[keybounce]

Ok, it is now time for some format logic.

First: The rules for logic, for proofs, especially for infinites, was changed after Godel demonstrated numerous flaws in the system of hand-waving that used to be used.

Second: The axioms of set theory are now vasty different than they used to be. For finite universes, they are equivalent -- Godel demonstrated that both basic arithmetic, and first-order logic, were self-consistent. Second order logic, however, can not be proven self consistent. That does not mean it was proven inconsistent.

Proving something true,
Proving that something cannot be proven true,
Proving that something is false

Are three different, valid, states of an idea.

The claim of "P or ~P", in other words, is *false* in a model/universe of unlimited things. There are restrictions on where you can just arbitrarily apply negation.

Self-referential with "not" in the infinite case is almost entirely invalid.

===

Proof by induction on positive integers:

To prove something true for infinite sets, the simplest way is proof by induction. For this, you have to first prove something true for some ground cases; then, you have to prove that if the previous was true, the next is true.

Example:
I know that F[0] is 0, and F[1] is one. I know that F[n] = F[n-2] + F[n-1]. Therefore, I know what F[x] is for any x.

In your case:

I know that item(0) is true. **I know that if item(n-1) is true, then item(n) is true**. Therefore, ...

What do you conclude? Proof by induction would say that they are all true, and there is no false.

Your flaws are:
1. That starred statement cannot even be made -- you made the assertion that each item is independent, and does not depend on the previous.
2. Even if you could, actually stating the proof formally, instead of hand-waving, leads to the conclusion that is opposite of what you said.

Please, rather than trying to be slick with english words, try to actually formulate what you are claiming in logic symbols.