Seth
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Every person that posts here writes down a paradox.

I'll go first,

"Thank you for saying your welcome"
Cutting in half forever.

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12th Archivist
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Seth wrote:
Cutting in half forever.

If you take 1/2 and add 1/4, 1/8, 1/16, 1/32, and so on, you will reach a definite answer. Well, not quite, but the idea behind it is quite interesting.

Wait, what grade are you in again?

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[Last edited by 12th Archivist at 04-19-2012 11:30 PM]

Seth
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Sorry That was silly of me.

Zeno's paradox?

6th

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[Last edited by Seth at 04-18-2012 06:07 AM]

Someone Else
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The king of France is bald.

skell
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This sentence is false.

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Seth
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The sentence below is true.

The sentence above is false.

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stigant
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2 graduates of medical school.

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Mouse
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Civil War

(or is that an oxymoron and not a paradox?)

Jacob
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stigant wrote:
2 graduates of medical school.

Don't get it.

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stigant
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Don't get it.

They're a pair of docs.

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Penumbra
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stigant wrote:

Don't get it.

They're a pair of docs.

How did I not see that one. This is like the time I finally got Beef Row's name....

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[Last edited by Penumbra at 04-19-2012 03:18 PM]

TFMurphy
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File: SeriesOver.jpg (159.8 KB)
Downloaded 656 times.
License: Other

May be relevant.

Keiya
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Does the set of all sets which do not contain themselves contain itself?

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Jatopian
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Keiya wrote:
Does the set of all sets which do not contain themselves contain itself?

No, it is an empty set because all of its components are empty sets too.

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stigant
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No, it is an empty set because all of its components are empty sets too.

I think we're about to get an earful from Trickster.

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Jutt
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Assuming we're working in ZFC, the correct answer would be: the class of all sets which do not contain themselves is not a set, but a proper class. Also, since the axiom of regularity forbids sets to be an element of themselves, that class is actually the class of all sets. And the class doesn't contain itself because it's not a set.

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Keiya
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Yes, I'm well aware that there's several formalizations that avoid it. They were designed to *specifically* avoid it... and as a result there's somethings they just plain can't describe.

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Trickster
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stigant wrote:
I think we're about to get an earful from Trickster.

Oh HELLZ yes.

Seth's "thank you": not a paradox, just a lie.

12th: you can't do anything forever, so calculus of limits is phrased in terms of values approached rather than definite results.

Someone Else: wat

skell and Seth: When truth is determined by statements, those statements can't make affirmative claims about their own truth values. The words "true" and "false" in that context aren't defined in the way we naturally treat them. Without realizing it, we're type-raising by imagining the statements refer to themselves one meta-level up. In logic, however, this is nonsensical.

To make this more clear, an example of this in everyday usage is the word "and". I may define something like "and" to mean "two nouns together", then apply it to verbs without realizing it. That's because human brains are great at generalization. But when you generalize logical discourse, you end up using things which not only weren't defined for a particular purpose, they cannot possibly be defined for that purpose.

The brilliance of Godel's breakthrough was this:

1) It is not possible to code the words "true" and "false" in any language powerful enough to encode facts about natural numbers, addition, and multiplication.

2) It IS possible to code the words "provable" and "not provable" in any system complex enough to express first-order logic, however.

He sidestepped the problem by changing "truth" to "provability". If you have a statement that says, "this statement is not provably true", then such a statement must be true (falsity would imply truth, which is a contradiction). But such a statement is also out of reach of any proof system humans can devise. Thus mathematics is incomplete.

TFMurphy: that's just fun.

Keiya and Jatopian and Jutt: put another way, the contradiction you arrive at with "set of all sets" simply illustrates that there is no set which contains all sets: such a set does not exist.

Sets are not "anything you can imagine", since you can easily imagine pseudo-paradoxes. We have to follow precise rules to decide if a particular set exists. And in the bounds of set theory, we often restrict things even further than this. We like to prove things without the Axiom of Choice (not called that because mathematicians prefer it, by the way) when possible, even though AC is obviously true on its face (so to speak). While set theory should imply a very rich Universe of sets, it's usually pared down to a scant little skeleton of "only those sets we can actually define with functions". Whenever we model it we use a countable model, which is teeny-tiny, and employ some frightfully clever if dirty tricks to get around the "but shouldn't there be more reals than naturals" issue.

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[Last edited by Trickster at 08-15-2012 05:27 AM]

Trickster
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Keiya wrote:
Yes, I'm well aware that there's several formalizations that avoid it. They were designed to *specifically* avoid it... and as a result there's somethings they just plain can't describe.

This is not entirely correct.

The changes made to set theory were not created specifically to avoid the set-of-all-sets issue. There are many, many ways around that particular problem.

The main problem is that, when Set Theory began, everyone was using the Axiom of Unrestricted Comprehension:

AUC (false): "For all predicates, there exists a set containing all and only members for which the predicate is true."

This Axiom is unfortunately false. It is self-contradictory. If you include this Axiom in any system, it always leads to a situation where every statement is provable (including true = false). There is no way to fix the axiom; it is irreparably broken. The set-of-all-sets was one of the first issues to highlight this, but there are other more damning problems.

Excluding AUC does not "restrict" things at all. I mean, it does make some things only true and other things only false, so compared to "everything is both true and false" that's a restriction I suppose.

The way the fix was done was to change it to the Axiom of Separation:

AS (okay, we believe): "If you have a set with things in it, and a predicate, taking only the stuff from that set for which the predicate is true also forms a set."

In other words, you can't make a set out of "anything", because if you could, you could just invent a paradox and call that a set. (Derp. It's kind of surprising it took so long for us to discover the error.) The things you take to form your set, under our new axiom, have to be members of a set already. This necessitates four new Axioms to describe succinctly how sets can be put together from the bottom up: Pairing, Union, Power Set, and Induction (Infinity). Any set you can possibly describe will be formed by these four axioms (this is simple to prove).

As for the other axioms: Extensionality was already being assumed in the older Naive Set Theory, Foundation (Regularity) and Replacement aren't really necessary for anything, and Choice isn't needed for most things either.

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[Last edited by Trickster at 08-15-2012 05:44 AM]

Trickster
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And now I'll introduce my own pseudo-paradox, in keeping with the thread.

The Cynical Child (an old tale rephrased)

As it turns out, the famous philosopher Zeno of Elea was thinking up logical paradoxes long before he was an adult. His view that "movement should not exist" from the apparent paradoxes of the infinite divisibility of space was but one of many logical inconsistencies he saw in reality. Even as a young boy, Zeno had quite a reputation for being cynical, dour, and altogether unpleasant.

To ease the youth's tortured mind, his father came into his room one day with an offering. "Zeno, my son," said he, "I see you spending all day in here with books and writings, and we worry about you. So I have prepared a special gift for you. Tomorrow is Sunday, so I will give it to you one morning this coming week, but I can promise that you will not know which morning until that morning arrives. You will be surprised," his father smiled.

Zeno furrowed his brow in deep thought, generally a grave omen. His father quickly retreated from his son's room as the child stewed in his own thoughts.

"One day this week?" pondered Zeno. "And I won't know which day in advance. Well, it cannot possibly be Saturday, for that is the last day of the week, and I would know with certainty Friday evening that I would receive the gift on Saturday morning," he realized. "So the gift must come between Sunday and Friday," he confirmed. Even Aristotle would have agreed with that conclusion, he decided.

"But, wait a moment. If the latest day in which I can receive the gift is on Friday, then it cannot be Friday either. I would know with certainty that, since Saturday is clearly impossible, going to sleep on Thursday would mean the gift must await me Friday morning," he concluded. "So I would know in advance in that case as well, and thus, neither can Friday be the morning in which I receive the gift."

Zeno quickly discovered that, by induction, there was no day in which the gift could be given, not even on Sunday! This means his father must have been lying about giving him a gift, or at the very least, was lying about Zeno not knowing the day in advance. Zeno wondered what his father was up to exactly.

Several days passed. Then, on Thursday morning, Zeno awoke to find his new pony.

And true to his father's word, he was surprised.

Where is the flaw in Zeno's logic?

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skell
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Trickster wrote: Where is the flaw in Zeno's logic?

He was just too full of himself clearly .

Anyhow wikipedia says that "this sentence is false" is a paradox and that's enough for me - if on some higher level of consciousness, knowledge or intelligence it isn't exactly true then heck, I am just too stupid to grasp it, at least for now. Then again I have encountered the provabilities, I think in Smullyan's books? Plus this goes dangerously close to philosophy for me. I am a simple guy!

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Trickster
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skell wrote:
Anyhow wikipedia says that "this sentence is false" is a paradox and that's enough for me -

In the sense of "apparent contradiction when described in ambiguous natural language", sure, but in the sense of "breaks all of logic", not so much.

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[Last edited by Trickster at 08-15-2012 02:01 PM]

skell
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Trickster wrote:
In the sense of "apparent contradiction when described in ambiguous natural language", sure, but in the sense of "breaks all of logic", not so much.

I guess we'd have to first take a step to the basics and get a common definition of a paradox... But nope, I don't wanna play this game, really, I am not educated enough in these fields to even try to bother .

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12th Archivist
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Trickster wrote:
Where is the flaw in Zeno's logic?

Was it that Zeno's father did not logic the situation as thoroughly as Zeno, and so was not constrained by the rules that Zeno thought he was?

Or was it that Zeno's father knew that Zeno would try to figure it out and come to the conclusion that no day was possible, so giving him the gift on any day would be surprising?

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Someone Else
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Trickster wrote:
Someone Else: wat

Oh, it's quite simple. There is no king of France. So is it a true statement that the king of France is bald, or not?

Jatopian
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Someone Else wrote:

Trickster wrote:
Someone Else: wat

Oh, it's quite simple. There is no king of France. So is it a true statement that the king of France is bald, or not?

Bald has extant as a prerequisite.

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rman
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Someone Else wrote:
Oh, it's quite simple. There is no king of France. So is it a true statement that the king of France is bald, or not?

I think it’s true.

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Trickster
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rman wrote:

Someone Else wrote:
Oh, it's quite simple. There is no king of France. So is it a true statement that the king of France is bald, or not?

I think it’s true.

You are correct, sir.

Another example of a vacuously true statement:

"If The Sun is green, then The Sun is not green."

These mainly sound nonsensical to us because we don't generally use English to communicate nothing useful.*

* except in American politics

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[Last edited by Trickster at 08-15-2012 11:37 PM]

Trickster
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12th Archivist wrote:

Trickster wrote:
Where is the flaw in Zeno's logic?

Was it that Zeno's father did not logic the situation as thoroughly as Zeno, and so was not constrained by the rules that Zeno thought he was?

Or was it that Zeno's father knew that Zeno would try to figure it out and come to the conclusion that no day was possible, so giving him the gift on any day would be surprising?

Those are both good guesses, but there's a more fundamental issue here, related to what I spoke about previously. What it means to "know" something is defined in an incomplete way here, and it keeps changing with each iteration of the "induction". It's the same paradoxical issue that hits when you try to define "truth" within a complex system...some statements will be outside your reach, so it can't be done.

In this case, what it means to "know in advance" includes information constructed from the analysis of the problem itself. You're using meta-information to change information that should be set in stone at the beginning of the tale, and this leads to contradictions. There's never a way to rigorously define "know in advance" here, and if you tried to do so, you would discover that you can't "know in advance" whether or not someone would "know in advance".

So, interpreted as best we can cram this ugly mess into logic, the initial declaration was false, regardless as to the outcome. (Just because the prediction came true doesn't mean the prediction itself was a valid one.)

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Someone Else wrote:

Trickster wrote:
Someone Else: wat

Oh, it's quite simple. There is no king of France. So is it a true statement that the king of France is bald, or not?

The orange clouds are purple

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