/sci/ - Science & Math » Thread #11116793

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Anonymous Mon 04 Nov 15:27:06 2019 No.11116793 View Reply Original Report

Quoted By: >>11116944 >>11117180 >>11117200 >>11117964 >>11118076 >>11118366 >>11118567 >>11118623 >>11118851

>he believes the well-ordering principle

Anonymous

Anonymous Mon 04 Nov 2019 15:31:36 No.11116803 Report

Quoted By: >>11116804 >>11116837 >>11117964

Show me a set that can't be well-ordered

Anonymous

Anonymous Mon 04 Nov 2019 15:32:28 No.11116804 Report

Quoted By: >>11116811 >>11116812

>>11116803
The reals.

Anonymous

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Anonymous Mon 04 Nov 2019 15:41:41 No.11116811 Report

Quoted By:

>>11116804
shut your mouth

Anonymous

Anonymous Mon 04 Nov 2019 15:42:02 No.11116812 Report

Quoted By: >>11116814

>>11116804
And? Why not?

Anonymous

Anonymous Mon 04 Nov 2019 15:42:44 No.11116814 Report

Quoted By: >>11116820

>>11116812
Give me a well-ordering of them, please

Anonymous

Anonymous Mon 04 Nov 2019 15:45:17 No.11116820 Report

Quoted By: >>11116825 >>11117964

>>11116814
I can't, provably. There are a lot of them, and I only have so much space.
Are you saying that something can't exist if it's too big for me to write down for you?

Anonymous

Anonymous Mon 04 Nov 2019 15:48:17 No.11116825 Report

Quoted By: >>11116851 >>11116932

>>11116820
You can construct a rule to well-order them at least, right? It's easy to well-order the naturals, even though you only have so much space

Anonymous

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Anonymous Mon 04 Nov 2019 15:58:07 No.11116837 Report

Quoted By: >>11120455

>>11116803
The set of all non-well-orderable sets

Anonymous

Anonymous Mon 04 Nov 2019 16:13:59 No.11116851 Report

Quoted By:

>>11116825
There are far less naturals than reals. There are only countably many finite length strings, and there are uncountably many reals. If I were to describe a well ordering rule on the reals to you, you could then write it out sufficiently for each real and produce a surjection from the finite length strings onto the reals. Such would be impossible.
This is only doable if you allow me to use an uncountable alphabet. But then the matter is trivial.
This is all obvious, is it not?

Anonymous

Anonymous Mon 04 Nov 2019 17:06:31 No.11116932 Report

Quoted By:

>>11116825
The rule is simple: they are ordered randomly. Done.

Anonymous

Anonymous Mon 04 Nov 2019 17:11:06 No.11116944 Report

Quoted By:

>>11116793
>He doesn’t take the logical chicanery of mathematicians as a means to construct a more effective art and body of quantitative tools for the sciences
>he trips over himself reenacting the fedora-religitard dialectic when mathfags generally understand platonism doesn’t real
>his motivations are purely computational and anti-artistic vendettas
Haha gay!

Anonymous

Anonymous Mon 04 Nov 2019 18:35:46 No.11117180 Report

Quoted By: >>11117188

>>11116793
why wouldn't I? lmao this nigga denies the continuum

Anonymous

Anonymous Mon 04 Nov 2019 18:38:47 No.11117188 Report

Quoted By:

>>11117180
The continuum doesn't require ZFC

Anonymous

Anonymous Mon 04 Nov 2019 18:43:45 No.11117200 Report

Quoted By:

>>11116793
If you "believe" in the induction principle, then that's equivalent.

Anonymous

Anonymous Tue 05 Nov 2019 00:23:39 No.11117964 Report

Quoted By: >>11117972 >>11118071 >>11118828

>>11116820
>>11116793
>>11116803

Retards. The well ordering principle only works on subsets of $\mathbb{N}$ , and this principle must hold for ALL subsets. Consider any open interval on $\mathbb{R}$ . The well ordering principle does not apply because that interval does not contain a smallest element. Even though the set of real numbers is a total order under $\leq$ , it is not well-ordered. Those are very different things.

Anonymous

Anonymous Tue 05 Nov 2019 00:25:58 No.11117972 Report

Quoted By:

>>11117964
>and this principle must hold for ALL subsets.
All non-empty subsets, that is.

Anonymous

Anonymous Tue 05 Nov 2019 00:54:55 No.11118071 Report

Quoted By: >>11118102

>>11117964
You can define more than 1 order on a set retard. They are not talking about the usual order.

Anonymous

Anonymous Tue 05 Nov 2019 00:57:00 No.11118075 Report

Quoted By: >>11118331 >>11118497

> the open interval (0, 1) ? [0,1] does not contain a least element. From the ZFC axioms of set theory (including the axiom of choice) one can show that there is a well order of the reals. Also Wac?aw Sierpi?ski proved that ZF + GCH (the generalized continuum hypothesis) imply the axiom of choice and hence a well order of the reals. Nonetheless, it is possible to show that the ZFC+GCH axioms alone are not sufficient to prove the existence of a definable (by a formula) well order of the reals.

some one explain this

Anonymous

Anonymous Tue 05 Nov 2019 00:57:12 No.11118076 Report

Quoted By: >>11118203

>>11116793
enjoy math without zorn's lemma bro

Anonymous

Anonymous Tue 05 Nov 2019 01:06:28 No.11118102 Report

Quoted By: >>11120038 >>11120295

>>11118071
Of course, but the well-ordering principle specifically deals with the usual order.

Anonymous

Anonymous Tue 05 Nov 2019 01:34:19 No.11118203 Report

Quoted By:

>>11118076
and without the fucking axiom of choice
how are you going to work with Z cross Z at all?

Anonymous

Anonymous Tue 05 Nov 2019 02:33:41 No.11118331 Report

Quoted By: >>11118344 >>11118351

>>11118075
There are countably many different strings of characters we can write, and hence countably many formulas. There are an uncountable amount of ways to order the reals, so there must be some ways to order the reals that are not definable. ZFC+GCH cannot prove that a well-order on the reals is one of the 0% of orderings of the reals that are definable.

Anonymous

Anonymous Tue 05 Nov 2019 02:40:46 No.11118344 Report

Quoted By:

>>11118331
[0,1][1,2][2,3]......
whats wrong with this ordering on the positive reals

Anonymous

Anonymous Tue 05 Nov 2019 02:42:15 No.11118351 Report

Quoted By: >>11118355 >>11118376

>>11118331
>ZFC+GCH cannot prove that a well-order on the reals is one of the 0% of orderings of the reals that are definable.
Then in what sense does it even exist? If it can't be defined, how can it exist?

Anonymous

Anonymous Tue 05 Nov 2019 02:43:23 No.11118355 Report

Quoted By:

>>11118351
axiom of choice/zorns lema/well ordering principle implies it exists

Anonymous

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Anonymous Tue 05 Nov 2019 02:47:50 No.11118366 Report

Quoted By: >>11118394

>>11116793
https://www.lesswrong.com/posts/zmSuDDFE4dicqd4Hg/you-only-need-faith-in-two-things

>You only need faith in two things: That "induction works" has a non-super-exponentially-tiny prior probability, and that some single large ordinal is well-ordered. Anything else worth believing in is a deductive consequence of one or both.
>(The reason why you need faith for the second case is that although more powerful proof systems - those with larger proof-theoretic ordinals - can prove the consistency of weaker proof systems, or equivalently prove the well-ordering of smaller ordinals, there's no known perfect system for telling which mathematical systems are consistent just as (equivalently!) there's no way of solving the halting problem. So when you reach the strongest math system you can be convinced of and further assumptions seem dangerously fragile, there's some large ordinal that represents all the math you believe in. If this doesn't seem to you like faith, try looking up a Buchholz hydra and then believing that it can always be killed.)

Is Yudkowsky right?

Anonymous

Anonymous Tue 05 Nov 2019 02:51:33 No.11118376 Report

Quoted By: >>11118386

>>11118351
It exists because its existence is implied by the axioms. It is not necessary to give an example of something to show that it exists, unless if you are a constructivist.

Anonymous

Anonymous Tue 05 Nov 2019 02:56:44 No.11118386 Report

Quoted By: >>11118390 >>11118856

>>11118376
If the axioms imply something stupid, then the axioms are bad choices and we should pick different ones
Example: Russell's paradox and the fact that it spawned ZFC

Anonymous

Anonymous Tue 05 Nov 2019 02:57:33 No.11118390 Report

Quoted By:

>>11118386
Its not stupid though

Anonymous

Anonymous Tue 05 Nov 2019 03:01:31 No.11118394 Report

Quoted By: >>11118434

>>11118366
>muh "induction just works bro"
Stop with this nonsense, and Yudkowski doesn't know what the fuck he is talking about. Saying that P(n) only holds up to some finite number n in the Naturals will get you a logical contradiction if you start out with the hypothesis that if P(k) is true, then P(k+1) is true for all k. Anyone who has been through an introductory proof writing class has seen this.

Although the well-orderedness of the naturals is essentially an axiom, I think it's a reasonable one to make. Here, try to construct a subset of the naturals where there isn't an obvious well defined smallest element. Protip: you can't.

Anonymous

Anonymous Tue 05 Nov 2019 03:24:28 No.11118434 Report

Quoted By: >>11118512

>>11118394
Not that guy but it seems the only reason such a set cannot be constructed is because ZF implies the Peano axioms. You're essentially trying to justify a set of axioms using a stronger set of axioms, which doesn't seem to be any more satisfactory than "induction is obviously a property of what we are trying to describe."

Anonymous

Anonymous Tue 05 Nov 2019 03:56:22 No.11118497 Report

Quoted By:

>>11118075
The usual notion of definability is in the projective hierarchy. It is consistent with ZFC+GCH that there no projective well-ordering of the reals.

Anonymous

Anonymous Tue 05 Nov 2019 04:01:57 No.11118512 Report

Quoted By:

>>11118434
>it seems the only reason such a set cannot be constructed is because ZF implies the Peano axioms.
Good enough for me. I mean, what more can you do? Gödel was pretty clear about the lack of completeness in such a formal axiomatic system as the ZFC. Your alternative is coming up with something that isn't axioms to justify all of math.

Anonymous

Anonymous Tue 05 Nov 2019 04:22:00 No.11118567 Report

Quoted By:

>>11116793
from the point of view of ZF, some sets can be well-ordered and some cannot
sets that can't be well-ordered are huge
the axiom of choice is like Christianity
it is a delightful dream of utopia
all of the bad evil sets have been destroyed
we can enjoy paradise on earth

Anonymous

Anonymous Tue 05 Nov 2019 04:23:04 No.11118569 Report

Quoted By:

Holy shit, what is with this thread? I leave the board for a couple years, and it fills up with retards. ZFC this, projective hierarchy that. /sci/ is and always has been a type theory board. Suck my excluded middle cock, setdrones.

Anonymous

Anonymous Tue 05 Nov 2019 05:01:34 No.11118623 Report

Quoted By: >>11118699 >>11118784

>>11116793
That's a slippery slope anon, soon you'll be saying a product of nonempty sets can be empty

Anonymous

Anonymous Tue 05 Nov 2019 05:44:07 No.11118699 Report

Quoted By: >>11118729

>>11118623
Not assuming a particular axiom is not the same as assuming it's negation.

Anonymous

Anonymous Tue 05 Nov 2019 06:14:30 No.11118729 Report

Quoted By:

>>11118699
gtfo filthy intuitionist

Anonymous

Anonymous Tue 05 Nov 2019 07:05:57 No.11118784 Report

Quoted By: >>11118858

>>11118623
A product of finitely many nonempty sets cannot be empty
A product of infinitely many nonempty sets, who knows?

Anonymous

Anonymous Tue 05 Nov 2019 07:35:56 No.11118828 Report

Quoted By: >>11120291

>>11117964
>t. computer science major taking discrete math

Anonymous

Anonymous Tue 05 Nov 2019 08:02:34 No.11118851 Report

Quoted By:

>>11116793
>believes

Anonymous

Anonymous Tue 05 Nov 2019 08:06:41 No.11118856 Report

Quoted By:

>>11118386
Russell's paradox is an actual contradiction, whereas no such contradiction is known to exist in ZFC. In fact, ZFC is equiconsistent with ZF, so if ZFC is inconsistent then so is ZF.

Anonymous

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Anonymous Tue 05 Nov 2019 08:08:27 No.11118858 Report

Quoted By: >>11120071

>>11118784
>A product of infinitely many nonempty sets, who knows?

Anonymous

Anonymous Tue 05 Nov 2019 20:19:50 No.11120038 Report

Quoted By: >>11120291 >>11120295

>>11118102
Try again, freshman.

Anonymous

Anonymous Tue 05 Nov 2019 20:32:49 No.11120071 Report

Quoted By:

>>11118858
To talk about them requires the axiom of choice, countable choice, or decidable choice

Anonymous

Anonymous Tue 05 Nov 2019 21:46:27 No.11120291 Report

Quoted By: >>11120295

>>11120038
>>11118828
hes literally right though

El Arcón

El Arcón Tue 05 Nov 2019 21:48:44 No.11120295 Report

Quoted By:

>>11118102
>>11120038
>>11120291
define the characters that you used to spell the word "ususal"

Anonymous

Anonymous Tue 05 Nov 2019 22:29:58 No.11120455 Report

Quoted By:

>>11116837
You can't construct this in zf you dumb fuck

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