/sci/ - Science & Math » Thread #14115353

75KiB, 1280x720, 1642447502612.jpg

View Same Google iqdb SauceNAO

Who is this madman?

Anonymous Mon 17 Jan 19:40:46 2022 No.14115353 View Reply Original Report

Quoted By: >>14131304 >>14131696 >>14136566 >>14139893

A new mathematics free from real numbers ... who is this madman? Where can I learn more about alternate mathematical systems?

Anonymous

Anonymous Thu 20 Jan 2022 20:10:33 No.14129128 Report

Quoted By: >>14129214 >>14129946 >>14131291

>>14128981
It is a good example, which exposes your confusion clearly.
First, it is obviously not a Dedekind cut, because 0 belongs to neither A nor B.
That slip aside, you are missing that there is no set B, assuming ZF or ZFC.
Set Theory is much more restrictive than you think, and you cannot define sets willy nilly.
You try to define B so that B is one thing if CH holds and another thing if CH is false.
But CH is not decidable, therefore you have not made any definition at all of B.

Anonymous

Anonymous Thu 20 Jan 2022 20:14:05 No.14129136 Report

Quoted By: >>14130015

>>14129084
>What's the criterion by which you decide whether a Cauchy sequence is "well-defined" or "ill-defined"?
By the criterion I gave. If we must make a Goodman's grue and bleen distinction for terms of a sequence then the terms are ill-defined. I believe you understand this, too, and are playing a devil's lawyer, which is good. So is there a further ZF construction which would allow sequences all of whose terms are undefined to still adhere to a single convergence?

Anonymous

Anonymous Thu 20 Jan 2022 20:16:41 No.14129145 Report

Quoted By:

>>14128986
You are right. I grew up during an age when there was only one Chaitin constant.
Now there are more.
But it does not matter at all for the argument, it only makes it stronger.

Anonymous

Anonymous Thu 20 Jan 2022 20:18:01 No.14129147 Report

Quoted By:

>>14129066
wtf no you have to list every component and when they were manufactured and who in the factory inspected them and where the materials were supplied from and where the metal ores were discovered and where the oils for the plastics were drilled and also know full family history of the first person to invent fire
otherwise how would you truly know it was a car?

Anonymous

Anonymous Thu 20 Jan 2022 20:39:38 No.14129209 Report

Quoted By:

>>14129066
>You seem to think that objects are defined singularly, one at a time
Where did I think so? You seem to misunderstand the problem, the problem is that the category if real numbers is too broad. Of course whatever number we would ever deal with is accounted for by the theory of real numbers, but almost every real number can't be defined or specified, this is why we should reconsider that theory. Because what is the purpose of a definition? As Spinoza said "omnis determinatio nagatio est", every definition is a limitation, which excludes other things. It is quite clear that if we found a way define numbers in some other way, so that vaguely speaking only definable or computable numbers would be accounted for, it would shed light on a lot of areas of mathematics. Think about the history of the parallel postulate, people didn't think it was an issue until they realized that there exist many other geometries, so defining things more rigorously here opened a way for whole new gigantic areas of mathematics. And if we were arguing 2 centuries ago, you would laugh at me asking me to show you two parallel lines which intersect.

Anonymous

Anonymous Thu 20 Jan 2022 20:42:58 No.14129214 Report

Quoted By:

>>14129128
>That slip aside, you are missing that there is no set B, assuming ZF or ZFC.
>Set Theory is much more restrictive than you think, and you cannot define sets willy nilly.
There is such a set, it is not a hard exercises to define it explicitly referencing ZFC axioms. Continuum hypothesis can be stated in terms of set theory, so where would a problem in defining such a set even arise?

Anonymous

Anonymous Thu 20 Jan 2022 20:44:12 No.14129218 Report

Quoted By:

Guenon wrote a good book on the subject btw.
https://ia801305.us.archive.org/30/items/reneguenon/1946%20-%20The%20Metaphysical%20Principles%20of%20the%20Infinitesimal%20Calculus.pdf

Anonymous

Anonymous Fri 21 Jan 2022 00:28:29 No.14129946 Report

Quoted By:

>>14129128
>First, it is obviously not a Dedekind cut, because 0 belongs to neither A nor B.
My bad, the inequalities in the definition of B should be non-strict.
>That slip aside, you are missing that there is no set B, assuming ZF or ZFC.
You're completely wrong, this is a valid set according to both ZF and ZFC. Its existence follows directly from the axioms.

Anonymous

Anonymous Fri 21 Jan 2022 00:48:57 No.14130015 Report

Quoted By: >>14131601 >>14131618

>>14129136
You didn't give any criterion. You're just saying that the terms are undefined, but they're well-defined according to the axioms of ZF. You clearly have a problem with the definition of a Cauchy sequence in set theory, I've given you an explicit example of a Cauchy sequence which ZF considers to be well-defined, but you
consider it to be ill-defined. Note that it probably means that you also consider "most" real numbers to be ill-defined. Welcome to the club.

Anonymous

Anonymous Fri 21 Jan 2022 10:47:35 No.14131291 Report

Quoted By:

>>14129128
found the brainlet. hint: it's you.
I suggest you take course in mathematical logic, you lack the knowledge needed to discuss these matters.

Anonymous

Anonymous Fri 21 Jan 2022 10:55:32 No.14131304 Report

Quoted By:

>>14115353
He's far from the first person to talk about the idea of basing math of and different foundation of numbers

Anonymous

Anonymous Fri 21 Jan 2022 11:01:20 No.14131314 Report

Quoted By: >>14131373 >>14131406

>>14119652
At some point math has to be self defined (circular) or definitions based on the real world and thus be intuitive. What ever is used fundamental to define everything else is going to be "shaky", but you have to start with something.

Anonymous

Anonymous Fri 21 Jan 2022 11:26:18 No.14131373 Report

Quoted By:

>>14131314
wtf no you have to play the game by my rules and only the things i say are well defined are well defined

Anonymous

Anonymous Fri 21 Jan 2022 11:42:48 No.14131406 Report

Quoted By:

>>14131314
>At some point math has to be self defined (circular) or definitions based on the real world and thus be intuitive.
Yes, exactly. However, "sets" are not based on the real world and are too vague to be understood intuitively. That has been demonstrated many times.
>What ever is used fundamental to define everything else is going to be "shaky", but you have to start with something.
No, it's going to be shaky only if you start with some vague undefined primitive. If you start with a notion such that there's no ambiguity about what constitutes this notion and what doesn't (for example if it's intuitive enough or if there's some definite method how to decide it), it's not going to be shaky at all. Nobody is playing the infinite regress game. I would be satisfied if someone could provide a characterization of a "set" that I can use to actually determine whether some thing, when put before me, is a set or not. Otherwise sets cannot be said to form a solid basis for rigorous mathematics.

Anonymous

Anonymous Fri 21 Jan 2022 13:22:46 No.14131601 Report

Quoted By: >>14131618 >>14131623

>>14130015
>You didn't give any criterion.
But yes, I did. My further criterion is that all outcomes of the sequence must also be Cauchy among themselves. Recall that the proof of convergence for a_n >>14124015 was by exhaustion: two outcomes are listed (call them A_1 and A_2), each is proven to get arbitrarily close to itself, then [{a_n}] is trivially proven as described in >>14125838 to be in a singeton equivalence class with itself. It is a simple patch to require a further proof that every outcome also gets arbitrarily close to every other outcome. In the case of A_1 and A_2 the epsilon is capped at 1 and so it is clearly impossible to prove that all outcomes of a_n get arbitrarily close to themselves thus a_n does not have the quality of being Cauchy, or rather Cauchier, since we've made a new restriction to the definition. (One could also pronouce the French way, as in the sequence a_n me fait Cauchier.) It is an easy fix and doesn't break any other properties of R that I can see.

Anonymous

Anonymous Fri 21 Jan 2022 13:32:27 No.14131618 Report

Quoted By: >>14131689

>>14130015
>>14131601
It also works equally well for sequences whose outcomes cannot be fully enumerated. For example, let c_n = 1/n if the nth coin is heads or -1/n if the nth coin is tails. Then by Kolmogorov we have that c_n converges with a probability of 1. However, it cannot be proven that every outcome converges with every other outcome, and so c_n is excluded from being Cauchier convergent. Indeed we can disprove in this case (though we needn't) by noting that the outcome C_1 = HTHTHT... gets arbitrarily close to every member of the equivalence class containing all the series representations of log 2 and that the outcome C_2 = THTHTH... gets arbitrarily close to [...] -log 2 and so the epsilon between C_1 and C_2 is capped at log 4.

Anonymous

Anonymous Fri 21 Jan 2022 13:33:45 No.14131623 Report

Quoted By: >>14131783

>>14131601
Can you formulate your criterion as a property of Cauchy sequences? As in

"A Cauchy sequence is well-defined if and only if....."

Anonymous

Anonymous Fri 21 Jan 2022 14:08:38 No.14131689 Report

Quoted By:

>>14131618
>unbounded sequence is cauchy

Anonymous

View Same Google iqdb SauceNAO 1580686909664.png, 343KiB, 1474x803

Anonymous Fri 21 Jan 2022 14:15:51 No.14131696 Report

Quoted By:

>>14115353
someone should notify the authorities... we just found a superspreader

Anonymous

Anonymous Fri 21 Jan 2022 14:51:19 No.14131783 Report

Quoted By: >>14131825

>>14131623
…every outcome of the definition can be proven to get arbitrarily close to all other outcomes of the definition.

Anonymous

Anonymous Fri 21 Jan 2022 15:03:12 No.14131825 Report

Quoted By: >>14131992

>>14131783
What is an outcome of the definition? Can you write it in the language of ZF?

Anonymous

Anonymous Fri 21 Jan 2022 15:55:21 No.14131992 Report

Quoted By: >>14132191

>>14131825
Set theory is not my area or interest so any formalism I write will be amateurish. Perhaps you can write it better? An outcome of the definition must surely be expressible by ZF, otherwise the original proof by exhaustion that a_n converges can't be expressed by ZF either which would obviate the need for an extra criterion.

Anonymous

Anonymous Fri 21 Jan 2022 17:03:19 No.14132191 Report

Quoted By: >>14132272

>>14131992
We should first answer the question whether you're proposing to remain within the boundaries of ZF and change the definition of a Cauchy sequence within ZF, or whether ZF should be replaced by a different formalism. I'm pretty sure it's the second option, because you don't really have a problem with the sequence {a_n} itself. The sequence itself (as a set) is equal either to the constant sequence of 1's, or to the constant sequence of 0's. Therefore if both (1,1,1,..) and (0,0,0,...) are Cauchy sequences, then so is {a_n}, and this is inevitable. You have a problem with ZF allowing us to define a set this way in the first place, i.e. you have a problem with the axioms of ZF.

Anonymous

Anonymous Fri 21 Jan 2022 17:26:18 No.14132272 Report

Quoted By: >>14132311

>>14132191
>if both [outcomes A_1 =] (1,1,1,..) and [A_2 =] (0,0,0,...) are Cauchy sequences, then so is {a_n}, and this is inevitable.
However, A_1 and A_2 can't be proven to get arbitrarily close to each other (in fact we can disprove it) so while a_n may be a Cauchy sequence it is not a "Cauchier" sequence. This doesn't prevent the set from existing in ZF or even from being a Cauchy sequence, but it does give us a fully remedied Cauchier equivalence class definition of R unless there is another angle to the argument which you have not yet described.

Anonymous

Anonymous Fri 21 Jan 2022 17:40:02 No.14132311 Report

Quoted By: >>14132340

>>14132272
If a_n is not a Cauchier sequence, then either (1,1,1,...) is not a Cauchier sequence or (0,0,0,...) is not a Cauchier sequence (or both).

Anonymous

Anonymous Fri 21 Jan 2022 17:51:46 No.14132340 Report

Quoted By: >>14132363

>>14132311
But 1_n = (1,1,1,...) has only one possible outcome which can be proven to get arbitrarily close to itself. (0,0,0,...) is not a possible outcome for 1_n whereas it is for a_n.

Anonymous

Anonymous Fri 21 Jan 2022 18:02:48 No.14132363 Report

Quoted By: >>14132411 >>14132443 >>14132536

>>14132340
There's no but. If P(a) is false and (a=b or a=c) is true, then P(b) is false or P(c) is false. This is basic logic. Your definition doesn't make any sense.

Anonymous

Anonymous Fri 21 Jan 2022 18:15:11 No.14132411 Report

Quoted By: >>14132422

>>14132363
>excluded middle
look at this amateur

Anonymous

Anonymous Fri 21 Jan 2022 18:17:20 No.14132422 Report

Quoted By:

>>14132411
He did not use the excluded middle in his argument.

Anonymous

Anonymous Fri 21 Jan 2022 18:19:26 No.14132431 Report

Quoted By:

>>14115445
radius times 3

Anonymous

Anonymous Fri 21 Jan 2022 18:23:36 No.14132443 Report

Quoted By: >>14132460

>>14132363
You haven't transcribed the basic logic correctly. That P(1_n gets arbitrarily close to 1 && 0_n gets arbitrarily close to 0) is true does not imply that P(1_n gets arbitrarily close to 0_n) is also true.

Anonymous

Anonymous Fri 21 Jan 2022 18:27:41 No.14132460 Report

Quoted By: >>14132477

>>14132443
What the hell is "P(1_n gets arbitrarily close to 1 && 0_n gets arbitrarily close to 0)" and "P(1_n gets arbitrarily close to 0_n)"? Is that capital P supposed to be probability?

Anonymous

Anonymous Fri 21 Jan 2022 18:30:53 No.14132477 Report

Quoted By: >>14132536

>>14132460
I'm using the formalism of the post I replied to. I assume his capital P identifies a proposition.

Anonymous

Anonymous Fri 21 Jan 2022 18:45:57 No.14132536 Report

Quoted By: >>14132668 >>14136167

>>14132477
You're not. In >>14132363, P(x) is an unary predicate about sets, in this case P(x) ? x is a Cauchier sequence.
{a_n} is equal either to (1,1,1,..), or to (0,0,0,...), thus ¬P({a_n}) implies ¬P((1,1,1,..)) or ¬P((0,0,0,...)) and this is inevitable. There doesn't exist a property of sets which would be satisfied by both (1,1,1,..) and (0,0,0,...), but not by {a_n}.

Anonymous

Anonymous Fri 21 Jan 2022 18:47:29 No.14132542 Report

Quoted By:

>>14120673
>you haven't seen something therefore it doesn't exist
literally kill yourself

Anonymous

Anonymous Fri 21 Jan 2022 19:14:53 No.14132668 Report

Quoted By: >>14132744

>>14132536
>P(x) is an unary predicate about sets
So as I said, you are not transcribing the logic correctly. Cauchier is clearly not a logic of first order.

Anonymous

Anonymous Fri 21 Jan 2022 19:32:46 No.14132744 Report

Quoted By: >>14132901

>>14132668
ZFC is a first order theory though. So at least we can agree that your remedy requires a different framework. Can you describe it?

Anonymous

Anonymous Fri 21 Jan 2022 20:07:20 No.14132901 Report

Quoted By: >>14133057

>>14132744
Why did you add the C? Did I not describe it well? What is wrong with my description?

Anonymous

Anonymous Fri 21 Jan 2022 20:53:46 No.14133057 Report

Quoted By: >>14136167 >>14136455

>>14132901
Not really. It's not clear what formal system are you trying to define Cauchier-ness in (if any, maybe this is all on a purely intuitive level), because it's not ZF(C). And it's not clear what is Cauchier-ness a property of, because it's not a property of sequences.

Anonymous

Anonymous Fri 21 Jan 2022 20:58:07 No.14133076 Report

Quoted By: >>14133108

>>14115381
>But who is this heretic?
no one. wildberger's methods are valid, but they don't invalidate the math he doesn't use.

Anonymous

Anonymous Fri 21 Jan 2022 21:06:49 No.14133108 Report

Quoted By:

>>14133076
They do, actually.

Anonymous

Anonymous Sat 22 Jan 2022 03:28:46 No.14134122 Report

Quoted By: >>14134841

>>14115366
he doesn't play devil's advocate he legitimately thinks that everybody else is retarded because he can't understand the concept of infinity.

Anonymous

Anonymous Sat 22 Jan 2022 10:28:10 No.14134841 Report

Quoted By:

>>14134122
He understands the concept better than you.

Anonymous

Anonymous Sat 22 Jan 2022 19:22:48 No.14136167 Report

Quoted By: >>14136455

>>14133057
>because it's not a property of sequences.
It's only not a property of sequences if we are treating the definition of a_n as a separate object from the sequence a_n because not being Cauchier is clearly a property of a_n. As you say here >>14132536
>{a_n} is equal either to (1,1,1,..), or to (0,0,0,...)
Even if we say that the sequence a_n contains only one of these sequences, the definition a_n contains both sequences and they do not get arbitrarily close to each other. Perhaps it would be more precise to treat Cauchier-ness as a property of sequence definitions rather than sequences qua sequence?

Anonymous

Anonymous Sat 22 Jan 2022 20:54:14 No.14136455 Report

Quoted By:

>>14133057
>>14136167
Upon reflection, if we are treating a_n as containing only one of the two sequences contained by its definition, then I don't see how the existence of the definition a_n can even be said to impinge on the well-definition of R since R is defined by the equivalence classes of sequences not by the definitions of sequences and so sequence a_n is then unambiguously either in the equivalence class of (0,0,0,...) or in that of (1,1,1,...). So the Cauchy requirement for sequences is fine as is and we do not even need to worry about the Cauchier requirement for definitions of sequences. "Problem" "solved."

Anonymous

View Same Google iqdb SauceNAO NJW.png, 324KiB, 882x889

Anonymous Sat 22 Jan 2022 21:23:31 No.14136566 Report

Quoted By:

>>14115353
He triggers /sci/ssies so he's based.

Anonymous

Anonymous Sun 23 Jan 2022 02:28:28 No.14137422 Report

Quoted By: >>14137439

Is Wildberger anti-vax?

Anonymous

Anonymous Sun 23 Jan 2022 02:35:18 No.14137439 Report

Quoted By: >>14139458

>>14137422
bad framing, the only thing that matters is anti bullying and pro bullying

Anonymous

Anonymous Sun 23 Jan 2022 17:29:44 No.14139458 Report

Quoted By:

>>14137439
>/pol/tard whining about bullying
That's rich

Anonymous

Anonymous Sun 23 Jan 2022 20:08:49 No.14139893 Report

Quoted By:

>>14115353
think you. for keeping to post him. i started watching him.

Capcode	All Only User Posts Only Moderator Posts Only Admin Posts Only Developer Posts
Show Posts	All Only With Images Only Without Images
Deleted Posts	All Only Deleted Posts Only Non-Deleted Posts
Ghost Posts	All Only Ghost Posts Only Non-Ghost Posts
Post Type	All Only Sticky Threads Only Opening Posts Only Reply Posts
Results	All Grouped By Threads
Order	Latest Posts First Oldest Posts First

Your latest searches