/sci/ - Science & Math

In particular I do not understand how to render "T is consistent" in mathematics. It is a purely philosophical matter, and I do not know how to use math to resolve "T is consistent" into a simple "yes-no" mathematical proposition.
I don't know what it means.
You have to actually explain how the axioms apply to the mathematical structures you've defined. You can't just say
>define some mathematical structures according to your whim
that isn't math
you have to go to /lit/ with that kind of content

The only thing that's "hard" to understand in Godel's stuff is Godel coding of syntax. The rest isn't any harder to understand than the concept of an interpreter in programming.

>>15212811
"T is consistent" just means that T does not prove (x and not x) for any x

>>15212917
You aren't really helping.
https://en.wikipedia.org/wiki/Rectification_of_names
>A superior man, in regard to what he does not know, shows a cautious reserve. If names be not correct, language is not in accordance with the truth of things. If language be not in accordance with the truth of things, affairs cannot be carried on to success. When affairs cannot be carried on to success, proprieties and music do not flourish. When proprieties and music do not flourish, punishments will not be properly awarded. When punishments are not properly awarded, the people do not know how to move hand or foot. Therefore a superior man considers it necessary that the names he uses may be spoken appropriately, and also that what he speaks may be carried out appropriately. What the superior man requires is just that in his words there may be nothing incorrect.

>>15212805
Unless you understand forcing you won't understand why AoC is independent of ZF. It's not something you can pick up from a wikipedia article.

>>15212871
>you're stupid if you don't accept Gödel
https://en.wikipedia.org/wiki/Argumentum_ad_baculum
it's a fallacy
it isn't math
it isn't science
you should go to /pol/ for this content
that is the only board where your content is accepted

>>15212921
>OP's pic doesn't refer to forcing...?
>How could Gödel write a correct proof without making reference to forcing??
Are you
>le implying
that Gödel's proof is
>le WRONG
le ?!?!

>>15212927
Godel didn't prove AoC is independent of ZF. It took results from Cohen to prove it.

>>15212921
I'm not convinced that forcing is math. There are ways to use symbols incorrectly (that is to say, incorrectly as far as mathematics is concerned) to fallaciously appear to make reasonable claims that "look mathy"
So forcing is now suspicious due to its association with Gödel.
So saying
>you have to understand this suspicious theory in order to understand this other suspicious theory
while simultaneously not demonstrating that you understand this suspicious theory yourself...
Are you going to demonstrate for us that you understand forcing?
>If a stranger on the internet claims that understanding something is required for understanding something else yet does not demonstrate understanding of anything...
Do you see the problem? Without demonstrating your understanding, you sound like you're playing tricks instead of demonstrating understanding.
If you're truly playing tricks, then you have no understanding to share with us.
And you're telling lies about your own understanding.

>>15212932
You know what I mean.
You're ignoring the issue: Gödel's use of "semiintuitionistic" and "constructible" are clearly philosophical and not mathematical.
Therefore, no matter what Gödel is saying, it isn't math.
He is playing word games.
He is pulling an Alan Sokal.
https://en.wikipedia.org/wiki/Sokal_affair

>>15212946
What is the difference between philosophy and math?

>>15212932
For the record, yes I fucked up. I confused
>AC is independent of ZF
with
>Con(ZF) -> Con(ZFC)
however it makes no difference, just accept the errata where I correct what I wrote earlier in this thread
If you demand details, then I'll FORCE you to find the details in Gödel's writings, you worthless idiot. Only a moron wouldn't be able to translate Gödel's enlightened vision into reality.

>>15212948
a matter of philosophy
take such questions to /lit/ or /his/
math is subordinate to philosophy
philosophy dominates mathematics
philosophy is constantly, everlastingly, without limit getting mathematicians out of ruts, supporting mathematicians, interpreting mathematics, separating mathematics from what is not mathematics, &c.

>>15212949
Which idea of Godel's are you rejecting exactly? Be specific

>>15212955
>idea of Gödel's
you're clearly indicating that you want to discuss Gödel's ideas rather than his (purported) mathematics
take such discussion to /lit/ or /his/ where ideas are welcome and people actually care about ideas
there are people qualified to talk about ideas on those boards
you won't find such people here, only retards

>>15212955
You realize that Gödel's use of the word "constructive" is extremely bizarre, right? He is using it like a magician instead of a mathematician. He's engaging in a bit of performance magic. To say this is philosophy is a stretch. This is grammatical sophistry, word conjuring. You realize that when we're doing mathematics you don't have to reject ideas. Rejecting ideas is part of philosophy.

>>15212954
I skimmed your ramble.
So you're saying that mathematics is subordinate to philosophy. Then you say the issue is that Godel's ideas are philosophical, not mathematical.
This doesn't make a clear argument. So I'll give you another opportunity, and i'll make it more explicit so you can hopefully understand.

Question:
You say Godel uses philosophical concepts instead of mathematical. What is the distinction here? What makes a concept mathematical instead of philosophical? And why does that matter?

If you don't give me a good reply, I'm just going to leave the thread. So if you want a real discussion, take your time to think about your reply

>>15212980
you have to go to /lit/ or /his/ to discuss this
you aren't using /sci/ correctly

>>15212990
ok, have fun buddyboy :)

>>15212980
>>15212990
>>15213019
meanwhile in reality

>>15213092
But in '41 East Germany (Japan) did attack West Germany (USA)
https://en.wikipedia.org/wiki/Attack_on_Pearl_Harbor

>>15213092
There is a political theory that you should violate the Logan Act as much as possible and use legal action targeting you or your campaign as free publicity.
In other words, the right thing to do is to conspire to break the law in order to organize some effort like forcing Russians to seek political freedom inside Russia by essentially taking away the human rights of Russians outside Russia and forcing Putin to convert Russia to an election system where the Russian people regularly select their leaders like we do in the West.
There is no political freedom for Russians outside Russia, there is no economic freedom for Russians outside Russia, there is no cultural freedom for Russians outside Russia, Russians have to conspire with the West to flip Russia to a Western Liberal Democracy™.
>Either that or a bullet to your head, ruskie
So, we're freeing the Russian people, and we're murdering everyone that doesn't like it, period.

>>15212805
Godel didn't show AC was independent, he showed it, along with GCH, was consistent by defining L.

>>15213201
what axioms do you assume to obtain this
in what formal system is it a theorem
are you actually going to point to a textbook where this theorem appears or simply claim that it's a relevant theorem that was proven in 1938 and for some (unexplained) reason does not appear in modern math texts?

>>15212805
>This is Gödel's "proof" that AC is independent of ZF
It's not.
It doesn't even claim to prove that ZF plus the negation of AC is consistent.
Have you even read your own pic?

>>15212871
Just to be sure, this thread isn't about his incompleteness result

>>15212946
>>15212974
What he's refering to in the pic is
https://en.wikipedia.org/wiki/G%C3%B6del_operation

and his L, which models all of ZFC in a class smaller than what's possible in principle. He's making that case that relatively conservative operations already give you an "object" (class that's a model) that validate the set theory in question.

I'd agree that "consistency" claims are problematic in the turtles-all-the-way-down sense, but you can still choose to understand what they mean when using those words. Replace it with model existence, based on somewhat the same logic, and you got your metamathematical theorem (even if it's not strong in a metaphysically convincing way)

I expected the constructible universe would be linked in the Gödel operations article, but that doesn't seem to be the case. So here
https://en.wikipedia.org/wiki/Constructible_universe

This is his ZFC world which lives inside ZF which validates that the axiomatic relations have a realization. The consistency proofs of this sort just always say: If the axioms of a theory T would imply absurd bullshit (be inconsistent), then the objects (for which we've previously demonstrated they realize and embody all the axioms of T) would necessarily also imply something absurd. So if the object aren't inconsistent (which we assume), then the theory isn't either.
The objects in this case are some part of sets sitting inside the universe of all sets, some "small cone" of sets, L. The theory we actually reason about is that theory with the same axioms as the one we use for the model, but we exemplify those axioms again in the relative conservative L.

Again to repeat, he proved that Choice is consistent (because it's validated when using ZF to look at the subclass L of sets, and verifying that it's a ZFC model).
Proving that ZF-Choice is also a consistent theory required harder techniques and came much later. But in principle goes by a similar logic.

>The evolution of mathematical proof is subject to formalism. MY FORMALISM.
the dude is a nazi thug
these people belong in computer simulations
fucking Star Trek: The Next Generation told you what to do with Gödel
there's an episode about simulating Moriarty in the holodeck
fucking simulate Gödel in your fucking computer fucking programs like fucking ELIZA
>fuck
fuck
>fuck
fuck
https://en.wikipedia.org/wiki/ELIZA

OP here
i am a fag
e.e. cummings tier fag
i apologize
i completely fucked up the op
>WHO
GIVES
>A
FUCK
you fucking people should know that I meant
>le Con(ZF) implies le Con(ZFC)
insteadof
>le ZF is le independent from le ZFC
which I FUCKING KNEW but FUCKING FORGOT because I'M FUCKING DRUNK and I'd rather listen to HEART than read your fucking posts,
>GOOF
BALL
https://www.archive.org/download/heart_of_darkness/heart_of_darkness_1b_conrad_64kb.mp3
https://soundcloud.com/d3nd3d3u77/chosen-few-nasenbluten-rave-on-the-eye-tech-07-07-95

Where are the inference rules?
Where are the variable scoping rules?
Where is inference rule correctness verified?
Where is it proven that formula schemas are equivalent to sets of formulas?
Where are the rules for expanding formula schemas defined?
The variable scoping rules for subproofs in Fitch derivations are complex. Yet this is the simplest formalism in terms of data structures and organization that could possibly be used to encode genuine proofs.
The "proofs are sequences of formulas" canard is easily exposed when investigating the relative ease with which ordinary proofs are translated into Fitch derivations and the relative difficulty which which ordinary proofs are translated into...this weird nonsense garbage that logicians claim is proof...where is the admission that genuine proofs have complex variable scoping rules that must be adhered to in order to be a genuine proof? Where is the admission that genuine proofs are not simply sequences of formulas with no complex variable scoping rules?

The simple fact of the matter is that modern textbooks don't actually give you insight into anything Gödel did. This is really fucking shocking: modern math texts simply omit Gödel. Why is this? It's due to Gödel's absolutely fucking bizarre relationship with the print publishing industry. It's like he gave the print publishing industry cancer. It's
>ABSOLUTELY
FUCKING
>INCREDIBLE
nobody really knows what's going on
it all seems very...miasma
https://www.youtube.com/watch?v=3Yi-xkJieaA

>>15213510
modern math texts are AFRAID to bite the hand that feeds them
they are afraid to go after the "Gödel idiot" subindustry of the print publishing industry
they are afraid that using math to study what Gödel did to the print publishing industry will kill the market, and it was a real market for part of the 20the century...perhaps even most of the 20th century...and now we have a "zombie print publishing subindustry" lurking in the aether...somebody must slay this zombie...

PNAS has never been a math journal.
Ever.
https://en.wikipedia.org/wiki/Proceedings_of_the_National_Academy_of_Sciences_of_the_United_States_of_America
The fact that you think they are qualified to review math submissions is 100% pure bullshit.
Gödel's results do not appear in a math context when they are published in print.
Let that sink in.

>>15213221
Its in Jech's Set Theory you dolt. Do you even read modern texts or do you just download pdfs and think that gives you knowledge.

>>15213550
seven fucking years and you finally admit the truth
Anon, you SUCK
go
>2
hell

>>15213559
Meds now

>>15212811
>that isn't math
and what the fuck is math for you?

>>15213231
>Just to be sure, this thread isn't about his incompleteness result
OP clearly has problems understanding the concept of "you can interpret this in that and that in this, so if either is a fuck up so is the other", so "it's like an interpreter in a programming language" is the correct response. If Godel coding doesn't even figure even better, since it's the only thing that's a bit fiddly.

>>15213550
>>15213559
I've got some bad news for you guys. Jech never defines "proof". Jech is proof that everything that I've been saying about Gödel is correct. And moreover, I have a hypothesis: German math practice is to assume a higher order logic context where certain rigid and non-evolving relationships hold between language and truth. English isn't like this. In English, we have the benefit of the pragmatic philosophy of William James, Charles Pierce, John Dewey and others.
https://en.wikipedia.org/wiki/Pragmatism
In English, we benefit from a theory of truth that is flexible. Drawing political parallels is too easy; I'll refrain from doing it here. The link between the intellectual and the biological is clear: this theory of truth depends on Darwin's discoveries. With no biological theory of evolution, there can be no intellectual theory of evolving truth.

>>15212940
Godel didn't have anything to do with forcing. He showed that AC is consistent with ZFC. Cohen used forcing to show (not AC) is also consistent with ZFC

>>15214863
He outsources the notion of proof to
>The Completeness Theorem states that every consistent set of sentences has a model
the (metamathematical) proof of which you can find in any introductory model theory textbook. "Proof" in the completeness theorem (and in Godel's incompleteness and ZFC consistency theorems) refers to a formal proof system, of which there are several that work, including Hilbert Systems and Gentzen's sequent calculus

>>15214876
Please demonstrate your understanding of
>He showed that AC is consistent with ZFC.
for this stranger on the internet. Thank you!

>>15214891
I think you have to be a German poet to make sense of this. We're going to tell you that Gödel is German poetry and that you have to go to /lit/ to discuss German poetry, German idealism, German lore, &c.

>>15214912
No, you just have to understand math. This involves reading an introductory textbook and working through all the exercises.

>>15214891
What's so bizarre about this comment is that I'm looking for a definition of proof, and you've directed me to a theorem, and then (without explanation) you suggested that you've "defined proof" by saying
>it's a formal proof system
So you're trying to demonstrate your understanding by saying "proof means a formal proof system"
it's another case of rectification of names
https://en.wikipedia.org/wiki/Rectification_of_names
you're just using the pumping lemma to coin new fake "definitions" that are just neologisms without explicit definitions, i.e. the head of a definition without a body, a Cheshire Cat definition!

>>15214919
No, you're using the pumping lemma together with an abuse of explicit definitions to sell me an infinite chain of definitions that go nowhere.
https://en.wikipedia.org/wiki/Pumping_lemma_for_context-free_languages
You aren't going to demonstrate that you have a definition of proof. You're just going to insult a stranger on the internet.
>you're stupid if you don't give me a definition of proof
no, I'm ASKING you NICELY for a definition of proof and you're INSULTING me and telling me that I'm STUPID

>>15214919
What's so completely insane about this is that you're trying to mindfuck me into doing work for you, and you're insulting me and suggesting that you're going to laugh at me if I do any work at all. It's really fucked up.

Why don't you look at a mechanized proof of Godel's theorems?

>>15214863
Its in chapter 13 retard.

>>15214955
There is a problem with the content of chapter 12. It is a matter of academic fraud, in particular fraudulently claiming that the complex variable scoping rules of formal proof don't exist.
So, Jech commits academic fraud in chapter 12.
I hope this clears the matter up.
Reading past chapter 12 is pointless, fren.

>>15214955
Insulting me and telling me to keep reading when I've identified academic fraud is really fucked up.
I know this guy isn't worth reading, and you're trying to mindfuck me.
Fuck you.

>>15214977
What I'm referring to is "guard variables" in Fitch derivations. You can learn about them in pic related.

>VERIFY
THAT
>ALL
GUARD
>VARIABLE
SCOPING
>RULES
HAVE
>BEEN
FOLLOWED
>IN
>ALL
SUBPROOFS
>AND
SURROUNDING
>PROOF
CONTENT
this kills the internet Gödel troll

people schizo out over the weirdest fucking shit sometimes man

>>15215000
It's academic fraud for Jech to ignore Fitch's contribution to mathematics through the concept of formally verifying that guard variable scoping rules have been followed in all subproofs and all surrounding proof content. Jech engages in an act of verbal conjuring that destroys the value of his work "Set Theory" past page 157.
see
>>15214863

>>15214937
Work for me? The person being enlightened will be you.

>>15215151
see
>>15194007

>>15214942
>mechanized proof
you mean a non-surveyable proof
https://en.wikipedia.org/wiki/Non-surveyable_proof

>>15215324
A mechanized proof of Godel's theorems is shorter than most human-written, human-legible programs.

>>15215324
https://arxiv.org/pdf/2104.13792.pdf
Here, page 21 contains the definition of Prf.

>>15215342
So he's definitely not increasing the reputation of Computer Laboratory by suggesting that he is too chicken to go through the peer review process.
Lawrence C. Paulson is too chicken to go through the peer review process.
If he wasn't, then an editor would put this in print or approve it for electronic publishing in an online academic journal.

>>15215338
Both of
>Gödel's theorems
>mechanized proof
are snake oil.

Feels good being a CS chad. So explicit even a piece of lifeless metal can understand it.

>>15215366
>non-sentient beings are capable of understanding
>Next the Master made a visit to Kuei-shan 11 and said to him, "I have recently heard that the National Teacher Chung of Nan-ch'üan 12 maintained the doctrine that nonsentient beings expound the Dharma.
https://terebess.hu/zen/dongshan-eng.html#r

Jean-Yves Girard proves it

>>15215375
>non-sentient beings are capable of understanding
>sentient being is uncapable of understanding
The absolute state of meat machines.

see aggregate Gödel index comment in /mg/
>>15216084

>>15214930
The word "proof" as in the theorem "there is no proof of AC or its negation from ZFC" is with reference to formal derivation in a proof system. As >>15214919 says, you will find what you're looking for in any introductory logic book. If you want a specific one, I know it's in Enderton.
If you aren't comfortable with formal proof, you could always interpret Godel/Cohen as "if there is any model of ZFC at all then there is a model of ZFC+AC and a model of ZFC+-AC"

>>15216171
You aren't demonstrating understanding. You're spreading cancer.

>>15216202
Actually you don't even need to go through the trouble of pirating the book, if you want to learn about formal proof systems this PDF introduces you to a bunch of them
https://www.cs.bu.edu/faculty/kfoury/UNI-Teaching/CS512/AK_Documents/Formal_Proof_Systems_for_FOL/main.pdf

>>15216171
I'm not quite sure if you understand that this is a case of breakdown in education philosophy
We can reasonably assume now that different people who have taken different courses, both of which use, say, Jech or any other text that makes reference to Gödel's beautiful poetry MIGHT MAKE DIFFERENT SUBSTITUTIONS FOR
>le formal proof
there is absolutely no standard as your comment attests
your comment attests an OPEN OPTION where we can be definitely sure that whoever graded the homework can make arbitrary decisions on whatever basis to decide what is and is not "proof" because it's an undefined term

>>15216210
you're selling snake oil

>>15216211
ok but in the completeness theorem the word "proof" refers to a specific sort of thing that everybody agrees about

>>15216171
you're articulating two different formal systems, and you're pretending they're the same
this is the essence of "one proof parameter Gödel poetry"
you can substitute any formal proof system you want
at that point you get math
however, Gödel's "theorem" will be true for some formal proof system arguments and false for others
it isn't like it's true for all formal proof systems
it's like a whole bunch of people are pretending that they aren't doing some weird linguistic
>hey, let's try to find functions from formal systems to {0,1}
>oh, great, Gödel's theorem is one such function
>if Gödel's theorem holds in S, then f(S) = 1, otherwise f(S) = 0
you really don't get that Gödel's theorems are functions from formal systems to {0,1}
it's one parameter math
you have to substitute a concrete "formal system proof" parameter to turn Gödel poetry into math

>>15216214
gotta keep those godelbucks rolling in, baby, these libgen pdfs aren't going to pay for themselves

>>15216218
that's education philosophy
that isn't math
you have to go to /lit/ or /his/ to discuss that

>>15216220
>you're articulating two different formal systems
Which two?

>>15216223
Your arguments against it are also education philosophy and as such do not belong on /sci/

>>15216224
Introductory logic book vs. model theory
Hodges "A Shorter Model Theory" p. 37

>>15216235
Why are you referring to textbooks as formal systems

>>15216236
see
>>15216171
>le ahem
This is sort of mind-blowing. We now agree that "proof" means one of
- le intro logic text
- le model theory |- (P |- Q iff A |= P implies A |= Q for all sigma-structures A for some signature sigma)
- le consensus education philosophy (as Anon [1] says, 'the word "proof" refers to a specific sort of thing that everybody agrees about')
and now I'm sure we'll get people who tell me
>all of these are the same, I won't elaborate, it's obviously, anyone can see it, you know what I'm talking about
[1] see
>>15216218

>>15216244
nobody said that proof means any of those things though. I said the notion of proof system can be found in an introductory logic text, not that it is the introductory logic text. There is a difference between "the ketchup is in the fridge" and "the ketchup is the fridge", ya dig?

>>15212805
Ah yes. The classic "it's not nonsense if we take it as an axiom" cope

>>15216248
you're really fucking stupid, aren't you

>>15212805
A proof is a sequence of statements where each one is either an axiom or arrived at from previous statements using predefined rules of inference. Why is this hard to understand, this is all very standard?

>>15216248
you're absolutely fucking insensitive to the fact that when the word "proof" hits education philosophy, the shit hits the fan and grad math texts turn into books of poetry
like
what the fuck
Between '38 and '76 these fucking krauts turned Gödel into the so-called "four color theorem" and that is quite a mess
the krauts are playing us
I mean...who the fuck blames 'em
nobody
absolutely nobody
we won WW II
so nobody blames 'em
...anyway

>>15217338
because I keep responding to either a bot or someone pretending to be retarded? Eh, helps me sort out my own understanding of math logic

>>15217341
I think you're suffering from like some accidental Buddhist voodoo mindfuck I did on you.
It's like you've got brain damage because Jech doesn't define proof on page 157 of "Set Theory" and now you're going to be a 4chan zombie that pukes candidate definitions of proof for the end of time no matter what the topic is
like
you definitely posted that in the wrong thread
that has nothing to do with the topic of this thread

>>15217344
Gödel wasn't even german you retard

>>15217356
what's so special about page 157?

>>15217350
no
we're deciding that "proof" simultaneously
- has multiple meanings
- is irrelevant to the topic
and we're making fun of people who think proof is related to the topic by rudely abusing them as if we were
>le raving schizo

>>15217363
see
>>15214863

>>15217364
Many words have multiple meanings, and it's hard to determine what "the topic" of this word salad thread is

>>15217367
ok but why does it matter that he doesn't define proof on that page? He doesn't define "symbol" or "technique" either, does that bother you just as much?

>>15217369
yeah, but it's because Jech doesn't define "proof" that his grad math text "Set Theory" degrades from perfectly reasonable math up until p. 157 to
>le Gödellian poetry
Clearly Jech is doing some of the work of figuring out that Gödel is
>le weird
and builds a kind of "makeshift sarcophagus" to contain math Chernobyl
but he doesn't do a very good job
and you can accuse him of academic fraud, too, by neglecting to define proof in a way that omits the hard work of others and makes it difficult to relate that work to mathematics

>>15217379
Every textbook in every subject necessarily contains an undefined notion though

>>15217379
Where exactly on page 157 does it switch from reasonable math to poetry? Can you pinpoint the sentence?

>>15217373
He uses italics with the word "consistent" to signal to the reader
>this is a precise mathematical definition
however the author fails to define what "consistent" means as a matter of using mathematical terms that have already been defined and rather appears to attempt to mislead the reader as to whether or not "consistent" has a precise definition
In short, the author fails to write a page of math free of errors, so at that page, all further pages must be discarded or checked for attempting to use one of the "false terms" or terms like "proof" and "consistent" that have no ordinary mathematical definition and are in fact loanwords from philosophy used as a chuck to define new math terms

>>15217388
see pic related
circled red sentence

>>15217407
But earlier on the page he says "A language is a set of symbols", without defining what he means by symbol. Is this legitimate math or poetry?

>>15217399
he does define it though
>if there is no formal proof of contradiction from Sigma
If there were an added parenthetical to the effect of
(for a definition of formal proof, see Enderton)
would it be reasonable math instead of poetry?

>>15217426
You don't seem to understand that the author has to define a term before it is used. Do you see the problem? For every other italicized term in this book, that comes before page 157, aside from the term "set" there is an unbroken chain of definitions going back to the beginning, just like every other ordinary math text

>>15217434
You didn't answer my question though. If he does not include the definition of formal proof but directs the reader to another math textbook containing such a definition, is that okay?

>>15217426
proof isn't defined, so the author is attempting and failing to provide a definition of consistent
the author includes an erroneous, fallacious candidate definition of consistent that uses an undefined term in what would have been an explicit definition if it had been correct

>>15217439
You also didn't answer the question. "formal proof of contradiction" is genuinely defined in Enderton -- if Jech directed the reader to that definition in a footnote, I don't see why that doesn't count as math

>>15217438
>>15217444
This seems to be part of the issue.
>Suppose an edit is made
>I'm not going to tell you the edit
>I will use some suggestive language
>I will speak poetically of the edit
>I will not actually make the edit
>The edited work does not exist in reality
>The edited work is a gleam in my eye
>Please imagine I have edited the work and give me a serious reply
no
make
>the
fucking
>edit
asswipe

>>15217438
>>15217444
do you realize that you're asking for an evaluation of something, but it's really just your imaginary hallucination
you have hallucinated these edits
you haven't actually prepared a page of math to be sent to the printer

>>15217457
I told you exactly the edit though, a footnote saying "for a definition of formal proof, see Enderton".
People with learning difficulties often struggle with hypotheticals, don't worry too much about it. You have to use your imagination and visualize a world where you are either talking to Thomas Jech on 4chan or some random anon has the ability to make edits to a published textbook; then the fourth edition comes out with that specified edit (which, repeated again for your own benefit, is a footnote saying "for a definition of formal proof, see Enderton"). In this hypothetical scenario, is this textbook legitimate math or is it poetry?
Perhaps you could even draw a cartoon illustrating this scenario if your imagination is too limited

>>15217461
you know you never answered my question here >>15217423. Why is his definition of language okay if he never defines what he means by symbol?

>>15217465
Well, first of all, you can't include the last name of the author in a real math text an expect readers to know which of the author's works you're referring to.
Second of all, you're insulting me.
Third, it seems to be you who is struggling with learning if you think it is a reasonable hypothetical for a math text to attempt and fail to refer to a work this way.

>>15217423
a symbol is a set
this is a model theory shibboleth
it's literally just an alias
every set is a symbol
everything is a set
everything is a symbol

>>15217479
>a symbol is a set
and just where in Jech does he make this definition?

>>15217476
I'm not insulting you, learning difficulties are a real thing and are nothing to be ashamed of. But you clearly have one, since you think the hypothetical in this situation is that the book would fail to refer to the work in this way. The hypothetical is that it actually did. You're right, to just say "Enderton" would be a bit sloppy, the correct thing would be to say "see [39] for a definition of formal proof", with bibliography entry [39] being A Mathematical Introduction to Logic by Enderton
>inb4 "but what if it isn't entry 39"
SO, clear your head, get some coffee, wrap your mind around this hypothetical and ask yourself: is this legitimate math, or poetry?

>>15217486
so, oddly enough
I think we can just assume that every set is a symbol because symbol never gets defined anywhere, by anybody
by the same token, every set is a proof (of every theorem)
:-)

>>15217501
and to be clear, this is acceptable, but it is not acceptable when it comes to the concept of formal proof? why?

>>15217497
you have to go back

>>15217512
cope seethe dilate cringe have sex

>>15217497
Well, I am insulting you for having a learning difficulty.
Learning difficulties are just like ADHD - made up nonsense.
I'm having more fun insulting you for having a learning difficulty.
Please, try learning something.
I want to watch.
IT'S
>SO
CUTE
>!!!!

>>15217515
What do you want me to learn?

>>15217497
>I'M
NOT
>INSULTING
(((YOU)))
https://www.youtube.com/watch?v=fcrexKS8kxA

Question for the Jech skeptic itt.
On page 74, Jech makes this definition of a set having "density 0". However, he does not provide any definition in this book of what it means for the limit of a sequence of numbers to equal zero. Is this okay?

>>15217517
I want you to learn learning.

>>15217515
>>15217519
triggered

>>15217524
That's really deep

>>15217497
>find a book by Enderton
which one?
>okay it's this book, now look through the entire book for the definition
the entire book?
>yes
You're being an idiot.

>>15217543
>what is an index

>>15217358
>Gödel
Not the anon you are responding to, but yes, he was ethnically german.
https://www.newworldencyclopedia.org/entry/Kurt_G%C3%B6del
And that's a good thing.

>>15217358
whoops, green text fail here
>>15217576
Should have been
>Gödel wasn't even german you retard
Not the anon you are responding to, but yes, he was ethnically german.
https://www.newworldencyclopedia.org/entry/Kurt_G%C3%B6del
And that's a good thing.

>>15217358
>>15217580
>the jooz
>>>/pol/

>>15217630
Ah yes, the jew reveals his anti-Gödelism is rooted anti-Aryanism. A mentally ill jew is the source of all of this.