It's official, set theory BTFO

go read a real book on set theory :
set theory book T (N. Bourbaki)

It's true that the foundational aspect is pretty hard at the beginning, but it's done, once and for all (as opposed to what he just said in the video) and afterward natural numbers are derived and all the "usual" arithmetic properties are recovered via PROOFS.

On another note, mathematics is the deduction of truths from a set of hypothesis so of course there is an axiomatic foundation for sets. Oh, and by the way, "relating to common usage" was what led the mathematics into contradiction walls in the XIXth. What he explains informaly is the theory of types, which is a set theory with "data types" and there is a full encoding of both theories into each others if you add a big cardinal axiom...
I admit that his dichotomy is the foundation of mathematics (which is "coded" in ZFC as existence of "set containing things" AND "set containing nothing")

>all mathematics is based on set theory

>set theory is the foundation of discrete math
>discrete math is the foundation of computer science

>all mathematics is based on computer science

It's simpler, more logical, and more intuitive.

>ok let's use another foundation for arithmetic because set theory complicates it too much
>to do this just assumed a priori the existence of lists, ordered sets, multisets and sets

ok bub

You're not assuming their existence a priori. You're defining what they are, and then coming up with examples.

THIS

>muh stick to reality

I don't agree with everything he says, but I'm really glad he's talking about it and trying to come up ideas from his own perspective, even if, he ends up arriving at the same conclusions, sometimes... without realizing it. It's much better than the herd like mentality of people like this

Who dismiss attempts at originality and prefer to mindlessly parrot whatever is en vogue.

Also, he really puts a lot of effort into contextualizing mathematical concepts in historical terms, which again, I enjoy. I think it's intellectually dishonest to assume (and often to want it to be true, regardless of contradicting evidence) that mathematics is created in a vacuum.

He has a video of him debating James Franklin on the existence of infinity. It's good to hear people argue from different points of view, see where they overlap, or disagree, allowing a third perspective.

Also, he's really not an ideologue, despite much of Veeky Forums's attempt to make him out to be one. He's clearly a calm person attempting to take an unfashionable point of view in order to get people thinking. If you want a hostile ideologue, you should see the kind of things that John Gabriel posts. He is blatantly hostile and despite making some points that I agree with (I like some of the points he makes about dedekind cuts and intervals), is mostly just an asshole to everyone on the internet.

>fails logic

>what is 78143?
It's just {78142} union 78142.
Is he bothered that nobody wants to write down thousands of curly braces? It's certainly possible to write the definition of this number down.

[eqn]78,143 \,=\, [\![0\,;\, 78,142]\!][/eqn]
Finite amount of work, perfect rigor, Wildburger BTFO at his own game.
>implying [math]\left[ ||||| ||||| ||||| ||||| ||||| ||||| \right][/math] is better than [math][\![0 \,;\, 29]\!][/math]

>convoluted definition

you must hold a PhD in modern mathematics.

fuck off with this crackpot

Spoken like a true CS autist.

so do ordinals exist in normanland?
does any notion of infinity exist?
also, how is his foundation any better than set theory?
does he believe in the incompleteness theorems?

all the butthurt ITT

...

He thinks we should all be searching for a suitable finite alternative to all concepts built on infinities, despite having no evidence that one exists. A truly noble effort to waste man-hours on a goose chase.

Why the FUCK can't finitistcucks just accept potential infinity into their lives?

>yes we can prove that for n=0
>yes we can prove that for n=1
>yes we can prove that for n=2
>yes we can generally prove that for n+1 assuming it for n
>BUT YOU CAN'T SAY FOR ALL NATURAL NUMBERS BECUZ UR REASONING ABOUT INFINITE COLLECTIONS DURR
how LITERALLY RETARDED do you have to be to believe this?

> muh axiom of induction

Well said Sir. I couldn't agree more.

Indeed, it was the shit flung at Wilderberger by the various menial minded arseholes on /pol/ that first got me interested in his work.

Whenever you have a pack of arsholes flinging shit at some guy with original ideas, or promoting for discussion the original ideas of others, then you know that they may very well be onto something.

Arseholes hate originality or any form of creativity. It offends their bland and mediocre minds.

I don't deny that alternative formulations are interesting and useful to science, but his attitude + suggestion that younger people should follow in his footsteps not out of personal interest but because "the standard way can't be valid" is infuriating to me. I might learn rational trig someday out of curiosity.

O always dislike things in pairs, can be modify to noting [ ] something [ | ], everything | ; without brakets because at that point don't matter is infinite.

> green text

that's like arguing that imperial units are better than metric units

fucking pointless

Stop posting this in every thread you inbred mongrel brainlet.

what do you mean

the theme in his videos isnt that he is rebuilding mathematics, its that rigor should be involved at every level of knowing it. it seems to be about the discipline of creating scrupulous proofs

>, but his attitude + suggestion that younger people should follow in his footsteps not out of personal interest but because "the standard way can't be valid" is infuriating to me. I might learn rational trig someday out of curiosity.

>his attitude

His calm and respectful tone? Have you watched any of his videos? He is, if anything, not aggressive enough. If you want aggressive, like I mentioned earlier, read up on some of John Gabriel's posts. He literally calls people "idiots" and "morons" when they disagree with him. Wildberger is like a ladybug, offering "alternative" (which are really just thinly veiled re-labelings/orderings of mainstream thought and not actually "different", ironically enough).

>the standard way *can't* be valid

Again, you have to actually look at him when he's speaking to infer the meaning. You can't just* look at it "rationally". He says this as a hook to get people thinking. It's a rhetorical device. He's clearly not an Alex Jones type self-righteous zealot. Watch his videos on mute if you're having trouble "seeing it". He is really harmless.

>I might learn rational trig someday out of curiosity.

I think the more important lesson to take from his videos is to think for yourself, not just mindlessly copy from other people. If you really want to learn about trigonometry, try to compute sin, tan, cos by hand, read about the history of astronomy, see how they did it before digital computers, etc. Actually try to read euclid's elements in the original greek. Take the holistic approach to learning. You'll get a much richer appreciation of trigonometry than just learning one more formula, one more interpretation, etc. Learn how to build and use your own machines, not just use other people's. You'll realize after trying to make something original that you can't, but that trying to teaches you about how things are related more than just following a set of instructions quickly.

I don't have a problem with the idea of infinity, but you can't call something an

-infinite set
-infinite collection
-infinite group
-infinite _____

without deriving a contradiction. Potentially infinity is great. Hilbert's hotel is a fun concept to think about, and surprisingly easy to use to explain to other people how infinity "works". I don't have a problem with it, but terms like set, collection, group, all refer to finite things, things with boundaries, and pairing them with infinity is always going to be a contradiction. Semantics are fundamental to communication.

One more thing. If you go far enough back, you'll start realizing that geometric reasoning is much more fundamental than numerical reasoning, and really where it comes from, intuition about space and boundary definitely came before representing amounts of objects, whether they're in a pile, heap, sack, etc, in the same way that perceiving things clearly came before writing systems. Older numerals were often constructed as configurations of some unit (a tally, a dash, a mark), so the geometric representation of the amount (some unit) was more directly related to the symbolic representation (which is now more removed from it). I think a lot of the confusion and frustration people have with too much of modern math (despite the claims of many of the educators that they're trying to make it MORE accessible) comes from the inability for people to connect hindu-arabic numerals to their 'naive' geometric intuitions of quantity. 3's don't necessarily look like what 3 refers to, neither does 2, 5, 6, 7, 8, 9, etc. Smart people tend to be dismissive of this viewpoint because they take for granted their ability to see the connection between things at a lower/"intuitive" level, but when you're not smart (like me, and most people) and have to explicitly describe relations between things to understand them, you're forced to see where the disconnect in mathematical reasoning starts.

...

Can someone post more videos like this? About understanding the foundation of mathematics?

Also, what's up with the memes of this guy and reals?

Every time I watch one of his videos and he mentions a problem with modern mathematics, it's not even a real issue. It sounds like he just doesn't understand things.

He claimed that the definition of a function is a rule that you apply to inputs, and that this is not rigorous. Of course it's not rigorous, which is why it's not the definition used by anyone.

>don't have a problem with it, but terms like set, collection, group, all refer to finite things, things with boundaries, and pairing them with infinity is always going to be a contradiction.
Where the hell are you getting this idea from?

So user, what.s the biggest natural number.?

>It sounds like he just doesn't understand things.
nigga is a professor, I'm pretty sure he understands.

number of things in the universe.

Are the irrationals always defined from the rationals? It's kind of a paradox if you think about it. Do we have a "Rational-free" definition of R/Q ?

The biggest number that we can call a natural is the biggest one that has been expressed.

This guy is a joke

>no rigour

I guess you don't

What? Have you seen a construction of the reals?

I take it as an axiom: every increasing sequence of rationals converges to something, and this something works the same way as rationals do when it comes to algebraic manipulations.

every bounded increasing sequence?

The unique complete ordered field.
Proving that it exists is another matter, but rationals are the smallest field of characteristic 0, you don't get any simpler than that.

I think some of us are being dishonest about our perception. As much as we would like to think of ourselves as being capable of imagining infinity, by it's own definition, you cannot. End of story. That's the entire purpose of the word. Something that cannot be captured, something that is unending, non-terminating. Sure, you could assume infinity, and work from there, but to claim that you understand infinity? That is a standard contradiction, A & ~A. Something does and does not have a boundary. Something is and is not terminating. The definition of the word implies it cannot be captured, or given a length. Anything you can imagine is, and always will be finite, terminating, truncated, possessing a boundary. Sure, you can tell yourself that there's something beyond the horizon, but you will never, ever know for certain, because, once again, if you knew, if you could give a certain answer, you would be talking about something that is finite... or you would be god, sitting on an perceivable plane of existence, looking down on us mere mortals. Unfortunately, unlike many of the people who browse this board, I am just a mere mortal and can only perceive finite things.

And before you begin the accusations (which I get frequently for even questioning the orthodoxy) I'm not saying infinity doesn't exist, just like I don't claim god doesn't exist, because, once again, I do not know, it's infinity- it's in the definition of the term. Let's not ignore definitions.

Also, it really pisses me off that no one believes me when I ask these questions. What's worse is that I can NEVER get a straight answer after someone "realizes that I'm serious". Never. Do you know what it's like to be told to take something on faith by people who are supposed to be the defenders of supporting their claims with evidence? It's tragically infuriating. I feel like I'm the one being trolled here. Like it's a big inside joke.

un-perceivable plane*

so why is there something rather than nothing?

>Bourbaki set theory
LEL
Go read about CZF

So you define things into existence?

>evidence
Kek, this is mathematics we're talking about here

Isn't this the same guy who thinks the square root of 2 is rational?

>What's worse is that I can NEVER get a straight answer after someone "realizes that I'm serious".
That's because perception and physical existence is completely irrelevant to math. Most people are probably completely fucking confused by you not understanding that.

No, you axiom things into existence

>That's because perception and physical existence is completely irrelevant to math. Most people are probably completely fucking confused by you not understanding that.
only undergrads think this

wait is this true, R is the only field that satisfies that? What if we look at only (R/Q, +, *,

your derivations from those axioms are then a tautology.

how do you justify your axioms?

>justifying axioms
Are you retarded?

Thanks for the rhetoric, feel free to supply an actual counterargument any time brainlet.

He hates the real number system and idea of infinite sets.

He's not really a crank in the traditional sense. He's dismissed because he doesn't like some of the philosophies of math and thinks that a lot of axioms are unjustified, which is retarded since that would mean dismissing entire fields of math. (He dismisses the entirety of basic trigonometry)

Basically, he's everything that Veeky Forums likes, some person who gets triggered by semantics.

this is how you get real

> He dismisses the entirety of basic trigonometry
and supplants it with his own better version.

Now analysis on the otherhand is completely useless from the way he does it

why do you prefer using this set of axioms instead of wildberger's`?

You're whitewashing his highly political rhetoric, probably because you're contrarian. Wildburger doesn't just say "I disagree", he claims mathematicians are misleading students and spreads outright lies about foundational mathematics. I don't care if he is attempting to be "original" (he's not saying anything original at all by the way). No one is dismissing constructivists and finitists. It's entirely his rhetoric that makes him a crank, not his mathematics.

There is nothing rigorous about his arguments.

Probably because Wildberger's math is decades behind modern math, and there is no obvious path to catching up. He can do what he wants, but it's frustrating for him to undermine the importance of existing math in the minds of his students. They should make the decision on their own to be interested in his math, not be lied to that anything else is unsound.

What are Wildburger's axioms?

Better is debatable, although I read his paper and its quite neat to see how much trigonometry can be done without the usage of irrational numbers. His Achilles heel is transcendental numbers though.

He knows what he's talking about without a doubt, he just has a problem with how math is taught is all. Hence why I say he's just some dude triggered by semantics.

> His Achilles heel is transcendental numbers though
But you don't need transcendental (or algebraic non-rational) numbers for rational trig. That's the beauty of it.

You don't need rationals to do trig with only integers either. But why would you want to?

Oh I didn't mean that transcendental numbers were the problem of rational trig. I meant that transcendental numbers aren't something he's fully addressed. He's also never really touched on how rational trig fails when you bring up examples of geometric constructs that require irrational numbers.

I'm just playing Devil's Advocate here, I actually like his philosophy of math.

>He's also never really touched on how rational trig fails when you bring up examples of geometric constructs that require irrational numbers.
like what?
if you use quadrature instead of length you shouldn't need square roots.
And calculating pi takes analysis instead of regular trig.

My problem is that you can't solve a common physical problem like y'=y without transcendental functions

you can't even into reading, so I'm just going to dismiss your post

Calculating pi? That's not what he said.

How do you eat a sound?

Or constructing pi.

Also not what he said. I can construct a circle with circumference pi. I can't construct a line with length pi.

what if you cut that circle, and pull on the ends?

Every Dedekind complete field F is isomorphic to R. Note first that the prime subfield of F is isomorphic to Q. Then every element f of F is the supremum of the set of rationals in F less than f, and can be identified uniquely by the supremum of the same set in R.

> I can construct a circle
bullshit

That's not a construction.

en.wikipedia.org/wiki/Compass-and-straightedge_construction

Another perspective: every complete ordered field must contain a copy of R, the Dedekind completion of Q. Let F be a complete ordered field with an injection from R to F. Then F is a field extension of R. By the Fundamental Theorem of Algebra, C is the algebraic closure of R and so it contains every field extension of R, in particular F. Then F is isomorphic to either R or C. If it's isomorphic to C it can't be ordered, as in an ordered field the square of every element has to be positive but i^2 = -1 can't be positive. So F is isomorphic to R.

every algebraic field extension of R*
If F is a transcendental extension, it is isomorphic to R(X) and so isn't Archimedean, let alone complete.

it is unsound.
of course it's behind, because no one bothers developping proper mathematics.

>not knowing about constructible numbers, constructibility

Please get out. You don't know math just because you watched some of berger-fag's vids and took a class in shitposting.

>R is the only field that satisfies that?
I found this post which try to characterize R.
math.stackexchange.com/questions/839848/category-theoretic-description-of-the-real-numbers

IT is odd that R has no universal property categorically, beyond the stuff about R is the terminal archimedean field

constructively-predicatively, R is not really a set, but sheaf topos, or rather a locale whose generalized elements are indeed real numbers.
ncatlab.org/nlab/show/real numbers object
here is how a theory of *one* real number is expressed
books.google.fr/books?id=LkDUKMv3yp0C&pg=PA367&lpg=PA367&dq=archimedean constructive mathematics&source=bl&ots=at40ELOwgl&sig=SdDpMvxOX2VHtaCJSBvG5qR4zA0&hl=en&sa=X&redir_esc=y#v=onepage&q=archimedean constructive mathematics&f=false

Retard here, is |powerset(O)| > |O| ?

there is also the definition of closed unit interval [0,1] thru coalgebras

ncatlab.org/nlab/show/coalgebra of the real interval

In an earlier posting I showed how to define co-inductively the closed
interval, in particular I showed that its elements are named by
sequences of 0s and 1s with the usual binary-expansion equivalence
relation. There is a well-known computational problem with this
approach, already in the definition of the midpoint operation: at what
point can you determine the first digit of the midpoint of .0000...
and .1111...?

At the Como meeting I learned from Andrej Bauer about a better
approach. Take the elements of [-1,1] to be named by infinite
sequences of _signed_ binary digits, that is -1, 0, +1.

[Just to confuse matters, Scedrov and I once used signed _ternary_
digits (n Cats and Allegators for the "Freyd curve"). The signed binary
expansions .+1 -1 and .0 +1 describe the same number, to wit, 1/4.]

Using signed binary expansions one can compute midpoints with a little
3-state machine that takes as input the sequence of pairs of signed
binary digits of the given numbers x and y, and produces as output
a sequence of signed binary digits for the midpoint x|y. (There may,
indeed, be momentary delays in the output, but there will not be an
indefinite delay -- indeed, the number of output digits will never be
more than one less than the number of pairs of input digits).

The challenge is to revise the co-induction so that it is this better
version that emerges.

That didn't display right, O should be the empty set symbol

en.wikipedia.org/wiki/Constructible_number#Geometric_definitions
Cool show me how to construct (pi, 0). Teach me something.