Logic - 2017-12-10 (page 1 of 3)

3:02 AM

@LeakyNun What element of that link are you referring to?

user131753

4:15 AM

@DavidReed: I have granted you access to the Philosophy of Mathematics room. Feel free to ask any question regarding Philosophy of Mathematics there.

David Reed

@user170039 Yes, thank you. Sry I didn't respond, I have a splitting headache.

@user170039 I very much appreciated your invitation

user131753

@DavidReed: No problem, although I did ping you there, since I was not sure where it would reach you, I though that I will let you know it.

Secret

4:49 AM

in Mathematics, 30 secs ago, by Secret

Random art ramble:
1. Art is a language of expression, which the audience are free to interpret however they like. It is the only known language without a syntax
2. Mathematics is a form of art
3. Logic is a foundation of mathematics
--------------------------------------------
Yet no syntaxless logic exists

David Reed

@Secret depends how strictly you want to interpret the word "syntax". The study of informal fallacies in logic I would consider to be largely devoid of syntax

Secret

i don't know, but every concept that we as a species try to express is often written in terms of a natural language (e.g. english) and that has syntax in the form of grammar and things like nouns should be placed after a verb.

Art is quite different. You basically have some object, some painting, or even just a dance, no notion of language needed, though people tend to come up related interpretations due to cultural norms and typical ways of what concepts associate to what experience, but even that varies from person to person

David Reed

5:05 AM

@Secret body language is syntaxless as well. I like that you're thinking abstractly though. Most people go through their whole lives without every having a single original thought--or even thinking for themselves in general.

user21820

@DuncanRamage You are the one that is rude. I already stated clearly that induction cannot be eliminated in a technical sense, but your response was that you're not convinced it's important. That shows sheer unwillingness to learn. You also have the sheer cheek to call others bad at presenting mathematics. Enough. You're not welcome here.

→ 7 messages moved to trash

David Reed

What question was this about?

user21820

@DavidReed That user posted this answer which is clearly wrong in that exact sense I stated, as any real logician will agree.

I ignored his ignorance, but after LeakyNun commented on his answer he came here to spew nonsense. So I stopped ignoring.

Secret

Hmm interesting, guess I am too focused on thinking about sign language and completely forgot body language in general. But yeah I like the abstract. Other than it is one of the most effective ways to access ideas that won't be as limited by our prior intuition and experience (one reason why mathematics is so good at coming up with ideas very removed from our experience, but then later feed back into our society
(take the invention of the computer and the internet as an example, it all starts with thinking about problem solving machines abstractly)), it, like category theory, highlights wha…

user21820

@Secret Mathematics is not an art. It is a cultural construct.

David Reed

5:15 AM

I consider mathematics to be a subfield of logic

One cannot say that mathematics is not an art

as what is and is not art is a completely subjective concept

user21820

Well it is subjective.

David Reed

I DEFINITELY AND WHOLLY OBJECT to mathematics ever being characterized as a science

user21820

But I was saying what I thought.

Just like you and Secret are.

And I don't see anyone except you having used the word "science", so what are you talking about?

David Reed

I understand. I probably was a little too indelicate in what I said. My head is splitting apart from a headache right now

user21820

@DavidReed You really need more sleep. It's bed time right?

Secret

5:18 AM

lol, then we differ from that point of view, cause I have no problem treating mathematics as the fourth pillar of science, especially now that we have computers to help on generating samples and library search through and classify a lot of algebraic structure, thus it gains a lab component

user21820

David probably has a different notion of "science" than you or I.

And we probably differ as well.

Secret

mhmm

David Reed

The fundamental pillar of science is the experiment

user21820

@DavidReed I disagree.

It is reproducibility of an empirically stable phenomenon.

David Reed

I refuse to call anything that does not involve experimentation and utilization of the scientific process science

user21820

5:19 AM

Experiments are no use if you don't believe it can be reproduced.

And also no use if you don't believe statistical significance.

Secret

e.g. my typical way of learning mathematics is very similar to chemistry: I generate a bunch of mathematical objects to be used, then mix them together and see what happens, and tweak the axioms on the fly and restart

David Reed

What is "empirically stable phenomenon"

Secret

This is one of the uncountably many reasons why my logic sucks, because I am not a very rule abiding mindset

David Reed

I certainly won't argue that mathematics is a critical component of science

user21820

You have to believe that there is some kind of order in the world, namely laws that govern everything.

Without that science is useless.

Because consider that it's entirely possible that you were created just a few seconds ago and implanted with memories making you believe that you have been alive for years.

You reject that (rightly) because you assume the simplest possible explanation via laws not fiat.

David Reed

5:22 AM

"Reproducability" is the word scientists use when describing the results of an experiment

user21820

Yes I know, but you have to believe that it means something.

Otherwise there is no point of experiments.

Just consider the above hypothesis. You cannot scientifically reject it.

But you can philosophically reject it, and that same philosophy actually is the underlying justification for (empirical) science.

David Reed

Its a thought experiment called the "brain in a box". I'm familiar with it

I don't reject it

user21820

If you don't reject it, then you will have trouble justifying science itself.

David Reed

I don't reject it as being impossible

user21820

You also will not be able to reject solipsism.

David Reed

5:26 AM

I view it as unlikely

user21820

Okay viewing it as extremely unlikely is still... acceptable, philosophically speaking.

But then you have even less reason to believe that mathematics is not a science, because the only reason for the axioms of PA is that they are empirically true at human scales.

There is nothing that logically demands PA to be meaningful whatsoever.

I should say I mean "discrete mathematics", not "modern mathematics".

David Reed

I think it may be better, in terms of me understanding what you're trying to get across, if you told me why you believe mathematics to be a science.

user21820

I believe the choice to investigate first-order Peano Arithmetic is based on its empirical validity.

So everything that stems from that (which give large areas of discrete mathematics) can be counted as a scientific endeavour.

I do not believe that modern mathematics (especially anything that invokes impredicative definitions of ZFC) is largely a science.

Is it clearer now?

David Reed

Yes

but I don't consider the methods used to investigate it scientific

user21820

Hmm.. you mean by proofs.

Secret

5:34 AM

Experimental mathematics

Experimental mathematics is an approach to mathematics in which computation is used to investigate mathematical objects and identify properties and patterns. It has been defined as "that branch of mathematics that concerns itself ultimately with the codification and transmission of insights within the mathematical community through the use of experimental (in either the Galilean, Baconian, Aristotelian or Kantian sense) exploration of conjectures and more informal beliefs and a careful analysis of the data acquired in this pursuit." As expressed by Paul Halmos: "Mathematics is not a deductive science...

David Reed

Interesting

Secret

My favorite method to explore maths (both on paper and in computers)

user21820

Well the stuff that Secret links to is one of the empirical ways to explore PA.

For example, we know that there is good empirical evidence for the Goldbach conjecture.

David Reed

one moment while I read

user21820

And also for certain other strange claims in number theory.

Secret

5:36 AM

Experimental mathematics also helped classify a library of topological and algebraic structures, as well analysing proofs

It's the branch of mathematics where machines are very good at

user21820

@DavidReed: I'll just highlight one prominent example of empirically found facts, that is on the list given in that article: Kepler's conjecture. Empirically conjectured, and there was a point where the proof was deemed probably correct by the experts but nobody dared to say they were absolutely sure, and then only recently with the help of computer proof assistants they finally verified the gigantic proof.

David Reed

I whole-heartedly endorse the view that experimental mathematics is science

user21820

So it really depends on your intended scope of mathematics.

David Reed

@Secret Nice finding

I really wish this headache would go away

user21820

@DavidReed Do you take paracetamol?

David Reed

5:40 AM

I took ibuprofen

user21820

I see.

Ibuprofen is indeed stronger.

David Reed

I take Norco that has paracetamol in it though

but I can't take it now, as it would render my last week of suffering a waste

user21820

If you need to rest, please go ahead. We could continue this interesting discussion next time.

(I sometimes get a headache when thinking too much mathematics. Then I take a nap.)

David Reed

It's likely a result of the 36 hrs I went without sleep followed by my excessive sleeping today

user21820

I see.

Secret

5:44 AM

If I think about too much mathematics, I usually dreamt about mathematics that night

user21820

Anyway another example is the 4-colour theorem, which I presume was conjectured because someone tried a lot of planar maps and couldn't find one that wasn't 4-colourable. The actual statement itself is a purely mathematical one, but wouldn't have been discovered without experiment.

And such examples are probably undiscoverable by blindly generating valid proofs.

Simply because the proof is too long.

Secret

I do wish other fields of science can learn something from mathematics though, such as the ability to come up concepts from scratch

user21820

@Secret I just wish that they would make all their assumptions clear.

Secret

If mathematics can learn the experimental side of other other 3 pillars of science, other pillars should also learn from maths on make all their assumptions and postulate in their models clear, and having some of their models generates something not inspired by observation, before then it is found to describe nature

I am a much worse physicists because there are too many assumptions not made clear (as well my personality of hate to take assumptions)

More generally, my thinking is inherently higher dimensional, I have extreme trouble seeing things from lower dimensional perspectives without sufficient training

(actually, 'extreme' sounds too strong, it's not that worse)

The tendency to avoid assumptions whenever possible is one reason when I learn new maths, I like to stick to first principle proofs, and as a reuslt become very bad at using theorems, as every theorem needs the input to fullfill some condition before it can be applied

Duncan Ramage

6:01 AM

@user21820 I'm sorry if we got off on the wrong foot; I am interested in why induction cannot be eliminated, I just tend to bristle at strong declarative statements.

David Reed

He can definitely be pretty blunt

user21820

@DuncanRamage It can be proven via proof theory, which is a branch of (mathematical) logic. This kind of fact is not so easy to understand, much less explain, which is why I did not provide any details in my comment.

If you are interested to know more, you can read the following posts first. Give me a while to dig them up.

Duncan Ramage

@user21820 Thanks. Could you give me a reference?

David Reed

That's my favorite part about him though.

user21820

@DuncanRamage Firstly, here is a conversation with another user called 123 about that, if I recall correctly.

Nov 14 at 10:09, by 123

It's not that I am complete blank slate about logic, I have attended some classes about logic, though they were very basic.@user21820

Click the link and read the following exchange.

It is a layman-level explanation of a proof that induction cannot be eliminated from a basic fact about odd and even natural numbers. Once you fully understand that fact, then it is much easier to grasp the phenomenon for more general things such as claims about arbitrary size matrices.

Because ultimately your claim is going to look like ( For every n ∈ N and A ∈ Matrix(n,n) we have ... ) and to prove it you either need external induction over n or you need an internal induction to justify the general form of entries in A.

David Reed

6:17 AM

@user21820 actually induction can be eliminated

Secret

is surprised

user21820

@DavidReed Prove it. It's impossible, so I'd like to see you try.

And anyway, before you start trying, what is clear is that the method in the question requires induction.

David Reed

by the halting problem, applied to Donald trump, I have created a model that can reason about programs and therefore my television is actually an alien from venus

user21820

@DavidReed What?? Please be serious for now, since you're otherwise going to simply confuse Duncan.

Duncan Ramage

@user21820 I can handle jokes, don't worry

David Reed

6:20 AM

That's true I suppose

Let me see if I can provide the context.

user21820

In particular, the friend is absolutely correct in stating:

> My friend says this is not a rigorous proof and that I have to use induction to prove det(A) = t^n + Sum { a[k] t^k : k from 1 to n−1 }.

This is what I was referring to as an internal induction.

Because you would be doing a sequence of row reductions and have to prove by induction that they result in the final column being as claimed.

:41646159 Just post the URL to the chat message. To get it, click the blue down-arrow and right-click "permalink" and copy URL.

David Reed

how do i do that?

nvm

yesterday, by user21820

@DavidReed If you interpret that as saying that the gods may not answer at all, then by unsolvability of the halting problem relative to the gods you will find that you cannot solve the puzzle at all either!!!

I give up

I'm a little loopy. Wound up taking the Norco actually. Please excuse the occasional silly outburst

user21820

@DuncanRamage: If you have questions about the stuff I explained to 123 (who, I realize, is the asker of that very question), please clarify. I think your point about possibility of t=0 is good and worth keeping, so please amend your answer because the claim that the solution given in the question is perfectly rigorous is incorrect and will mislead others.

@DavidReed I'm not aware that paracetamol causes loopiness?

David Reed

Norco is a mixture of hydrocodone and paracetamol

user21820

Oh.

Duncan Ramage

6:29 AM

@user21820 Give me a day to mill it over. I'm terribly busy at the moment and I'd like to give it a close reading.

user21820

Okay sure.

@DavidReed This just means that you cannot figure out whether the god will answer eventually or not, even if you were as powerful as them.

David Reed

I know what it means, I just thought the whole conversation was hilarious

user21820

Lol!

So I presume you know about relativization?

David Reed

I was basically trying to emulate you with what I said a few seconds ago

user21820

Har har.. too bad what I said is actually formalizable and provable, whereas I can't make head or tail of yours. =)

David Reed

6:34 AM

sry comp briefly died

user21820

Nobody was the wiser... until you said it. =P

Oh you remind me, have you scanned your computer for malware?

David Reed

Yes

nothing found

user21820

Okay. If it's not malware, it could be a RAM issue causing your random glitches.

David Reed

I briefly recall relativization from when I went through complexity a lifetime ago

user21820

I see you must have taken a couple of computability courses.

David Reed

6:36 AM

not in school no

user21820

Oh? Interesting then. Highly motivated.

Haha..

David Reed

But during the aforementioned rough period I went through I spent roughly $30,000 on approx. 200 txtbooks on all different fields of academia

user21820

I see, that's a lot to read.

David Reed

When I told you my kindle account was a pantheon of knowledge I was being quite serious

Yes, unfortunately when it comes to memory, what you don't lose you use

Every year I lose more and more of it

user21820

Wait isn't it the other way around?

What you use you don't lose?

David Reed

6:39 AM

yes

yes typo/norco

user21820

Do you know enough to comment on whether 'regularization' in physics is actually sound or wishful-thinking?

David Reed

there is nothing about quantum mechanics that is anywhere near sound

user21820

I don't know enough physics, and it's a headache (as Secret put it, too many assumptions not made clear) to try to understand what they are doing. But I got the von Neumann feeling:

David Reed

The stupidest possible approach to quantum theory one can take is to view it as being an accurate reflection of reality

user21820

> With four parameters I can fit an elephant, and with five I can make him wiggle his trunk. − Attributed to von Neumann by Enrico Fermi, as quoted by Freeman Dyson in "A meeting with Enrico Fermi" in Nature 427 (22 January 2004) p. 297

David Reed

6:41 AM

I take it you are familiar with Richard Feynman

Arguably one of the greatest contributors to quantum theory and physics in general

user21820

I'm not too familiar. I only have 'popular-science' knowledge of quantum mechanics in anything beyond the basic Schrodinger equation and solutions to basic problems.

At least there I know all the assumptions that go into it.

David Reed

Many believe him to be the 2nd greatest physicist of the 20th century (Einstein first)

anyways heres an anecdotal story

Secret

you really should take a visit to h bar some day, cause a couple of quantum people of SE resides there

David Reed

One of my dad's friends is a theoretical chemist that was a student at Caltech while Feynman was a professor there. One night he ran into him at a bar, and said "I'm taking quantum mechanics, and although I think I understand the math behind it something about it just doesn't make sense to me"

user21820

And he said you don't understand things but just get used to them?

David Reed

6:47 AM

Feynman responded "Don't worry, I can promise you, it doesn't make since to your professor either. You know how I know that? Because I've spent my entire life studying it and it doesn't make sense to me either"

user21820

Oh.. different quote.

David Reed

this isn't a famous quote, this is from a private conversation he had with a friend of my dad's

Secret

quantum mechanics has well established mathematics framework, but quantum field theory is another story

David Reed

They've settled for finding something that accurately describes the outcomes of experiments

Secret

it is perhaps one of the most mathematical of the physics theories we have besides general relativity, with few assumptions

David Reed

6:50 AM

To try and read more into it than that is to try and venture beyond the realm of human intelligence

user21820

So do you believe the Copenhagen (wavefunction collapse) interpretation?

David Reed

This is the genesis and thought process behind quantum mechanics:

Secret

I am more of the objective collapse camp

user21820

I don't believe any kind of 'observational collapse' makes sense.

Secret

well, you can take humans out of the picture, and replace it with some detector, same results will occur

user21820

6:52 AM

@Secret Yes, but I'm trying to say I don't believe 'collapse' even makes sense.

David Reed

Young's double split experiment- Light demonstrates wave-like properties. Light must be a wave,

Is there a way to press enter to go to a new line without having it send?

user21820

Shift-Enter

David Reed

ty

Secret

I am also quite fond of another interpretation that is kinda opposite: Epistemics, where the wavefunction is just a summary of what we know about the quantum system

and thus quantum mechanics becomes something like doing statistics on noncommutative probabilities

user21820

@DavidReed This does not actually follow, because nothing prevents a global controller from controlling particles to behave statistically as if governed by wave mechanics.

And if we take the Schrodinger equation seriously, well the wavefunction is omnipresent, so this notion is not so far-fetched.

Secret

6:57 AM

To work out which quantum mechanics interpretation you are in, the first step is to ask yourself whether you believe the wavefunction is a physical entity or just a construct for summarising information. The next step is to ask yourself about what do you think it happens at the moment of measurement

So from the above, you are clearly not cophenhagen nor objective collapse models

that leaves a lot of other choices, including many worlds interpretation, epistemics, shut up and calculate, qbism and others

user21820

There is Bohmian mechanics or pilot wave theory, that has recently gained more attention:

Pilot Wave Theory and Quantum Realism | Space Time | PBS Digital Studios

►

quantamagazine.org/…

I'm not saying I buy it, but it's one of the other choices.

What I can say is that we can safely rule out MWI, as more ridiculous than solipsism and 1-minute-ago-creationism.

Because what governs the rules for each world and the interactions between worlds? Still an overarching framework, so we can't ever dismiss the fact that there is still a single reality at the most fundamental level.

@Secret I currently believe that it is reasonable to treat the wavefunction as an actual entity, even as the particle itself, or at least as the controller of the particle. At the point of interaction (including measurement), the many particles involved (in proximity) interact in a continuous manner (no collapse) resulting in drastically different resulting wavefunctions.

David Reed

2: Black-body radiation: 2 different laws exist to describe it: Both are extremely accurate for a small interval of wave-lengths and diverge greatly from experimental observations outside of the interval. Max Planck invents a regression constant, h, to
make the data fit an equation he uses to describe it

2. Photoelectric effect: Wave model of light makes predictions inconsistent with observed reality. Einstein uses h to introduce the notion of a photon. This view accurately predicts photoelectric effect. Light must therefore be both a wave and a particle

Secret

ah yes, I almost forgot bohmian mechanics. They are fine for non relativistic quantum mechanics, but they still need to work on how to include relativistic scale stuff in it.

I also like Bohmian mechanics because the wavefunction is nonlocal, which makes it easy to explain entanglement

user21820

My current view seems to be able to account for the various experiments, including the quantum eraser and things like that. It is similar but distinct from pilot-wave theory.

The standard no-go theorems such as Bell's inequality will not apply to my view because it's inherently non-local.

In short: Particle = Wave-function.

Secret

My view is similar to yours in that the wavefunction is physical, but differences include the wavefunction is neither particle nor wave entity and measurement is govern by some interaction (but in my view, I sorta cop-out and be agnostic about that we don't know what it is).
I have never thought about the requirement for the interaction to be continuous, so that's an interesting thought and fits quite well with how physics treats interactions (since rarely singularities arises in most real life phenomenon)

user21820

7:11 AM

@DavidReed Your point 5 is slightly sarcastic, right? =) But so far empirically it works, and the continuous differentiability is a crucial assumption that lets us solve the equation.

David Reed

yes its sarcastic

user21820

Somehow that continuous differentiability assumption picks out a single family of solutions from infinitely many families, and the one it picks seems correct.

David Reed

the whole series of thought processes that led to quantum mechanics is so absurd

user21820

At least we are aware it's an assumption, which is what I was referring to by "I know what assumptions go into it.".

Secret

well, you need something that can interfere like waves, yet does not change in intensity, the only way to do that is to use complex numbers

David Reed

7:13 AM

If you ever go through the process of finding the wave function for a system you'd be shaking your head

After you find it, you literally integrate it over all space and then divide by it's value so it normalizes to 1

user21820

@DavidReed I did. It's quite amazing actually when you see the quantum levels match up with the spectra lines (with slight discrepancies of course).

And then I say, forget about anything more complicated than one ideal neutral hydrogen atom.

David Reed

For me, I view it as a model that is useful for predicting the outcomes of observable events and don't bother reading anything more into it than that

You can't solve for anything but the hydrogen atom

user21820

But the collapse interpretations don't even allow you to predict with certainty in the sense of a logical deduction.

David Reed

The other electron's ruin your ability to specify a potential

user21820

Yeap. It actually ties in with my view that every particle is a wave-function that spans the universe, since it means it gets exponentially harder to analyze interactions of multiple particles.

David Reed

7:21 AM

@Secret Your "cop out...we don't know what it is" parallels my "beyond the realm of human intelligence"

LastIronStar

Is anyone here familiar with Category Theory?

David Reed

@Secret I enjoy talking to you btw. Just thought I'd mention

Isn't there a cat theory chat?

I'm familiar, though its been awhile

Secret

@user21820 We computational chemists need to routinely deal with systems more complicated than a hydrogen atom all the time, which is why quantum computing will be really useful for us

LastIronStar

well there's no-one there

Secret

another group that will massively benefit from quantum computing are the condensed matter physicists (which has huge overlap with material science nowadays)

David Reed

7:24 AM

Was a joke

Sry I can see how it would come across passive agressive

The rest of society will suffer tremendously as RSA will be rendered obsolete

They're out now though, 8 qubits atm i believe

user21820

@LastIronStar You could try inviting people from the main chat-room there, as well as David, since he says he's familiar. I'm not because I don't quite see much benefit for me to learn it right now.

@DavidReed Lol.

Secret

I think the latest from (forgot company) have passed the 50 qubit mark. During the development, various groups have been slowly rolling out quantum cryptography, thus we won't all be crashed withotu RSA I think

user21820

Needs thousands of qubits to break 4096-bit RSA, right?

LastIronStar

ok thanks

David Reed

I'm not sure.

Soleil

7:28 AM

@LastIronStar Yes, I have few notions about categories

But I guess the McLane should be the reference here.

LastIronStar

ok cool, i'm not sure how to invite to the cat theory chat room, here's the link to it -> chat.stackexchange.com/rooms/68360/category-theory

user21820

@LastIronStar Click the user and then "invite this user ..."

Linking works too.

LastIronStar

thanks for the heads up, next time

user21820

@DavidReed: It would be thoroughly amusing if someone discovers some physical barrier to going beyond a certain number of qubits.

I would laugh.

David Reed

In November 2011 researchers factorized 143 using 4 qubits.[81]

@secret are you familiar with what they're doing with graphene in computing?

Secret

7:32 AM

@DavidReed Not really, I don't follow much of the quanutm computing news in journal paper depth

David Reed

I found it quite amazing that the nobel prize was awarded to the two people that first isolated it using scotch tape

I just thought since you were a chemist

They're using it to make transistors that are substantially more efficient than current silicone ones

Secret

I am familar with some news on the electronic properties of graphene, including better and better methods of manufacturing large sheets of it, its various magnetic and electronic properties, use in water splitting, but otherwise, I am not very familar with much else (or perhaps given the amount of information I need to track from many domains of knowledge, I don't recall what I remember without a suitable memory jot)

I have MASSIVE procrastination problems reading all sorts of things from the four pillars of science, to arts, to politics, to some traditional cuture, gossips and so on

not to mention extra brain juice is used to track the ideas in all my dreams, which I had (nearly) every single night

If I were running a company, I would be running an idea repository of the human spcies (almost)

David Reed

@user21820 You should familiarize yourself with category theory. A large push for remformulization comes from the problems they're having with statements from cat theory not being first-order expressible

That being said, it's so abstract one could almost call it mental masturbation

user21820

@Secret Yea you haven't gotten around to solidifying your grasp of logic.

Secret

indeed, so little time to summarise all knowledge of humanity :P

David Reed

7:39 AM

Diagram chasing is useful though

user21820

@DavidReed If people so like higher-order stuff, just go to higher-order logic, Henkin style. Doing that to ZFC gives MK for second-order, which some set theorists prefer over ZFC itself.

David Reed

I would not feel comfortable even going to 2nd order

user21820

Do you know about ACA?

It's a predicative second-order extension of PA.

Very concrete in that sense.

12

A: Are sets and symbols the building blocks of mathematics?

The things you actually write on the paper or some other medium are not definable as any kind of mathematical objects. Mathematical structures can at most be used to model (or approximate) the real world structures. For example we might say that we can have strings of symbols of arbitrary length,...

Got to go for a while.

David Reed

ok

Secret

My current thoughts about the foundation of mathematics: No matter how much I like the weirdness of infinite sets and actual infinities in general, they are probably an artifact of ZF

and thus a practical foundation will not have actual infinities built in

David Reed

7:43 AM

The difference between countable and uncountable sets is outrageously important

Secret

As for potential infinity, I suspect we still need it, though I need to finish reading both forallx and ACA to figure out whether it is really the case

David Reed

Whether you want to actually think about them being different sizes of infinity, or densities doesn't really matter. You can just think about it strictly from the definition in terms of bijections

Secret

@DavidReed I had some discussion with user21820 and Leaky about this in the past months, it seems there is no way to make the notion of uncountable cardinality predicative

We also come to a consensus that $\omega_1$ may not be even constructive

The best we can do is we have an algorithm to prove there is a surjection, but we cannot prove whether there will be an injection predicatively, and we need that injective map to establish the different levels of infinity

David Reed

What I mean is that there are really important theorems that distinguish between the two

Secret

hmm?

David Reed

7:48 AM

maybe I'm not following what you mean by predicative

Secret

in Mathworks (Not the main chat!), Oct 14 at 15:47, by user21820

@Secret Typically predicative mathematics assumes as given a fixed collection that satisfies the axioms for naturals. And then everything else is built predicatively on top, meaning that you don't assume anything for granted until you have justified its existence.

You build things using only what is given in a fixed collection (such as the natural numbers and PA)

David Reed

Then he's asserting that you can't build the reals

which is nonsense

LastIronStar

what's the difference between computability and constructibility?

Secret

well, we cannot build all the reals, but we can build the computable (to some finite number of oracle) subset of it. We still trying to work out how much of real analysis we can preserve

In ZF, there are uncountably many reals, but we only have a countable language, thus uncountably many of them cannot even be wrote down and use in proofs

David Reed

I should probably mention that 21820's views on this subject differ remarkably from 99.9% of the mathematical community

Construction of the real numbers

In mathematics, there are several ways of defining the real number system as an ordered field. The synthetic approach gives a list of axioms for the real numbers as a complete ordered field. Under the usual axioms of set theory, one can show that these axioms are categorical, in the sense that there is a model for the axioms, and any two such models are isomorphic. Any one of these models must be explicitly constructed, and most of these models are built using the basic properties of the rational number system as an ordered field. == Synthetic approach == The synthetic approach axiomatica...

Secret

7:53 AM

btw, you mentioned there are theorems that can distinguish betwen countable and uncountable, what are these cause I am always struggling to figure out the essence/minimal criteria of uncountability?

David Reed

I'm fond of the Dedekind construction myself because it better settles the question of whether there are holes in the number line

LastIronStar

but what if we don't need all those uncountably many reals to prove things

David Reed

@LastIronStar sry computability generally refers to computable functions. Construction can refer to virtually anything

LastIronStar

i asked because normal computers cannot work with real numbers while quantum computers in theory can

David Reed

Heres one: A function is Riemann integrable if and only if it has at most countably many discontinuties

Another one: Any countable set has measure zero

it pops up all the time

Topology: A topological space is separable if it has a countably dense subspace

@LastIronStar Interesting. I didn't know that.

Transcript for

♦ $Mathematics$ Logic

Transcript for

♦ Logic

♦ $Mathematics$ Logic