Logic - 2017-12-10 (page 2 of 3)

« first day (480 days earlier) ← previous day next day → last day (2337 days later) »

8:02 AM

Here's a big one: You need the fact that the rationals are countable to prove the heine-borel theorem, without which virtually all of real analysis is impossible

at least for anything more than one variable

Yeah, user21820 has some possible counters on that, he then showed me a MO question (which I can link here if you want), but most importantly, that caused me to recently spent some time meddling with Lebesgue measure, and thus showing the countable set result

1. A function is Riemann integrable if and only if it has at most countably many discontinuties
->That might be sidestepable by using other notion of integration by changing the measure, or redefine continuity on the new topology of the n-computable reals

So whatever uncountability is, will probably be tied to measure theory somehow

@DavidReed Your claim does not follow.

yes you define the lebesgue integral

ACA can do essentially all real analysis that pertains to the real world.

i.w.c. uncountability is still important

8:06 AM

No click the link to see what I objected to.

You said that I claimed that you can't build the reals.

That has to be clarified before it makes sense, but on the surface interpretation it is not so.

If you're saying you can't create an uncountable set I don't see how you could say that doesn't also imply you can't build the reals

Did I say you can't create an uncountable set?

Don't forget that what Secret says I say may not be what I say. =)

eh?

I got something wrong?

@Secret I didn't read carefully, but you didn't say that either.

17 mins ago, by 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

That's how I interpreted that statement

8:08 AM

@DavidReed He was paraphrasing and incredibly compressing several hours of conversation into one comment.

It's understandably vague.

Since you want to know my viewpoint, I might as well say it myself.

I would like to hear your thoughts on transfinite induction at some point though

I'll adopt a form of type theory. Let func(S,T) denote the type of functions from S to T. This will contain every object that you can justify to be a function that maps any input of type S to an output of type T.

Then we can prove that there is no surjection (usual meaning) from S onto func(S,bool).

Via the standard diagonal argument.

This I fully accept.

What I don't accept is the follow-up claims that you can prove in ZFC, namely that there is no injection from func(S,bool) into S.

And the informal interpretation of this fact as stating that the size of func(S,bool) is strictly greater than that of S.

Following so far?

@DavidReed I'll get to that later. Simple stuff first.

no

What is your definition of the Boolean type

It consists of exactly two objects, called true and false.

What you expect it to have, basically.

Ok

8:15 AM

Which point needs elaboration?

I haven't looked at types since I did computational complexity

with lambda calculus stuff

It would be better to formulate what you're saying strictly in FOL fo rme

Hmm... it's better not to, since type theory is not tied to any particular logic, and may not even be compatible with some. Simply take a type S as accepting some objects, rejecting some others, and not answering on the rest.

We start by assuming that we already have the type of natural numbers with the standard properties.

The best way to present this would be to tell me why you cannot prove in ZFC, that no bijection exists between $\mathbb{R}$ and $\mathbb{N}$

@DavidReed But you can in ZFC prove that. So what's your point?

And what I said earlier implies that I accept that there is no surjection from N onto func(N,bool).

where N is the type of naturals.

So to be clear then you don't reject the existence of an uncountable set?

8:23 AM

Depends on what you mean by uncountable. That's why I did not use that term just yet.

39 mins ago, by 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

Cardinality is a completely different matter.

There is no surjection from N onto func(N,bool). Is that sufficient for your idea of uncountable?

However, I claim that it cannot be well justified that there is no injection from func(N,bool) into N.

My idea of an uncountable set is one in which there is no bijection between it and the naturals

And what is "bijection". If it's an injection that is also a surjection, then my statement above should have sufficed for you.

If I'm following you, you view that as being distinct from it having strictly greater cardinality then the naturals

8:27 AM

Sorry got to go again.

np

Continue later!

Look forward to discussing this later

@Secret What I meant is the only proofs of the heine-borel theorem I have seen rely first on reducing to a countable cover before getting a finite one

hi @user21820 @DavidReed

hey man

820 just departed

8:37 AM

talking about cardinality?

i honestly don't know what we're talking about

since you are apparently a party to what this discussion has been maybe you can clarify

am I?

54 mins ago, by 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

in my past week, I have not went through in detail of the Heine Borel theorem, despite have do the exercise of proving nested interval lemma and the compactness of closed real intervals, I should look at it again

Also another possible reason for my inaccurate translation is because ever since user21820 explained his type model to me, I semiconsciously shift my definition of uncountable set to "something that does not inject into the naturals" hence the confusion

You don't need it for those theorems

You don't need it for calculus in one variable. For more than one variable, you need it quite frequently

actually you'd need it for compactness of closed real intervals

but you don't need the notion of compactness for calculus in one variable

8:44 AM

well @Secret I don't recall much; what did I say about predicativity?

@LeakyNun What resulted was a long period of me believing that 820 was trying to prove to me something that he was not

Namingly that one cannot find a bijection between the reals and naturals

well

that's unfortunate

@LeakyNun I cannot find any hits in mathworks, probably I mixed you all up

or rather that one cannot show that such a bijection doesn't exist

1 hour ago, by 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

1 hour ago, by David Reed

The difference between countable and uncountable sets is outrageously important

Is this where the trouble started?

8:48 AM

yes

he's right

you're also right

look up Skolem's paradox

I remember this

countable model of a theory that expresses uncountable sets

I said this though, in terms of its appliations:

Since the discussion with user21820, and also after my exposure to infinite dedekind finite sets, I have a very narrow definition of uncountability.

We don't call dedekind cardinality uncountable because they are not even comparable to the aleph cardinals, similarly, (if I recall correctly) the surjection map notion in user21820 type theory I don't call it uncountable either, because the surjection map only give a notion which term is more computationally complex than the other, and not a notion of size

1 hour ago, by 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

so that refines my understanding of the minimal characteristic of an uncountable set being dictated by an injection

8:52 AM

That is, one can ignore the meaning of them entirely, from a theorem proving perspective all that matters is the definition

but tbh, I don't know what uncountability will mean beyond set theory

I figured that after you said you were a chemist. What I aimed to inform you is that the distinction matters in mathematics enormously because there are theorems that apply to one but not the other

I suspect the Lebesgue measure argument you raised (countable set have zero lebesgue measure) may be one of the important justification on why we need axioms to ensure uncountable objects exists

@DavidReed so what you said and what he said are not incompatible

It was my interpretation of secret's translation of 820's previous statements that caused the confusion

8:59 AM

translation of a translation plus distortion along the way

ugh

o btw, turns out defining the notion of finite in ZF without using any set theory is extremely hard (for me)

David Reed: uncountability is very useful in mathematics, e.g. in measure theory and topology
Secret: uncountability is a construct of ZF(C)

51 mins ago, by user21820

What I don't accept is the follow-up claims that you can prove in ZFC, namely that there is no injection from func(S,bool) into S.

The notion of uncountability predates ZFC

ugh

9:03 AM

We can get rid of $\omega_1$ easily and most of the maths will be intact (and thus most topology theorems will lack counterexamples as a result), but I strongly doublt we can do measure theory or even probabiltiy measure without uncountability

I've had it with these discussions that go nowhere

lol

@Secret we only need the beth numbers

but to justify beth, we need powerset, which is impredicative (in both 820's sense and also the sense of predicativity of the predicative maths community)

oh god

9:05 AM

some elaboration: Predicativity in predicative mathematics community means the definition is not circular

@DavidReed

everyone here knows that the reals are uncountable

40 mins ago, by user21820

There is no surjection from N onto func(N,bool). Is that sufficient for your idea of uncountable?

39 mins ago, by user21820

However, I claim that it cannot be well justified that there is no injection from func(N,bool) into N.

I was fixing to add to that but decided to remove it

this is his viewpoint; everything else is strawman

Yeah, that is why, whenever I said the word "reals", I mean the reals we all understood. Any alternate defintion of reals I add descriptors e.g. n-computable reals

I'm discussing secret's viewpoint now, as to what elements of math require the notion of uncountability

9:07 AM

Footnote: It might seems to you guys I try so hard to get rid of infinity, but my motivation is the opposite: I am trying to slap mathematics very hard so it can tell me they cannot be removed and exactly why

@DavidReed You mention uncountability predates ZFC, do you mean it is already there before cantor times?

Simple answer: You lose calculus

no

cantor predates ZFC

ooops

uncountability originates with cantor

ZFC comes later

I see, so it arises with the diagonal argument

that was one of his proofs yes

9:11 AM

one thing that ponders me, though, is it possible to have cantor theorem outside set theory, and in particular without powersets?

Even in 820's theory only one direction of it (surjection) can be recovered

but we need both directions to get bijective mappings hence uncountability

I now understand exactly what he was saying earlier with bool

@DavidReed bool is a finite type lol

it's the easiest of all

@user21820 he meant the powerset of the naturals

$2^\mathbb{N}$

(hoefully I have not mistranslate this time, my fascination of infinity can often distort my translation without me noticing...)

(especially having been exposed to things like amorphous, infinite dedekind finite, bersteid, vitali etc. sets in the span of just one month, my mind can get a bit wacky with all these infinities buzzing around)

Infinite dedekind finite and other quasiinfinite sets are kinda fun. They behave like onions which shrink indefintely in size as you peel away part of it (Dedekind cardinality decreases as you take away elements from the set)

@user21820 is this what you mean? : ZFC cannot prove that there does not exist an injection $\mathbb{N} \to \mathbb{2^\mathbb{N}}$

9:20 AM

Btw here's what I had the other day when I tried to define finite in the same way in set theory but without using set theory

It's just a ping that he will be able to read later without having to go through the whole discussion

(NB I use --- as conversaton separator so that people won't mistaken one block of my message is related to another)

in Rambles, Dec 6 at 16:54, by Secret

1. There is a maximum and a minimum
2. The maximum is a successor of some number
3. Every number is a successor of some number, except the minimum
4. There are only successors and the minimum, and between any two successor there are no numbers
5. No number can have a sequence of predecessor (if any) that continues indefinitely

very nice!

I am not sure if I cut off all nonstandard naturals, though, they are very hard to chop away

I spent 2 nights figuring out how to rule out something like this, before I come up with point no. 5

{0,1,2,3,4,...,...,5,6,7,8,9}

(where ... means it increases indefinitely and decreases indefinitely respectively)

The challenge is that I cannot use something like "every subset has a minimum" because that implies membership which is a feature of set theory

I also try to do away bijective mappings completely as otherwise we will need to fist definte a section of naturals as finite since finite in ZF means it bijects with a section of the naturals, which sounds too circular

I think you did quite well

9:26 AM

what's interesting about this experiment is that I found it is much easier to come up infinite numbers than finite ones

I also owe my thanks to a bunch of people that lead me to this:
1. user21820 for introducing me to predicative mathematics and thus make me aware of the highly nonconstructive and impredicative nature of $\omega_1$ and uncountable well orderings
2. Asaf Karagila and Noah Schweber in MSE for telling me that even in set theory $\omega_1$ being neither projective and/nor borel under some models
3. Alessandro for some discussion on the use of $\omega_1$
4. Leaky for telling me that I can think of $\omega_1$ as an unreachable ordinal from below

All of these help me to identify on what is essential in capturing the properties of the natural numbers, what potential and actual infinities are and the above 5 rules came up (which should be axiomisable under some foundation system)

(the list is subjected to refinement as my understanding of logic becomes better)

9:43 AM

I'm back. Nice to see a discussion going, but yea better not assume I said something unless you quote me directly. =)

@user21820 let φ : func(N,bool) → N be an injection. Construct a function f : N → bool as follows: given n : N, if φ(g)=n for some g, then f(n) = ¬g(n); otherwise, f(n) = 0.

@Secret In my opinion Cantor's diagonalization argument merely shows the surjection part.

Then, let φ(f) = m. f(m) = ¬f(m) by construction of f.

The injection part is simply an artifact of classical set theory, namely the idea that for every set S there is a powerset that decidably splits the universe and can recognize whether any object is a subset of S or not.

yeah, and that is precisely how the proof by contradiction step is used, which is not allowed in predicative mathematics if I understood correctly

9:45 AM

@DavidReed No ZFC can prove (I purposely emphasized that "can").

@user21820 reread that lol

28 mins ago, by David Reed

@user21820 is this what you mean? : ZFC cannot prove that there does not exist an injection $\mathbb{N} \to \mathbb{2^\mathbb{N}}$

@DavidReed you reversed the arrow

Ugh.

Let me quote myself. It's clearer.

2 hours ago, by user21820

What I don't accept is the follow-up claims that you can prove in ZFC, namely that there is no injection from func(S,bool) into S.

Just take a look at the proof in ZFC.

@DavidReed what he does not accept is not the claim that "you can prove xxx in ZFC", but the claim (that you can prove in ZFC) that xxx is true

it took me a second to parse that

but that isn't an excuse for strawman :P

Crap.

You are right; I was ambiguous.

Wait.

According to standard English grammar, my sentence was uniquely parseable.

@user21820 I didn't say it isn't

I just said it's harder to parse

9:48 AM

I know; for a moment I thought it wasn't.

@user21820 now what's wrong with my proof?

5 mins ago, by Leaky Nun

@user21820 let φ : func(N,bool) → N be an injection. Construct a function f : N → bool as follows: given n : N, if φ(g)=n for some g, then f(n) = ¬g(n); otherwise, f(n) = 0.

@Secret $\omega_1$ is nothing more than the set of all countable ordinals

its the bool part that's throwing me, are you trying to refer to powersets when you say that

@LeakyNun The if-then-otherwise part.

$\varepsilon_0$ is the set of all ordinals that you can reach from $\{0,1,\omega\}$ by exponentiation, multiplication, and addition

$\omega_1^{CK}$ is the set of all recursive ordinals

@user21820 why is it wrong?

@LeakyNun Warning: In predicative set theory, there is only the ordinal class $[0,\omega_1^{CK})$, there is no set of all countable ordinals

thus $\omega_1$ is not a set in predicative mathematics

9:51 AM

@LeakyNun We got to decide whether we want to talk predicatively or not. In ZFC your sentence is true (assuming canonical ordinals).

Otherwise you do have the type of countable well-orderings, but don't have a single uncountable well-ordering.

Let me get back to this later.

@LeakyNun A simple motivational argument on that is you can take any sequence of computable increasing functions on the computable ordinals (those that have recursive ordinal notation), it will always be computable, thus you can never reach, let alone exceed $\omega_1^{CK}$ so unless you know the existence of $\omega_1^{CK}$ via other means, there is no way to know from below that there is an uncomputable ordinal

In fact, this issue arises as early as $\omega$ (as no increasing (insert suitable something) function on the finite ordinal will give something that is not a finite ordinal), but $\omega$ survives because the default model of ZF has the axiom of infinity

Wait wait I don't think LeakyNun or David would know what you're talking about now, and let's not go there yet.

ok

@LeakyNun The issue here is that you need to know that you can determine whether "φ(g) = n for some g ∈ func(N,bool)". Depending on your personal philosophical view, this may or may not be justifiable.

If you take a strict constructive view, then this is probably not justifiable, since you are now (you hope) constructing a member of func(N,bool) and ought not to be able to ask questions about the whole type func(N,bool).

hmm

9:58 AM

Going this way leads to strictly predicative type theory, which is PA at 1st-order and ACA at 2nd-order and can easily be extended to higher-order.

More precisely, you can construct an object as long as its defining expression only quantifies over objects of lower order.

I personally don't know whether it's necessary to be so strict.

Though what we can be sure of is that predicative higher-order arithmetic can do practically all real-world mathematics.

So if you are an unimaginative realist then this is pretty much all you ever need. =P

This claim I'm making can be justified by the results from the field of Reverse Mathematics, that even mere ACA suffices for lots of real analysis.

@user21820 what's ACA an acryonym for?

Arithmetic-Comprehension-Axiom, which is actually just one axiom schema of that system.

Reverse mathematics

Reverse mathematics is a program in mathematical logic that seeks to determine which axioms are required to prove theorems of mathematics. Its defining method can briefly be described as "going backwards from the theorems to the axioms", in contrast to the ordinary mathematical practice of deriving theorems from axioms. It can be conceptualized as sculpting out necessary conditions from sufficient ones. The reverse mathematics program was foreshadowed by results in set theory such as the classical theorem that the axiom of choice and Zorn's lemma are equivalent over ZF set theory. The goal of reverse...

And as linked from my post on sets and building blocks, there is a nice introductory article on Reverse Mathematics.

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,...

And for me, the main reason I entered this foundation of mathematics investigation is not about reformulating mathematics, but merely tried to understand all possible notions of infinity so as to understand the true nature of infinity

My post gives philosophical justification for believing that ACA is reasonable as a foundations. Of course, it's of great interest to attempt to justify further extensions, even to impredicative systems. But I stopped at ACA because I am quite confident in its justification.

But predicativity interested me because my mindset is not built for proof by contradictions and I like to be able to have a math object on my hands when I do proofs

10:05 AM

@Secret Ironic, isn't it? You asked about all kinds of infinity, but I kept telling you I don't believe half of them exist, and the others I believe split into a myriad of different things that ZFC doesn't realize are distinct.

@DavidReed @LeakyNun: Suppose we take a less strict view, namely that the function types built from naturals are somehow there already.

And so it is acceptable to quantify over them in defining new objects.

@user21820 yeah, but regardless of my fascination, the truth is the most important, and my belief by interrogating all systems enough, I will be able to figure out the origin of every concept including how to justify them, which is a lot more important than being able to play with infinities. it is one reason why I have stronger doubts about uncountable cardinality than before I entered predicative mathematics

@LeakyNun <− Then your construction here is valid.

However, it does not generalize to arbitrary type S.

2 hours ago, by user21820

What I don't accept is the follow-up claims that you can prove in ZFC, namely that there is no injection from func(S,bool) into S.

In particular there is a very very interesting type for which we expect this not to hold.

This is interesting, I hadn't heard of this project

Namely, if S = Obj, the universal type containing every object.

@DavidReed what project?

10:11 AM

It doesn't mention whether you get heine-borel for $\mathbb{R^n}$ with aca or n>1 though

for*

reverse mathematics project

(cont.) ... For me, the most important question to my personal aim in the foundation of mathematics is the following: What is the minimal requirement for a foundation to capture all mathematics that are not circular in some notion, and if infinities does not exists, which of these and in the most detail breakdown possible, describe how they ceased to exist

@user21820 I haven't seen anything more beautiful than ordinals and big numbers, in all seriousness

I am ok to have something shown to not exists, or exist, or know that I cannot prove that it exists or does not exist. The only thing I am not ok with is the inability to show any of the above possibilities

in Simply Beautiful Art's realm of calculus and analysis, 9 hours ago, by Leaky Nun

@SimplyBeautifulArt I mean, groups, topology, cohomology, algebra, they're all cool

in Simply Beautiful Art's realm of calculus and analysis, 9 hours ago, by Leaky Nun

but nothing compares to ordinals

@LeakyNun Lol!

10:13 AM

in Simply Beautiful Art's realm of calculus and analysis, 9 hours ago, by Leaky Nun

it's the most beautiful thing I've ever encountered

they're really beautiful

Then you can go play with Buchholz hydra.

And don't forget infinite dedekind finite sets, they are literally onions the ROUNDEST thing in existence

(these will be gone when uncountability ceased to exist though, sadly, but truth always comes first, regardless)

> By definition, the truth cannot lie

Buchholz hydra can't be killed by well-orderings that can be proven to exist by the impredicative system called Π[1,1]-CA...

Anyway I haven't finished what I was saying.

Let id = ( func(obj,bool) x ↦ x ). Then it appears that id is an injection from func(obj,bool) into obj.

Random comment: It would be scary to think if the universal type does not inject into itself

But by the earlier result we know that there is no surjection from obj onto func(obj,bool).

The only objection against this is that the universal type does not exist.

But so far nobody has given any cogent defense of such an objection.

10:20 AM

how do I invite a user to chat that's not currently in chat

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

Oh.

If the user is not in any chat-room, then you can't invite them.

You can only comment on their post to link them to the room.

NB: Asaf has very bad experience in chat rooms in the past thus he never uses chat

what if they have reputation 6

@Secret Actually he does drop in once in a while, but not in this kind of chat-room. Only in administrative-type of rooms. You can see from his chat history.

@DavidReed Then you've to ask a moderator to grant them permission.

The interface says owners can, but I've tried before and it doesn't work and moderators tell me that I actually can't.

I'm able to create private chatrooms for people, but cannot do so if their rep is too low?

10:23 AM

chat need at least 20 rep

Yes. It's a bad design.

well SE chat is not the main focus of SE, it is the main that is the main focus

@Secret Too bad for main.

Anyway, the standard 'reason' (that most non-logicians give) for there to be no universal type is, guess what? Because Cantor 'proved' that the powerset of any set is bigger than itself.

But clearly, that 'reason' depends on something like Z set theory.

@user21820 is it?

there is no universal type because it would have to belong to itself

Yes. Everyone does that. Including mathematics professors in my university.

@LeakyNun And what exactly is wrong with this?

10:27 AM

@user21820 because the whole premise on type theory is to avoid Russell's paradox by having nothing belonging to itself

I actually don't see the problem with just the universal type to belong to itself, but for other types to belong to itself, it can generate a circular loop very quickly

And that's why I said before that even Russell's type theory does not satisfy me.

Russell's type theory is necessarily stratified, because it is based on classical logic.

(btw russel paradox does not necessary prevent me from going against it, because I like everything that is weird)

If on the other hand you switch to my kind of view, where types may not decidably split the universe, then Russell's paradox goes away naturally.

let T be the type of types that do not belong to themselves. Does T belong to itself?

10:29 AM

->it cannot proven that T does/does not belong to itself QED

Let T = { S : S ∈ type such that S ∉ S }.

Ok. How do I delete a chatroom I created?

@DavidReed you can't

bummer

Then T ∈ type and hence T ∈ T ≡ T ∉ T. Nothing wrong with that, since ( T ∈ T ) could be null (neither true nor false).

10:30 AM

mods can I assume?

@DavidReed Yes they can but no need to bother them, since it will auto-delete itself after a while.

Unless you detest the room's existence for some reason, then by all means tell them.

Not being able to show something is (insert truth value) is not the worst. The worst (and thus the only thing I hate about this world I live in) is the refused to be explained, which in a logical statement: You don't know that you are/are not being able to show something is (insert truth value)

@DavidReed If you don't say anything inside the room, it will be deleted after 7 days

the name of the room is "David's test chat" purpoe 'tmp"

Hahaha..

You could rename the room and description and keep it as your personal chat-room!

10:32 AM

ah ok. I thought I had read six months. I think that may be tags I was thinking of

Null does not deter me because it still explains itself as being "I cannot be explained"

Like SBA does with his "realm".

@Secret I was explaining to LeakyNun why Russell's paradox goes away.

I don't care if you cannot be explained, as long you let me know you cannot be explained

It's just infinite loop in the computability perspective.

@user21820 I know, my rambles is just brought up "memories"

Which then explains my interest in the predicative project in a short sentence: I would rather be informed that infinities cannot exists and why, than to be left in the blue and having a false hope that they exist

false hopes ARE THE WORST!

So I enjoy this journey and the discussions so far

10:35 AM

@LeakyNun @DavidReed: And in case you are wondering whether my strange kind of type theory is meaningful at all, there is a very simple model that should satisfy you, namely Turing machines or ideal programs.

The notion of infinity exists in my mind. Therefore, infinities exist

@DavidReed You joking right?

@user21820 NB I have 3 pesonal chat rooms, btw

The notion of a flying spaghetti monster the size of a meatball that controls the US president exists in my mind. Therefore... =D

Not really, the class of natural numbers exists in my mind, and it is an infinite class. Regardless of what formal theory its expressed in, I view that in and of itself to be sufficient to justify its existence

10:39 AM

@DavidReed You're only justifying the notion's existence. Not the entity itself.

Yes

However in this case the entity itself is an idea

If you treat "infinity" as just the idea then I've no problems with that.

But of course people who read your sentence will think otherwise.

For me in a predicative sense: Exists = can be justified to spend effort to define it into the system via axioms and build from the bottom

I don't know whether there is something in the physical universe that would be considered infinite

Just like if you say "The God of Judas exists." or something like that, you would be understood differently from if you said "The idea of the God of Judas exists.".

10:41 AM

@user21820 therefore a flying spaghetti monster the size of a meatball that controls the US president exists!

@DavidReed No disagreement there.

That's also why you should be happy with the model I just proposed.

@DavidReed Some say black hole singularities, other says quantum fields, jury is still out

Where obj is simply the collection of all finite binary strings.

And func(obj,bool) would be the collection of all programs that recursively split the universe.

And in general a type is simply a program that says "yes" or "no" for every input or does not halt.

but in the realm of mathematics, when somebody asks me if an infinite class exists, I interpret the entity to be the idea of an infinite class

@DavidReed Hmm you're going to run into trouble if you think that way and talk to mathematicians for too long.

10:43 AM

I use the "idea" interpretation whenever I want to build an algebraic structure from scratch (which then I don't need to worry about whether it has ties with reality)

@DavidReed: What if they ask you whether CH is true? It's a question of whether some set exists.

It's also 5 am and I took a pain pill

@DavidReed Umm I really think you should not neglect your sleep.

my ability to articulate what I'm thinking is moderately compromised

I like to think of (the ordinals in) $\varepsilon$ as ordinals of hereditary-base $\omega$ (heavily inspired by Goodstein of course): every "digit" is finite; the number of "digit"s is also finite

10:45 AM

So the predicative project will give a yes/no answer to this predicate: P: infinity as an entity = infinity as an idea

I don't want to sound like an old grandmother, but I hope each one of you gets enough sleep per day. Mathematics can wait.

@LeakyNun You mean ε0.

The problem is I can't take my sleeping medication when I take pain medication this late or I'll go apnic and die.

Hmm..

if the answer is no, then infinity only exists as an idea and I will just treat as extra building blocks I can add to my constructs that has no ties with reality

Sorry got to go again haha..

10:47 AM

@user21820 yes

however, reality is cunning and known to give more than yes/no answers

but regardless of what answer it gave me, the last answer I want to see is: it cannot be shown, and you don't know it cannot be shown

and should that happens, reality will be destroyed, lol :P

the only way reality can get away from giving me that answer is to append it with an explanation on why this is the answer

> My philosophy: Everything has to be explained, if they don't explain, then explain why they don't explain. Anything that refused to explain itself without valid grounds will be erased from existence

My dark side in a nutshell

Back to Rambles

(Actually, trash my rambles about myself, they cluttered up the flow of the messages too much)

3 hours later…

1:36 PM

Just figured out a way to get an uncountable language:

Take a square as our letter $\square$

put a vertical line in it.

Shift the vertical line uncountably many times to get uncountably many letters

In practice, one can only make some multiple of avogadro number of letters

well then that's not even an infinite number of letters :)

Are you mentioning this based off my question the other day?

nah, it a random thought popped up when a ask a MSE and someone point out to me I forgot the symbol < in my language

Ah

which then I saw the word alphabet, noting there are only 26, and then look at chinese characters, noting how much more complicated they are compared to english letters, simplifying it into a box and then the stroke idea above came up

The fact that my life philosophy is so abrasive (to require everything to be explainable) means over time I have learnt to track my stream of consciousness to some extent

Almost every emotion I felt, has an associated chain of thoughts linked to them, thus they are traceable in principle

That's interesting

2:48 PM

@Secret: If you want, I could just move your 'rambles' to your room. =)

« first day (480 days earlier) ← previous day next day → last day (2337 days later) »

Transcript for

Dec '1710

♦ $Mathematics$ Logic

This room is meant for discussion about logic, including found...

formal-systems foundations logic no-trolls

join this room
about this room

00:00

06:00

12:00

18:00

all times are UTC

$Mathematics$

site design / logo © 2024 Stack Exchange Inc; legal

mobile