Logic - 2017-10-22 (page 2 of 2)

user21820

14:00

@Mathmore: Are you an undergraduate student now?

Mathmore

@user21820 Nah I have done graduation. (MSc)

user21820

Hmm I don't exactly recommend PhD unless one really really wants to do research.

But if you do want to, then my advice is to look for a school with professors that work in your preferred area of logic.

Mathmore

@user21820 I am enthusiastic about research.

Okay

Secret

protip: whatever you do (unless you love computers) don't do a phD involving coding hours of code

user21820

Recently I attended a talk by Slaman; he was really good.

Secret

14:03

though I guess it is unavoidable nowadays...

user21820

But my knowledge was too little and I only could follow a third of his 5-day talk.

Mathmore

Mathematical logic involves coding?

user21820

In modern logic, hardly any. My viewpoints are very computability-related because I'm from a CS background.

Slaman works in recursion theory, which studies the Turing jump operation and iterates and various related problems.

user400188

I have a book on modling semantic web servers rtight in front of me.

Mathmore

To be honest I am enthusiastic about whatever I have learnt till now. Sometimes I am attracted to Topology, sometimes to abstract algebra and now with this discussion I have started liking this subject too. Why this happens?

user21820

14:06

@user400188 That sounds like practical application to me. It is widely believed by most proof theorists that ACA suffices for nearly all practical mathematics. =)

user400188

because math is interesting

you never actauly told me what that stands for by the way

or maybe i missed it

oh wait I saw it

user21820

Note that the constructive incompleteness theorem (which we didn't discuss) shows that ACA can't prove its own consistency, which we should believe if we use it!

Secret

until we discovered an uncountable entity in our universe, I don't think we will ever need anything beyond second order arithmetic

user21820

Ok I got to go now. See you all! Feel free to post follow-up questions on the write-up.

user400188

Thank you for the class @user21820 . I appreciate it :)

Mathmore

14:09

@user21820 Thanks a lot for this discussion. Bye! :)

Raute

Thanks for letting me spy on you, guys. I'll invite myself another time. :-)

user400188

Bye everyone

Simply Beautiful Art

bye

user21820

14:34

@Raute You're all welcome to come by any time.

@SimplyBeautifulArt: Hey you come only after the class! Lol.

@user400188 Thanks for all your participation too!

Simply Beautiful Art

@user21820 lol

I was actually explaining how to use ordinals in the context of extremely large ordinals to some programmers.

user21820

@SimplyBeautifulArt Why do those programmers need them?

Simply Beautiful Art

@user21820 Someone was interested in understanding TREE(3), as per my recent answer.

user21820

Ah no wonder.

Simply Beautiful Art

They say they've got a headache now lol.

user21820

14:45

Lol.

Simply Beautiful Art

I've only explained up to zeta naught so far, and briefly flew by everything up to ϑ((Ω^ω)ω)

user21820

@Secret: My core reasoning is briefly sketched here.

Simply Beautiful Art

I hope my analysis is correct.

user21820

I think you will find it consistent with your belief that there is a unique reality and "it doesn't lie" in the sense of every sentence about reality is either true or false and not both.

@SimplyBeautifulArt Haha yea.

Simply Beautiful Art

I used to have h(n,a,b) reduce to h(n,a,b-1) + h(n,a,b-1) + ...

Now I have h(n,a,b) reduce to h(n,a,b-1) + h(n,a,b-1).

user21820

14:47

@Secret: It's also the standard way I use to explain classical logic as being meaningful for sentences about reality:

4

A: Is Godel's modified liar an illogical statement?

Your question has two main facets. The first is that you did not grasp the way logic does not fall to the liar paradoxes. The second is that there are deeper reasons as to why we have such apparently innocuous sentences in natural language that seem to defy assimilation into formal logic systems....

Simply Beautiful Art

And I believe the relationship is h_new(n,a,ω^b) ≈ h_old(n,a,b)

Secret

@user21820 My first impression on this: It is possible that L bootstrap itself to existence

user21820

@Secret No. If nothing existed at all then nothing would exist ever.

English fails because it's time-dependent, but I mean independent of time.

Secret

True, so L came to exist because of (?)

user21820

@Secret It is fallacious to assume that L came to exist at some point.

It is possible that L is its own reason.

Not in any sense of generation or causality.

Secret

14:50

But how is L's reason is L itself different from "L bootstraps itself?"

or rather, L is tautological?

user21820

The English word "bootstrap" has the connotation of "extend/improve itself".

Yes I was just going to say that we have to distinguish between facts and causes.

It is a fact that something exists.

Nothing caused it. It just is the case in our reality.

Secret

Ah, I must be reading too much about bootstrap paradox, thus I end up using bootstrap to mean something self referential

@user21820 That is a valid statement

it could be possible that ultimately, the laws of physics exists because it is a fact they do

user21820

@Secret <− And the statement you made here is also a fact that is not caused by anything.

@Secret This is possible, however it is not the simplest explanation. That is the reason for favouring the belief in existence of a conscious lawmaker.

For example, it is possible that your computer is a product of random putting together of parts.

But it is not the simplest explanation.

Secret

yup, there's the multiverse model, which is quite popular among parts of the physics circle

it has its attractive and unattractive features, thus I am kinda neutral about it

user21820

@Secret Actually if you read the comments I linked, you will see that it's a useless hypothesis.

Secret

14:58

Btw, I am ok to accept that it is a fact that A exists, but I am more happy if we can show it is the case, and not just believe it

More generally, if something has some property A (be it undecidable, true, false etc.) it is always nice we can show it, or show that we cannot did so either because it is the negation, or a limiting rule such as incompleteness theorem prevents it.

I am also ok with, known known, known unknown, and unknown unknowns, as long I am somehow aware it.

However... there exists one single concept that I detest a lot about this reality...

The only entity I hate (let's hope I actually wrote this properly because my logic kinda sucks) can be expressed as follows:

user21820

@Secret What is A?

Secret

A is any predicate, including "this is undecidable", "this is a contradiction" etc.

user21820

Wait how is it related to the current discussion?

Secret

sorry, I accidentally gone off a tangent because the discussion reminds me the thing I hate the most (which has nothing to do with the discussion). Let's get back on topic then

user21820

Hahaha..

I thought you wanted to ask how to show existence of a lawmaker and not just believe it...

Secret

15:02

I am ok to accept that we cannot show existence of a lawmaker if it is a fact that there exists some limitation we cannot show it

(argh, this is so screwy to read in english!)

user21820

It's okay I got the meaning.

The limitation is that we can't use any formal system otherwise it just pushes the issue into showing that the formal system is sound for reality.

Which of course we can't. Clearly circular.

But we can justify it philosophically.

Secret

Well, reality is probably more than any formal system can capture, otherwise we would have known all the unknown unknowns already

and unknown unknowns are precisely the exciting things that reality has

Unknown unknown is actually pretty weird if you look closely: It can have any truth value we can and cannot imagine, or no true value at all, or something more bizarre than words can capture, and we will not know until we make them known

user21820

@Secret I don't know about that, but formal systems cannot access reality even if it can capture it. For example even if you believe in an infinite world with classical mechanics, you still can't show that PA captures reality even if it does, because you need an interpretation mapping from sentences over PA to sentences about reality.

It is possible that some formal system captures reality but whatever the case we will never be able to non-circularly show that.

Secret

hence the word "probably"

though to be fair, whoever can lay their hands on such formal system will be omniscient in some sense

user21820

@Secret Why so? They still can't solve the halting problem.

Oh.

Hmm the problem is that the formal system may be such that we can't enumerate all theorems using a computer.

Never mind. Really too big can to open now.

Secret

15:10

Some more wild thoughts:

(This is sounding a bit like worldbuilding):

user21820

As for unknowns, you may be referring to quantum effects. It's a common idea that quantum mechanics has convincingly shown that the world is not deterministic, but that is not at all a valid conclusion. Note that Bell's inequality only rules out certain classes of local hidden variables, and only under the assumption that particles actually have classical quantities that we can measure.

In my opinion (not sure how many physicists share it) every particle is literally its wavefunction, not a point entity.

So Bell's inequality says nothing and it is entirely possible that everything is deterministic.

In that case there would be no sufficiently precise sentence that does not have a truth value. Here the phrase "sufficiently precise" is in fact very vague, but I can't help it.

And yet I do not believe that everything is deterministic, because we can't exclude the possibility of certain kinds of free will.

Secret

nah quantum is nothing compared to unknown unknowns. Unknown unknown is by definition, something we don't have ideas on until we discover them and make them known (e.g. can you imagine some stone age people talk about the internet)

The physics community agrees on that particles are not pointlike entities. For $\psi$-ontic interpretations, the wavefunction is a physical entity uniquely associate to any quantum state including particles. But there are also $\psi$-epistemic interpretations which said $\psi$ is just a summary of our knowledge about the quantum system

user21820

@Secret I thought neuroscientists in general believe in full physicalism?

At least they think that all thoughts are 100% a product of biochemistry?

Secret

They mostly do, but they dared not to say anything about hard consciousness, though they believe they will eventually pin it down as some brain function

user21820

Hmm.

I don't know much about what they say haha..

Secret

15:20

I am ok with either possibilities. If our consciousness is all due to biochemistry, then we will have a full understanding of our brains.

user21820

I'm also not sure how convincing it would be to you, but I personally believe that the same Occam's razor (choosing the simplest accurate explanation) points strongly towards the existence of free will.

Just considering the nature of consciousness may not be sufficient for this.

Secret

But suppose the very slim possibility that consciousness is nonphysical, then we need to start worry about when beliefs directly interact with each other as if they are individuals, it could be exciting but I am not sure (I actually have discussed about this with a philosopher in a NewScientist panal event back in August)

user21820

I believe that once you accept the existence of a conscious lawmaker, the case is more or less decided concerning many things including free will.

Because then the attributes of the physical world can be used to infer (Occam's razor) some aspects of the intended purpose.

Secret

I like to think the answer to freewill is yes freewill exists, which is why my religious belief is agnostic.

user21820

If we ignore purpose, then we may get nowhere in trying to figure out whether consciousness is purely biochemical.

Secret

15:24

And in fact, if a conscious lawmaker does exists, my primary concern is want to have a discussion with him/her/it and learn about its worldviews

user21820

@Secret Exactly.

I can't say I can prove it, but I do believe that same lawmaker has guided me in many specific ways.

Secret

My viewpoint in general is that if the supernatural exists, our primary concern is not to (insert suitable sentence), but to figure out how we can coexist and work together

user21820

Anyway for ease of reference I can quote the main points from my version of the golden rule:

in This is the Realm of Simply Beautiful Art, Jul 6 at 10:37, by user21820

> Basically, we talk about conscious beings here. Don't ask me to define "conscious"; use your 'common sense'. We define a being to be moral iff it does not intentionally cause harm to moral beings, and minimizes harm to immoral beings. This is a recursive and not obviously well-founded definition, so it gives us conclusions in some cases and is inconclusive in others.

in This is the Realm of Simply Beautiful Art, Jul 6 at 10:37, by user21820

> Now "harm" is defined slightly differently in the two contexts. "harm to moral beings" is defined to be what those moral beings perceive to be harm to themselves. "harm to immoral beings" is defined to be what the being in question perceives to be harm to them. Namely, moral beings decide what is harm to them but immoral beings don't have that right.

in This is the Realm of Simply Beautiful Art, Jul 6 at 10:37, by user21820

> I welcome any probing or critique of my ethical framework, as I apply it in real life and wouldn't want to make ethical errors. Though rather tangential, I believe to a certain extent logical reasoning helps to dissect ethical frameworks for possible inconsistencies.

Notice that nowhere does my moral code involve the lawmaker that I personally believe exists!

However, from observation of the laws over the physical world I come to the conclusion that the lawmaker himself is moral according to the above definition.

Nevertheless there are tons of immoral beings and hence the lawmaker does not always get what he wants.

Secret

My bottomost line for morality (I have other standards of course) is as long reality don't bore me, I will consider it as moral

Btw, I remember in a philosophy society I am in in uni, I once talked about what I think universal evil might be like:

It is something that suppress other worldviews (beliefs, individuals, objects etc.) to oblivion, and it will continue to do so until it is the only existence left, then it destroys nothingness and finally itself

The moral standard I used when talking about this entity is deutonological ethics, which said every individual has the right to fullfill their purpose or task, and something is considered immoral if they are prevented from completing their purpose

user21820

@Secret I don't think that makes much sense. No entity can destroy itself without relying on some more fundamental entity or laws by which it triggers and achieves that destruction.

That is the whole point of the lawmaker argument; it shows clearly that ultimately there is no valid explanation for the laws than a lawmaker, and also it shows a clear asymmetry between stability and chaos.

Do you get my point?

Secret

15:40

how about a law such that when everything is gone, it ceased to exist?

so you can had it to execute the destruction of universal evil defined above, and then it fizzle itself out of existence

user21820

@Secret That doesn't work. You need that law to be such that when everything else is gone, then ...

Secret

ah yes, the else is important

otherwise Russel Paradox horray... :P

user21820

The problem is that there is no way for a law to vanish by itself, for the reason I stated above.

Granted, it's less obvious to see why than physical entities.

Secret

so it cannot be preprogrammed with its existence some conditional expiry date or something similar?

user21820

Well think about it; existence of a law actually means existence of something that institutes and enforces the law.

In layman terms, yes food can have expiry dates, but who will throw the expired food away?

Secret

15:45

can a lawmaker destroy itself?

user21820

Ah. My answer is "no".

The universal lawmaker cannot make it such that he/she/it never existed. This is non-debatable.

The universal lawmaker also cannot destroy itself at some point. This is my answer to your question.

Secret

but why, it is the bottommost of that chain of command, surely it has something akin to root access and can do rm -rf / if it wants to?

user21820

@Secret No. Even in the computing world you know that is false. The program has to run on the hardware. That hardware is the thing that carries out the command.

Secret

Hmm... so the lawmaker is the only entity that no other entity including itself, can destroy

user21820

I think so.

That said, I think it makes sense to believe that the lawmaker can voluntarily restrict himself.

Not in an absolute way, but as a matter of choice.

That is one of the numerous reasons to believe in free will, because it seems the lawmaker did indeed restrict himself in order to permit other conscious beings to have free will.

Secret

15:50

except that it does not have the choice to destroy itself (though why would one destroy itself if it has all the power...)

user21820

Yea it doesn't have the choice to destroy itself, but if it is moral as I believe, then it wouldn't want to either.

Secret

Interesting, so metaphysically speaking (assuming there are no alien logic that is incompatible with the logic we humans used) then it is consistent for our universe to have a law that cannot be violated: Thalt shall not destroy itself

user21820

"Thalt"?

Leaky Nun

@user21820 thou

Secret

I fail at spelling

user21820

15:54

@LeakyNun I know haha..

Leaky Nun

@user21820 then it was mean of you :c

Secret

Anyway, the above will certainly rule out the existence of my notion of universal evil, but can universal evil and universal good exists at all?

user21820

But anyway that conclusion doesn't follow. Just because one being cannot destroy itself does not imply that everyone else cannot.

As you stated, a program can very well erase itself, by relying on lower-level entities.

@Secret I'm not sure what "universal" here is intended to denote, but I believe that my moral code is close to the the unique framework that maximizes stability preservation in the human world.

The universal moral code that extends beyond the human world is simply that a moral being is one that maximizes stability preservation.

Leaky Nun

@user21820 do you have an actual example of an independent statement in the theory of concatenation?

Secret

what will be considered universal good and evil under the above moral code?

user21820

15:57

Any being that attempts to perpetuate chaos would be evil under that code.

Secret

Hmm, then the devil will fall under that.

user21820

I believe the universal moral code more or less logically leads to the human-specific one when we give high priority to mental and emotional stability.

Leaky Nun

why are we talking about morality here lol

Secret

but what about universal good, I consider that to be a harder question

user21820

@Secret Aha. Yes I believe that the evidence quite clearly points to at least two opposing conscious beings, one being the lawmaker and one being a conscious being that tries to mess things up.

That's why everything at the lowest levels seem to be stability preserving, while a lot of things at slightly higher levels seem chaos inducing.

For instance the deepest physical laws seem to make it such that one needs to continuously expend energy to maintain instability.

Whereas if one expends zero energy then the entire system will settle into a (possibly new) stable state.

For the physical world one could use a rigorous definition of physical stability to make sense of what I'm saying. I'm not too sure how to pin down this vague notion for the non-physical world.

@LeakyNun You would encode the consistency of TC as a sentence about strings. If you carefully work through the section about encoding program execution as a string, you would be able to do this!

Leaky Nun

16:03

@user21820 do we have any... more real... example?

user21820

@LeakyNun It came up due to a couple of @Secret's comments in his room, and we decided to discuss here instead.

@LeakyNun Sure. Take the program that computes the well-ordering of ε[0]. TC can't prove that it is a well-ordering.

Secret

Hmm, so the lawmaker is the universal good and instability will be the universal evil under this moral code...

user21820

If you don't like ordinals, take the Hydra game. TC can't prove that the Hydra game always terminates no matter the starting tree.

@Secret Essentially. But I want to point out that its is a non-trivial assertion that the lawmaker is moral, in my viewpoint.

Secret

That means I am an immoral person, because I like a bit of both, lol

user21820

@Secret As long as the instability you like to cause does not cause harm to moral beings, it is fine.

Note that clause in my moral code that mainly requires moral beings to not intend to harm moral beings.

For example, plotting a chaotic function in a graph software is fine. So is designing a chaotic PRNG.

Secret

16:08

yeah, chaotic process is actually fun, and quite orderly

user21820

@Secret That's another interesting topic, how order always emerges despite all attempts at causing chaos.

I admit I don't understand that aspect of dynamical systems.

Secret

Well, for some systems, such as far from equlibrium system, there is simply more ways to make orderly things than randomise homogenous things, hence entropy production lead to emergent patterns (cf Belousov-Zhabotinsky chemical reaction)

and life itself, at least partly, is an entropy production machine

user21820

@Secret Wow very interesting.

Secret

And that's one of the many examples that shows entropy counts number of ways to satisfy some given constraint, and not a measure of disorder or chaos

Or in more laymen terms, entropy and the second law is capable of both what we traditionally called chaos and order

user21820

That's quite true. Even in the limit of maximum entropy (aka heat death) we have 'order', though not a very nice one.

Secret

16:14

Heat death is a bland and rather boring state, where nothing happened

user21820

Which comes full circle to one of my first few comments that I don't believe it's necessary for entropy to continually increase to have conscious life with free will.

Secret

My original idea of universal evil (which is now debunked by the lawmaker argument) is partly modelled based on the heat death, the concept of nothingness and then make it eternal. Imagine a heat death where no quantum fluctuation exists, thus no possibility to restart back into a lively state and cause another big bang

This is one example I commonly quote to my friends and tell them no matter what, don't let your future kids to get involve in bullying, because it will permanently scar your mind

My mind is scarred in that I actually really want to simultaneously destroy the world, to observe the world, and to maintain the world

user21820

@Secret You can turn your thoughts about destroying the world to good use; namely you can devise ways to avert the methods of destruction that you can foresee.

Secret

and there are neuroscience researches showed how bully victim and bullies have permanent changed in their brain structure, so... bullying is one of those things which everyone loses

They actually pop up together, and in fact, a lot of my personality traits are synergistic relations between my dark and light personalities

For example, my dark side desires to destroy and torture those who wronged me allows me to came up with very unusual monster barring techniques that allow me to explore mathematical structures in interesting ways

user21820

I wouldn't say you have a dark personality; that may be a bad way to view yourself that causes you to be unable to become completely free from the effects of the past.

Secret

16:22

Everybody has a dark and light side. what is important is how not to be consumed by the darkness and learn how to face both and (ideally) make them work together

user21820

@Secret Hmm.. this I disagree with. =)

Secret

For example, a lot of weird ideas that Leaky and the main knew about me stems from the dark side, but with the actual darkness harming stripped bare, leaving behind the neutral part

user21820

It is true that everyone does some bad things, but if you look at the real desires alone, some people's deepest desire is for a perfect world in which no immorality resides.

Secret

That's true. As for my deepest desires, it is a neutral one: To mix ideas and different thinking, generate new ideas, and not get bored

That's what I called Idea Mixing, and it can arguably be called the most powerful part of my light side, as it allows me to naturally acquire a lot of traits that the society said a good person should have

such as teamwork, communication, open mindness etc.

Or more precisely, most of my good personality traits are a byproduct of pursuing this ideal

user21820

But why stay neutral? One day you will need to make difficult choices and it would be better if you had already thought through the reasons for it.

Secret

16:28

You can thought through some of them, but there are many which you cannot anticipate until when it happens. And when it does, it is a split second decision. I'll let my intuition (and whatever left of my rationality at those points) to deal with that one

user21820

Anyway I don't wish to push you in any direction unless you are personally convinced that it is correct. And even then, it doesn't exclude the possibility of changing one's mind in the future.

Secret

That's true, as far I am aware, my worldviews have at least 4 major changes:

user21820

I personally believe that the most meaningful life is a moral one, and so I choose that path.

My viewpoints were built slowly after I left a suppressive religious group.

Because I was worried of making mistakes again.

Secret

< 2011: Scientism
2011: First time truely open up to interpersonal emotions
2014: Agnostic
2015: More artistic and intuitive thinking opens up
2016: Esoterics, and how not to get in trouble with the sciences

Secret

I see, it is good you left that

user21820

16:33

Is scientism the same as scientology?

Secret

Of course NOOOO!

user21820

Oh.

Hahahaha..

Secret

Scientism is the belief that science solves everything

Scientology is some kind of pseudoscience mix paranormal mix quark

user21820

Yeap.

Secret

(and tbh I have not read much about it, so its true nature I don't understood)

I will be able to talk more about charkras, crystals auras, buddhist spin offs, newage, quanutm mysticism and others (and I know how not to mix them with science) than scientology or satanism

user21820

16:35

@Secret There is one more curious thing I thought about when you mentioned this. The whole underlying framework of science is based on physical laws being there and remaining the same (stable world).

Secret

> The laws of physics does not change over time

Is indeed a central postulate

user21820

And though it is possible that the world is fooling us at every moment, it is empirically debunked.

Secret

The alternative is very wild, where no experiment is reproducible

(wilder than any esotericts can dreamt of)

user21820

Because it is an empirical observation that assuming that laws of physics do not change actually produces empirically verifiable theories.

So it is curious that empirical verification empirically verifies itself somewhat...

That's how I justify science in general over quackery.

Secret

Rather, if the supernatural exists, then since it is so hard to reproduce it, science can say nothing about it and thus excluded from science, so science naturally filters only the phenomenon that are stable in some sense

which is fine, because it is so successful

user21820

16:39

@Secret This is why I defer to logic instead of empirical science when discussing non-physical issues. Because the argument for one reality is sufficient to justify classical logic and hence we can use it for reality even regarding the non-physical aspects.

Secret

Since I am agnostic, I do not deny nor confirm the existence of the supernatural, but when working with scientists, unless we are having a chat on worldbuilding and philosophy, supernaturals will be out of the discussion.

The logical viewpoint does not readily came to me, partly because I am so bad at logic until in recent years when you and leaky and others trained me from the discussions

user21820

Yea for me also the default is to try to find a physical explanation.

Simply because it is almost always the case.

Secret

Anyway its 3:41 now and I need to sleep. Hopefully tomorrow, I will have a fresher mind to figure out how to code this (insert swear word) thing

user21820

@Secret Sure. I'm going off soon too. Sorry for keeping you up late! All the best with your coding and see you!

(Thanks for not swearing here.) =)

Secret

It's ok, I really love these discussions

but anyway, night

user21820

16:46

@LeakyNun: Just so that I don't forget, all the examples I gave you for sentences about strings that TC cannot prove are actually also unprovable in TC+induction.

Leaky Nun

@user21820 I see

user21820

Here induction (for binary strings) is ( P("") ∧ ∀x ( P(x) ⇒ P(x+"0") ∧ P(x+"1") ) ⇒ ∀x ( P(x) ) ).

The reason is that TC+induction is as strong as PA. TC itself is as strong as PA−. "As strong as" meaning that there is a computable translation from one to the other that preserves provability.

So if you want an even simpler one that TC cannot prove but TC+induction can, just choose some suitably hard instance of the induction axiom.

If you recall we previously discussed how to define unary.

Then ∀x ( unary(x) ⇒ ∃y ( x=y+y ∨ x=y+y+"1" ) ) cannot be proven by TC! (I think!)

I will sketch how to prove that it cannot be proven. Remember we had this model of TC that I called CBLLO (countable binary labelled linear orders)?

Oct 1 at 8:10, by user21820

A countable binary labelled linear order (CBLLO) is a relation < on a subset S of N that satisfies reflexivity and transitivity and anti-symmetry plus a binary labelling of S (namely a function from S to {0,1}). Let C be the set of all CBLLOs. Given any X,Y in C, we define X~Y iff there is an isomorphism from X to Y (namely a bijection f from X to Y such that for any a,b in X we have f(a).label = a.label and ( f(a) < f(b) iff a < b )). Then ~ is an equivalence relation on C.

Oct 1 at 8:15, by user21820

Given any G,H in C/~ define G+H = { X+Y : X/~ = G and Y/~ = H }. Then you can prove that G+H is a member of C/~. So now L is C/~ with the + we have just defined on it.

Oct 1 at 8:17, by user21820

If you're lazy to think in terms of equivalence classes (like me), then just think in terms of the original CBLLOs and show that the axioms hold if you interpret "=" as "~", and then apply the general theorem that you can mod out ~ to get a first-order model as long as + respects ~.

Oct 1 at 8:21, by user21820

(1) ∃e ∀x ( x+e = x = e+x ).
(2) ∀x,y,z ( (x+y)+z = x+(y+z) ).
(3) ∃x,y ( x≠y ∧ ¬∃u,v ( u≠e ∧ v≠e ∧ ( x=u+v ∨ y=u+v ) ) ).
(4) ∀a,b,c,d ( a+b = c+d ⇒ ∃x ( a+x=c ∧ b=x+d ∨ a=c+x ∧ x+b=d ) ).

CBLLO does not satisfy cancellation, namely the sentence "∀x,y,c ( x+c = y+c ⇒ x=y )". And I just realized that cancellation can be proven by TC+induction. So there you have another concrete example!

But CBLLO also does not satisfy ∀x ( unary(x) ⇒ ∃y ( x=y+y ∨ x=y+y+"1" ) ), because the linear order W (for "weird") on the rationals in (0,1] ⋃ {2} ⋃ [3,4) all labelled with a "1" is a counter-example.

Namely, in CBLLO we have unary(W) is true but ¬∃y ( W=y+y ∨ W=y+y+"1" ).

The reason is that any such y must have a gap (two elements with nothing in-between), and W has exactly 2 gaps, so y must have exactly one gap. But then you can check the cases and both fail.

Though it's true that ∃y ( W=y+"1"+y ).

@LeakyNun: Interesting? But I shall really be off now. We can continue discussing TC next time. =)

Leaky Nun

17:12

@user21820 sure

shredalert

20:45

@user21820 all fixed.

Transcript for

♦ $Mathematics$ Logic

Transcript for

♦ Logic

♦ $Mathematics$ Logic